sign in sign up
<<
Home , Jordan normal form, Square root of a matrix

38 LA09 Abstracts

IP1 ing impact of her reduction result. Particular attention Preconditioning for PDE-constrained Optimization will be paid to bidiagonal factorizations of totally positive matrices, along with a number of related applications. For many partial differential equations (PDEs) and sys- tems of PDEs efficient preconditioning strategies now exist Shaun M. Fallat which enable effective iterative solution of the linear or University of Regina linearised equations which result from discretization. Few Canada would now argue for the use of Krylov subspace methods [email protected] for most such PDE problems without effective precondi- tioners. There are many situations, however, where it is not the solution of a PDE which is actually required, but IP4 rather the PDE expresses a physical constraint in an Opti- Model Reduction for Uncertainty Quantification mization problem: a typical example would be in the con- and Optimization of Large-scale Systems text of the design of aerodynamic structures where PDEs describe the external flow around a body whilst the goal For many engineering systems, decision-making under is to select its shape to minimize drag. We will briefly de- uncertainty is an essential component of achieving im- scribe the mathematical structure of such PDE-constrained proved system performance; however, uncertainty quan- Optimization problems and show how they lead to large tification and stochastic optimization approaches are gen- scale saddle-point systems. The main focus of the talk will erally computationally intractable for large-scale systems, be to present new approaches to preconditioning for such such as those resulting from discretization of partial dif- Optimization problems. We will show numerical results ferential equations. This talk presents recent advances in in particular for problems where the PDE constraints are model reduction methods — which aim to generate low- provided by the Stokes equations for incompressible viscous dimensional, efficient models that retain predictive fidelity flow. of high-resolution simulations — that make possible the characterization, quantification and control of uncertainty Andy Wathen in the large-scale setting. Oxford University Computing Laboratory Karen Willcox [email protected] Massachusetts Institute of Technology [email protected]

IP2 IP5 Exploiting Sparsity in Functions, with Ap- plications to Electronic Structure Calculations Recent Advances in the Numerical Solution of Quadratic Eigenvalue Problems In this talk, some general results on analytic decay bounds for the off- entries of functions of large, sparse A wide variety of applications require the solution of a (in particular, banded) Hermitian matrices will be pre- quadratic eigenvalue problem (QEP). The most common sented. These bounds will be specialized to various ap- way of solving the QEP is to convert it into a linear problem proximations to density matrices arising in Hartree-Fock of twice the and solve the linear problem by the and Density Functional Theory (assuming localized QZ algorithm or a Krylov method. Much work has been functions are used), proving exponential decay (i.e., ‘near- done to understand the linearization process and the effects sightedness’) for gapped systems at zero electronic tem- of scaling. Structure preserving linearizations have been perature and thus providing a rigorous justification for the studied, along with algorithms that preserve the spectral possibility of linear scaling methods in electronic structure properties in finite arithmetic. We present a new calculations for non-metallic systems. The case of density class of transformations that preserves the block structure matrices for disordered systems (Anderson model) and for of widely used linearizations and hence does not require metallic systems at finite temperature will also be consid- computations with matrices of twice the original size. We ered. Additional applications, possible extensions to the show how to use these structure preserving transformations non-Hermitian case, and some open problems will conclude to decouple symmetric and nonsymmetric quadratic matrix the talk. This is joint work with Paola Boito and Nader polynomials. Razouk. Francoise Tisseur Michele Benzi University of Manchester Department of Department˜ofMathematics˜ and Computer Science [email protected] Emory University [email protected] IP6 IP3 Definite versus Indefinite Matrix Factoriazations and Total Positivity Matrices with symmetries with respect to indefinite inner products arise in many applications, e.g., Hamiltonian and The concept of a grew out of three symplectic matrices in control theory and/or the theory separate applications: Vibrating systems, interpolation, of algebraic Riccati equations. In this talk, we give an and probability. Since them, this class of matrices con- overview on latest developments in the theory of matrices tinues to garner interest from both applied and theoretical in indefinite inner product spaces and focus on the sim- mathematicians. In 1953, a short paper appeared by Anne ilarities and differences to the theory of Euclidean inner Whitney in which she established a ”reduction” result that product spaces. Topics include normality, polar decom- may be one of the most powerful results on this topic. My positions, and singular value decompositions in indefinite plan for this presentation is to survey the relevant history from Anne’s paper to present day, and to discus the result- LA09 Abstracts 39

inner product spaces. University of Maryland, College Park Department of Computer Science Christian Mehl [email protected] Technische Universitaet Berlin Institut f¨urMathematik [email protected] IP10 Sparse Optimization: Algorithms and Applications

IP7 Many signal- and image-processing applications seek sparse Beyond the Link-Graph: Analyzing the Content of solutions to large underdetermined linear systems. The Pages to Better Understand their Meaning basis-pursuit approach minimizes the 1-norm of the solu- tion, and the basis-pursuit denoising approach balances In recent years the Web link graph has attracted much at- it against the least-squares fit. The resulting problems tention. However once we start analyzing the content Web can be recast as conventional linear and quadratic pro- pages many other fascinating graphs reveal themselves. In grams. Useful generalizations of 1-norm regularization, my talk I will discuss some of the information structures however, involve optimization over matrices, and often lead that we can construct from different Web content analysis to considerably more difficult problems. Two recent exam- sources, such as: anchor text, query and click logs, social ples: problems with multiple right-hand sides require joint- tagging, natural language processing of text, and user pro- sparsity patterns among the solutions and involve mini- vided semantic data. mizing the sum of row norms; matrix-completion problems require low-rank matrix approximations, and involve min- Hugo Zaragoza imizing the sum of singular values. I will survey some of Yahoo! Research Barcelona the most successful and promising approaches for sparse [email protected] optimization, and will show examples from a variety of ap- plications. IP8 Michael P. Friedlander Numerical Linear Algebra, Data Mining and Image Dept of Computer Science Processing University of British Columbia [email protected] In this talk, I will present two applications in data mining and image processing. The first application is related to the problem where an instance can be assigned to multi- IP11 ple classes. For example, in text categorization tasks, each Using Combinatorics to Solve Systems in M- document may be assigned to multiple predefined topics, matrices in Nearly-linear Time such as sports and entertainment; in automatic image or video annotation tasks, each image or video may belong Symmetric M-matrices arise in areas as diverse as Sci- to several semantic classes, such as urban, building, road, entific Computing, Combinatorial Optimization, and Ma- etc; in bioinformatics, each gene may be associated with a chine Learning. In this talk, we survey recently developed set of functional classes. The second application is related combinatorial algorithms for preconditioning these matri- to multiple class image segmentation problem. For ex- ces. These algorithms enable one to solve linear equa- ample, foreground-background segmentation has wide ap- tions in these matrices in asymptotically nearly-linear time. plications in computer vision (scene analysis), computer The problem of solving equations in symmetric M-matrices graphics (image editing) and medical imaging (organ seg- leads to questions in such as: What is the mentation). Both applications share the concept of semi- best approximation of a graph by a sparser graph? What supervised learning and involve several numerical linear al- about by a tree? And, what is a good decomposition of a gebra issues. Finally, I will present some matrix compu- graph? We will provide answers and explain their conse- tation methods for solving these numerical issues. Exper- quences. Finally, we will survey a randomized combinato- imental results are also given to demonstrate the applica- rial algorithm for scaling a symmetric M-matrix to make tions and computation methods. it diagonally dominant. Michael Ng Dan Spielman Hong Kong Baptist University Department of Mathematics Department of Mathematics MIT [email protected] [email protected]

IP9 CP1 Confidence and Misplaced Confidence in Image Re- Preconditioning Systems Arising from Electron construction Structure Calculations Using the Korringa-Kohn- Rostoker Green Function Method Forming the image from a CAT scan and taking the blur out of vacation pictures are problems that are ill-posed. By Electron structure calculations are known to be computa- definition, small changes in the data to an ill-posed problem tionally expensive and arise in different fields of physics. make arbitrarily large changes in the solution. How can we Recently a linear scaling method for metallic systems has hope to solve such problems using noisy data and inexact been developed by Zeller at Forschungszentrum Jlich. This computer arithmetic? In this talk we discuss the use of side method uses QMR to solve a linear system with multiple conditions and bias constraints to improve the quality of right-hand sides arising from the discretization of the prob- solutions. We discuss their impact on solution algorithms lem. The convergence rate depends on the temperature of and show their effect on our confidence in the results. the underlying physical problem. We will present first re- Dianne P. O’Leary 40 LA09 Abstracts

sults on preconditioning these systems. tion Matthias Bolten Existing papers show that, after finite difference and fi- University of Wuppertal nite element discretization of incompressible Naiver-Stokes [email protected] equation, we will get special linear system with struc- ture of saddle point problem. GMRES(m) with newly de- veloped preconditioner Hermitian/Skew-Hermitian Sepa- CP1 ration (GMRES(m)-HSS) provides an efficient way to solve On Preconditioners for Saddle Point Problems the saddle point system. In this paper, firstly we discuss Arising in Implicit Time Stepping of Maxwell’s saddle point system from finite volume discretization of in- Equations compressible Naiver-Stokes equation. Then, we solve the system by GMRES(m)-HSS. Furthermore, if the size of the Time integration of space-discretized Maxwell’s equations original PDE model is large, the solution speed of the linear is often carried out by second-order time integration system is still slow. Since large portion of computational schemes, treating the curl terms explicitly and obeying the effort of GMRES(m)-HSS is spent on building the orthog- CFL stability restriction. In different applications, for in- onalization basis, in this paper, by newly developed high- stance, when locally refined unstructured meshes are em- level parallel programming language, we develop a method ployed, the CFL restriction can be restrictive, so that un- to accelerate convergence of GMRES(m)-HSS by parallel conditionally stable implicit time integration becomes at- Gram-Schmidt (pGS) process. Theoretical analysis shows tractive. In implicit time integration of space discretized that pGS can decrease time complexity of GMRES(m)- Maxwell’s equations large-scale saddle-point a linear sys- HSS from O(m) to O(m). Computational experiments tem has to be solved at every time step. As our recent show that pGS can significantly increase solution speed of research shows, an powerful preconditioner is then a must. GMRES(m)-HSS. We report on recent results on preconditioning for this type of saddle-point problems. Di Zhao Louisiana Tech University Mike A. Botchev [email protected] University of Twente [email protected] CP2 Jan Verwer On a stable variant of Simpler GMRES and GCR CWI The Netherlands We propose and analyze a stable variant of Simpler GM- [email protected] RES and GCR, which is based on an adaptive choice of the Krylov subspace basis at a given iteration step using an intermediate residual norm decrease criterion. The new CP1 direction vector is chosen as in the original implementation A Two-Level Ilu Preconditioner for Electromag- of Simpler GMRES or it is equal the normalized residual netism Applications vector as in the GCR method. Such an adaptive strategy leads to a well-conditioned basis of the Krylov subspace, In this work, we consider the solution of the linear systems which results in a numerically stable and more robust vari- arising from electromagentism applications by precondi- ant of Simpler GMRES or GCR. tioned Krylov subspace methods. Sparse approximate in- verse preconditioners based on Frobenius norm minimiza- Pavel Jiranek tion are quite effective for this type of problems. We ex- CERFACS periment with a two-level ILU preconditioner with graph [email protected] partioning techniques, and discarding small entries which correspond to far field interactions. Preliminary results show that this two-level preconditioner is competitive. CP2 Progressive Gmres: A Minimum Residual Krylov Rafael Bru Method Which Is Equivalent to Minres in the Sym- D. Matematica Aplicada, ETSEA metric Case Univ. Politecnica de Valencia [email protected] We study Beckermann and Reichel’s Progressive GM- RES (ProGMRES) algorithm, a minimum residual Krylov n × n Natalia Malla method for approximating solutions to nearly sym- Instituto Matematica Multidisciplinar metric linear systems, with skew symmetric part of rank s<

CP2 satisfied by the sequence of monic polynomials orthogonal GIDR(s,l): IDR(s) with Higher-Degree Stabiliza- with respect to a measure. The basic Geronimus trans- tion Polynomials formation with shift α transforms the monic Jacobi matrix associated with a measure dμ into the monic Jacobi matrix Recently Sleijpen and van Gijzen have proposed IDR(s) associated with dμ/(x − α)+Cδ(x − α), for some constant with stabilization polynomials of higher degree, named C. In this paper we examine the algorithms available to IDRstab, which is an efficient method for solving real large compute this transformation and we propose a more ac- nonsymetric systems of equations. Their derivation of curate algorithm, estimate its forward errors, and prove IDRstab is based on a complex version of the so-called that it is forward stable. In particular, we show that for IDR principle, although they consider real systems of equa- C = 0 the problem is very ill-conditioned, and present a tions. In the present talk we introduce a generalized real new algorithm that uses extended precision. IDR principle, which enables us to incorporate stabilization polynomials of higher degree into IDR(s). Maria Isabel Bueno Cachadina The University of California, Santa Barbara Masaaki Sugihara, Masaaki Tanio [email protected] Graduate School of Information Science and Technology, the University of Tokyo Alfredo Deano m [email protected], masaaki [email protected] University of Cambridge, UK tokyo.ac.jp [email protected]

CP2 Edward Tavernetti University of California, Davis Notes on the Convergence of the Restarted Gmres. [email protected] Our first result concerns the convergence of restarted GM- RES for normal matrices. We prove that the cycle- CP3 convergence of the restarted GMRES applied to systems with a normal coefficient matrix is sublinear. The second Spectral Methods for Parameterized Matrix Equa- result concerns the convergence of the restarted GMRES tions in the general case. In the general case, we prove that We apply polynomial approximation techniques to the any admissible cycle-convergence curve is possible for the vector-valued function that satisfies a linear system of restarted GMRES at a certain number of cycles. This re- equations where the matrix and the right hand side de- sult holds for any eigenvalue distribution, therefore eigen- pend on a parameter. We derive both an interpolatory values alone do not determine the convergence of restarted pseudospectral method and a residual-minimizing Galerkin GMRES. method, and we show how each can be interpreted as solv- Eugene Vecharynski ing a truncated infinite system of equations; the difference University of Colorado Denver between the two methods lies in where the truncation oc- [email protected] curs. Paul Constantine Julien Langou Stanford University Department of Mathematical & Statistical Sciences Institute for Computational and Mathematical University of Colorado Denver Engineering [email protected] [email protected]

CP3 David F. Gleich Stanford Interpolation of Reduced-Order Linear Operators [email protected] on Matrix Manifolds

A new method is presented for interpolating reduced-order CP3 models on matrix manifolds. This method relies on the logarithm and exponential mappings to move the interpo- Riemannian Optimization Algorithms for Matrix lation data to a tangent space where standard multivariate Completion interpolation algorithms can be applied and bring it back We present new algorithms for the problem of recovering a to the manifold of interest, respectively. The proposed large low-rank matrix from a subset of its entries, known as method is illustrated for several engineering applications matrix completion. The methods minimize the least-square associated with the orthogonal manifold and non-compact error of the projection of a matrix directly on the Rieman- Stiefel manifold. nian manifold of matrices of fixed rank. The optimization David Amsallem, Charbel Farhat algorithms are the generalization of steepest descent, non- Stanford University linear cg and Newton’s algorithm to this Riemannian man- [email protected], [email protected] ifold. We will demonstrate the numerical performance in comparison with some recent algorithms.

CP3 Bart Vandereycken Katholieke Universiteit Leuven A New Algorithm for Computing a Darboux Trans- [email protected] formation with Large Shifts

A monic Jacobi matrix is a which con- Stefan Vandewalle tains the parameters of the three-term recurrence relation Scientific Computing Research Group KULeuven 42 LA09 Abstracts

[email protected] Ljiljana D. Cvetkovic, Vladimir Kostic University of Novi Sad, Faculty of Science Department of Mathematics and Informatics CP4 [email protected], [email protected] Generalized M-Matrices

Various classes of generalized M-matrices are discussed in- CP4 cluding the new class of GM-matrices. The matrices con- QR Decomposition of H-Matrices sidered are of the form sI − B where B has either the Perron-Frobenius property or the strong Perron-Frobenius The hierarchical (H-) matrix format allows storing a vari- property but it is not necessarily nonnegative, I is the iden- ety of dense matrices from certain applications in a special tity matrix, and s is a nonnegative scalar not exceeding the data-sparse way with linear-polylogarithmic complexity. A of B. Results that are analogous to those new algorithm to compute the QR decomposition of an H- known for M-matrices are shown for these classes of gener- matrix will be presented. Like other H-arithmetic opera- alized M-matrices and various splittings of a GM-matrix tions the HQR decomposition is of linear-polylogarithmic are studied along with conditions for their convergence. complexity. We will compare our new algorithm with two older ones by using two series of test examples and discuss Abed Elhashash benefits and drawbacks of the new approach. Drexel University Department of Mathematics Peter Benner, Thomas Mach [email protected] TU Chemnitz Germany Daniel B. Szyld [email protected], Temple University [email protected] Department of Mathematics [email protected] CP5 Eigenspaces of Third-Order Tensors CP4 Mixed Class of H-Matrices and Schur Comple- Building upon a recently described tensor-tensor multipli- ments cation [Kilmer, Martin, and Perrone; submitted, October 2008], we show that, on a particular of n-by-n ma- It is well-known that there exist nonsingular H-matrices trices, every linear transformation can be represented by such that their comparison matrix M(A) is singular. All tensor- by a third-order tensor. In equimodular matrices related to this case form the mixed this context we investigate invariant sub-modules and de- class of H-matrices, HM . Here, we study their Schur com- fine tensor eigenvectors and their corresponding eigenma- plements. In particular, it is determined that the Schur trices. In addition, we examine an Arnoldi-like method for complement of a nonsingular matrix in HM exists and is computation of tensor eigenvectors and eigenmatrices. an H-matrix, but it can belong to HM or to HI (M(S)is nonsingular). Conditions to distinguish these two cases are Karen S. Braman given. South Dakota School of Mines [email protected] Isabel Gimenez, Rafel Bru, Cristina Corral Instituto de Matematica Multidisciplinar Universidad Politecnica de Valencia CP5 [email protected], [email protected], Combinatorial Applications of [email protected] Multidimensional Matrix Theory Recently the multidimensional matrix theory have found Jose Mas series of applications mostly in all areas of mathematics. Institut de Matematica Multidisciplinar In our talk we will be constrained only by our own findings Universitat Politecnica de Valencia concerning some new approaches in solving problems by [email protected] means of multidimensional matrices, matrix constructions and functions. Moreover, due to time restrictions, we will CP4 report only on combinatorial applications explaining our Matrix Network Method and its applications in combina- Numerical Approximation of the Minimal torial enumeration, in graph and hypergraph coloring, in GerˇsgorinSet symmetric function theory, and in substructural combina- One possibility to find a good numerical approximation torics. to the minimal Gerˇsgorin set (MGS) is presented in Armenak S. Gasparyan [Varga, Cvetkovi´c,Kosti´c,Approximation of the minimal Program Systems Institute of RAS Gerˇsgorinset of a square complex matrix, ETNA 30 (2008), [email protected] 398-405]. Here we will show some improvements of this al- giruthm, plotting an area which contains MGS itself, and for which a limited number of boundary points are actual CP5 boundary points of MGS. Fourier Basis Regression Against Multicollinearity

Richard Varga Multicollinearity is a long-standing problem in Statistics. Department of Mathematical Sciences, Kent State This paper presents an alternative solution to deal with University multicollinear observations. In this paper, we propose to [email protected] use orthogonal Fourier bases F to obtain an estimate of β LA09 Abstracts 43

from the subspace spanned by the orthogonalized columns CP6 of the X matrix. We also demonstrate that the orthogonal- Inverse Sherman-Morrison Factorization and Its ized independent variables have research-related physical Related Preconditioners interpretations which arise not just because of mathemat- ical convenience but because of the inherent information We analyze the Inverse Sherman-Morrison factorization of contained in these new bases. Finally, we show the mathe- an , in particular its relations with the matical elegance of using Fourier bases in regression analy- LDU factorization. Since the ISM factorization is highly sis as opposed to the often messy mathematics of ordinary parametrizable we analyze also some relations of the fac- regression. It is demonstrated that the two definitions of tors for different parameter choices. Also the block version multicollinearity, namely, that based on variance inflation of the factorization is studied. The main motivation for factors and condition numbers are in fact equivalent defi- introducing the ISM factorization is to use it as a precondi- nitions. tioner. Some existence properties of these preconditioners and some numerical results are shown. Manuel Beronilla, Eduardo Lacap University of Rizal System Jose Mas, J. Cerdan, JOSE Marin [email protected], Institut de Matematica Multidisciplinar lacap [email protected] Universitat Politecnica de Valencia [email protected], [email protected], [email protected] CP5 Scalable Implicit Symmetric Tensor Approxima- Miroslav Tuma tion Academy of Sciences of the Czech Republic Institute of Computer Science We describe some algorithms for producing a best multilin- [email protected] ear rank r approximation of a symmetric tensor, useful e.g. for working with higher moments and cumulants of a large p-dimensional random vector. Since the approximated d- CP6 pd p way tensor has entries, for large it is critical never to The Importance of Structure in Algebraic Precon- explicitly represent the entire tensor. Working implicitly ditioners and approximately where appropriate, this approach en- ables the use of higher order factor models on large-scale In recent years, algebraic preconditioners have achieved a data. considerable degree of efficiency and robustness. Incom- plete factorizations represent an important class of such Jason Morton preconditioners. Despite their success, for many problems Stanford University further tools are required to enhance performance and im- [email protected] prove reliability. We propose an approach that is based on forcing the original matrix structure, or an appropriately CP6 modified structure of powers of the system matrix, into a value-based incomplete . Our im- Combinative Preconditioning Based on Relaxed plementation will be included within the HSL mathemati- Nested Factorization and Tangential Filtering Pre- cal software library. conditioner Jennifer Scott The problem of solving block tridiagonal linear systems Rutherford Appleton Laboratory arising from the discretization of PDE on structured grid Computational Science and Engineering Department is considered. Nested Factorization preconditioner intro- [email protected] duced by Appleyard et. al. is an effective preconditioner for certain class of problems and a similar method is imple- mented in Schlumerger’s widely used Eclipse oil reservoir Miroslav Tuma simulator. In this work, a relaxed version of Nested Fac- Academy of Sciences of the Czech Republic torization preconditioner is proposed. Effective multiplica- Institute of Computer Science tive/additive preconditioning is achieved in combination [email protected] with Tangential filtering preconditioner. Pawan Kumar CP6 INRIA Futurs Optimal Block Method and Preconditioner for [email protected] Banded Matrices Optimized Schwarz methods (OSMs) for PDEs use Robin Laura Grigori (or higher order) transmission conditions in the artificial INRIA interfaces between subdomains, and the Robin parameter France can be optimized. We present a completely algebraic ver- [email protected] sion of the OSM, including an algebraic approach to find the optimal operator or a sparse approximation thereof. Frederic Nataf We can thus apply this method to any banded or block Laboratoire J.L. Lions banded linear system of equations, and in particular to dis- [email protected] cretizations of PDEs on irregular domains. With the com- putable optimal operator, we prove that both the OSM and Qiang Niu GMRES with OSM preconditioning converge in no more School of Mathematical Sciences, Xiamen University than two iterations for the case of two subdomains. Very [email protected] fast convergence is attained even when the optimal opera- tor is approximated by a sparse transmission matrix. Nu- 44 LA09 Abstracts

merical examples illustrating these results are presented. CP7 Fast Iterative Solver for Incompressible Navier- Daniel B. Szyld Stokes Equations with Spectral Elements Temple University Department of Mathematics We introduce a block preconditioning technique for spec- [email protected] tral element discretization of the steady Navier-Stokes equations. This method is built from subsidiary Domain Martin J Gander Decomposition solvers which use the Fast Diagonalization Univesity of Geneva Method to eliminate degrees of freedom on subdomain in- Section de Math teriors. We demonstrate the effectiveness of this precondi- ’ematiques tioner in numerical simulations of several benchmark ?ows [email protected] including Kovasznay ?ow, ?ow in a lid-driven cavity and ?ow over a backward-facing step. S´ebastien Loisel Temple University P. Aaron Lott Department of Mathematics National Institute of Standards and Technology [email protected] [email protected]

Howard C. Elman CP7 University of Maryland, College Park Sublinearly Fast Solvers for Finite Difference Op- [email protected] erators on Mostly Structured Grids

For boundary value problems associated with constant co- CP7 efficient finite difference operators on structured grids with- Numerical Linear Algebra Applied to the Simula- out body loads, we will demonstrate there exists solvers tion of the Dynamic Interaction between a Railway whose complexity is lower than O(N), where N denotes the Track and a High Speed Train number of grid points. We analytically construct the fun- damental solution for the difference operator, then use fast In the simulation of the dynamic interaction between a techniques for computing convolutions involving the funda- railway track and a high speed train, many useful linear mental solution. The resulting methods scale on the order algebra techniques have to be applied effectively in order of the number of grid points on the boundary. to execute it as efficiently and accurately as possible. At first, in the eigenvalue analysis of the large model consist- Adrianna Gillman ing of two subsystems of the vehicles and the track, it is an University of Colorado at Boulder interesting problen how we should deal with the interface of Department of Applied Mathematics contact between a wheel and a rail. Contact points of the [email protected] interaction move along the track and therefore the charac- teristic of the system depends on the time. An appropriate Per-Gunnar Martinsson methodology of eigenvalue analysis of these dynamically University of Colorado at Boulder interacting system is needed. Secondly, for railway engi- [email protected] neers it is very important to know the influence of the sys- tem structural components on a variation of contact force between a wheel and a rail. Because a large variation of CP7 contact force causes the early deterioration of the track ge- Linear Systems with Large Off-Diagonal Elements ometry. In this study, including the investigation of the and Discontinuous Coefficients above problems, an application of numerical linear algebra and related techniques such as singular value decomposi- Previous work showed that “geometric” scaling of the equa- tion, inverse analysis method are described. tions is useful for handling nonsymmetric systems with discontinuous coefficients. It is shown that after such Akiyoshi Yoshimura scaling, AAT contains relatively large diagonal elements. Tokyo University of Technology These two operations are inherent in the sequential CGMN Emeritus Professor and the block-parallel CARP-CG algorithms, making them [email protected] very effective for systems with large off-diagonal elements and discontinuous coefficients. Examples involving hetero- geneous media include convection-diffusion equations with CP8 large convection and the Helmholtz equation with large Cyclic Reduction and Multigrid wave numbers. Given a discretized PDE on a uniform mesh, one step Dan Gordon of cyclic reduction amounts to ordering the gridpoints in Dept. of Computer Science red/black fashion and eliminating the ones that correspond University of Haifa to one of the colors. The resulting operator is spectrally [email protected] better than its original (unreduced) counterpart, but a drawback of this approach is that implementation is not Rachel Gordon easy, in particular in three dimensions, due to the irregular Dept. of Aerospace Engineering structure of the associated mesh. In this talk we examine The Technion - Israel Inst. of Technology cyclic reduction in the context of multigrid, analyze the [email protected] smoothing properties of cyclically reduced operators, and discuss ways of efficiently implementing solvers based on LA09 Abstracts 45

this approach. Point Problems Chen Greif Bai, Parlett and Wang recently proposed an efficient pa- University of British Columbia rameterized Uzawa method for solving the nonsingular Canada saddle point problems (see Numer. Math. 102(2005)1-38). [email protected] In this paper we investigates the W− norm convergence of the parameterized Uzawa methods for saddle point prob- lems under the a suitable weighted norm of the iterative CP8 matrix. Based on the estimates of the weighted norm of Adaptive Algebraic Block Iterative Methods the iterative matrix, an new inexact parameterized Uzawa methods for saddle point problems is presented. This talk will present a new method of adaptively con- structing block iterative methods based on Local Sensitiv- Bing Zheng ity Analysis (LSA). The main application is the develop- School of Mathematics and Statistics ment of adaptive smoothers for multigrid methods. Given Lanzhou University, Lanzhou, P.R. China a linear system, Ax = b, LSA is used to identify blocks [email protected] of the matrix, A, which are strongly connected in a sense similar to the traditional measure of strength in algebraic multigrid methods. Block iterative Gauss-Seidel or Jacobi CP9 methods can then be constructed based on the identified Implementing the Retraction Algorithm for Fac- blocks. The size of the blocks can be varied adaptively al- toring Symmetric Banded Matrices lowing for the construction of a class of block iterative alge- braic methods. Results will be presented for both constant Problems arising when modeling structures subject to vi- and variable coefficient elliptic problems, systems arising brations or in designing optical fibers wrapped around a from both scalar and coupled system PDEs, as well as lin- spool give rise to eigenvalue problems and linear systems ear systems not arising from PDEs. The simplicity of the with symmetric, banded matrices. We present various vari- method will allow it to be easily incorporated into existing ations of the retraction algorithm which factors a matrix multigrid codes while providing a powerful tool for adap- into a product of matrices while preserving symmetry, the tively constructing smoothers tuned to the problem. bandwidth, and element growth even when the original ma- trix is indefinite. The factorization may be used to solve Bobby Philip a system or compute the inertia of a matrix. It is based Los Alamos National Laboratory on the algorithm of Bunch and Kaufman for symmetric [email protected] nonbanded systems.

Timothy P. Chartier Linda Kaufman Davidson College William Paterson University [email protected] [email protected]

CP8 CP9 Halley and Newton Are One Step Apart Compact Fourier Analysis for Deriving Multigrid As Direct Solver In this talk we onsider numerical solution of a nonlinear system of equations F (x) = 0 where the function is suffi- A Compact Fourier Analysis (CFA) for designing new effi- ciently smooth. While Newton’s method which has second cient multigrid (MG) methods is considered. The principal order rate of convergence method, is usually used for solv- idea of CFA is to model MG by means of generating func- ing such systems, third order methods like Halley’s method tions and block symbols. We utilize the CFA for deriving will in general use fewer iterations than a second order MG as a direct solver and give necessary conditions that method to reach the same accuracy. However, the number have to be fulfilled by the MG components for this pur- of arithmetic operations per iteration is higher for third pose. General practical smoothers and transfer operators order methods than second order methods. We will show that lead to powerful MG algorithms are introduced. that for a large class of problems the ratio of the num- Christos Kravvaritis ber of arithmetic operations of a third order method and Technical University of Munich Newton’s method is constant per iteration using a direct Department of Informatics method to solve the linear systems including the cost of [email protected] computing the function and the derivatives. We also dis- cuss replacing the direct method using an iterative method. Thomas K. Huckle Trond Steihaug Institut fuer Informatik University of Bergen Technische Universitaet Muenchen Department of Informatics [email protected] [email protected] CP9 Geir Gundersen, Sara Suleiman University of Bergen Diagonal Pivoting Methods for Solving Unsymmet- [email protected], [email protected] ric Tridiagonal Systems This talk concerns the LBMT factorization of unsymmet- CP8 ric tridiagonal matrices, where L and M are unit lower B × On the Parameterized Uzawa Methods for Saddle triangular matrices and is block diagonal with 1 1 and 2 × 2 blocks. In some applications, it is necessary to form 46 LA09 Abstracts

this factorization without row or column interchanges while of software tools, which include FEMLAB, Mat-Lab and the tridiagonal matrix is formed. We propose methods Simulink, and SCILAB. for forming such a factorization that both exhibit bounded growth factors and are normwise backward stable. Kumud S. Altmayer University of Arkansas Jennifer Erway kumud [email protected] Wake Forest University [email protected] Kamran Iqbal University of Arkansas Roummel F. Marcia Little Rock Department of Electrical and Computer Engineering [email protected] Duke University [email protected] CP10 Eigenvalue Localization for Matrices with Constant CP9 Row Or Column Sum Orthosymmetric Block Reflectors and Qr Factor- izations Motivated by some research on synchronization on net- works of mobile agents, we are interested in finding a good Reflectors and block reflectors are widely used tools in nu- eigenvalue localization for a square, real-valued, non sym- merical linear algebra. Both types of reflectors are needed metric matrix with the number k on the main diagonal and for the construction of non-Euclidean variants of the QR exactly k ”−1” on off-diagonal places (in random position) factorization, for example, in hyperbolic and symplectic in each row. According to Gerˇsgorintheorem, all the eigen- scalar products. We shall present implementation details, values of such a matrix are located within the circle with perturbation and error analysis of the hyperbolic and the center in k and radius equak to k, but this localization is symplectic QR factorization. not good enough. We are interested in tighter localization of all eigenvalues, different from zero. More generalized Sanja Singer resuts for matrices with constant row or column sum are Faculty of Mechanical Engineering and Naval obtained. Architecture, University of Zagreb [email protected] Ljiljana D. Cvetkovic University of Novi Sad, Faculty of Science Sasa Singer Department of Mathematics and Informatics Department of Mathematics, University of Zagreb [email protected] [email protected] Juan Manuel Pena Vedran Novakovic Departamento de Matem´aticaAplicada, Universidad de Faculty of Mechanical Engineering and Naval Zaragoza Architecture, University of Zagreb [email protected] [email protected] Vladimir Kostic University of Novi Sad, Faculty of Science CP10 Department of Mathematics and Informatics Solutions of Systems of Algebraic Equations from [email protected] a Physical model of Induction Machine

This work deals with the study of a system of algebraic CP10 equations arising in the mathematical modeling of an elec- A Hadamard Product Involving Inverse-Positive tric machine. The machine considered may be an Induction Matrices with Checkerboard Pattern machine, a synchronous machine, or a transformer. Our model is primarily based on a transformer. - The trans- A nonsingular real matrix A is said to be inverse-positive former may be used to transfer large amount of power, if all the elements of its inverse are nonnegative. This voltage, or current. In the present case we consider the class contains the M-matrices. The Hadamard prod- machine windings for homogeneous or non-homogeneous uct involving M-matrices has studied for several authors, medium with Dirichlet and Mixed Boundary conditions. Markham, Johnson, Neumann and other. Recently Chen We study time independent equations first, and later com- in 2006 prove a conjeture concerning the Hadamard pow- pare it with time dependent equations. Many cases of time ers of inverse M-matrices. In this work we study the dependent as well as time independent Maxwell’s equations Hadamard product of certain classes of inverse-positive ma- have been studied before. A comparison of exact solutions trices. And we prove that for any pair A , B of triangu- to numerical solutions is established for different layers of lar inverse-positive matrices with the checkerboard pattern, transformer windings under dissimilar structures. We fur- the Hadamard (entry-wise) product A◦B−1 is again trian- ther look for extra eddy currents produced and the loss gular inverse-positive matrices with the checkerboard pat- of power therein. A Power-current or skin depth plot and tern. Some interesting inequalities about of the minors of Torque-skin depth plot will be done to find out how to con- this class of matrices are presented in this work. trol and minimize the power loss and increase amount of torque required to start the machine or stop the machine Maria T. Gass´o in minimum time by using transient analysis and classical Instituto Matematica Multidisciplinar Nyquist and Bode plots. An example is considered that Universidad Politecnica Valencia depicts how equations and solutions behave differently in [email protected] different media/materials with respect to power and torque control. The analysis is being done by using a combination Juan R. Torregrosa, Manuel F. Abad LA09 Abstracts 47

Instituto de Matem´aticaMultidisciplinar trix. Unfortunately, this algorithm is unstable and, for ill- Universidad Polit´ecnicaValencia conditioned matrices, the resulting vectors do not form an [email protected], [email protected], orthonormal basis. Several variants have been proposed, all requiring the eigenvalue decomposition of the symmet- ric matrix AT A. We propose shifted Cholesky to cure the CP10 problem of breakdown in Cholesky, and a subsequent re- Spectral Inclusion Sets for Structured Matrices orthogonalization step.

We derive eigenvalue inclusion sets for centrohermitian, Matthew W. Nabity, Julien Langou skew-centrohermitian, perhermitian, skew-perhermitian, Department of Mathematical & Statistical Sciences and persymmetric matrices, that can be considered struc- University of Colorado Denver tured analogs of the Gerˇsgorinsets. We also present the [email protected], nonsingularity conditions associated with these spectral in- [email protected] clusion sets. In addition, we derive sets that can be consid- ered the structured analogs of the Brauer sets. Our tech- niques are general enough to allow for other symmetries as CP11 well. Resolution of Large Sparse Non-Hermitian Com- plex Linear Systems on a Nationwide Cluster of Aaron Melman Clusters Santa Clara University Department of Applied Mathematics To solve a large sparse non-Hermitian complex linear [email protected] system Ax=b, we asynchronously compute in parallel the dominant eigenelements of A and a preconditioned restarted GMRES method on GRID5000, a Nationwide CP11 cluster of heterogeneous distributed clusters. We ex- Parallel Preconditioning for Block-Structured CFD ploit the computed eigenvalues to accelerate the GMRES Problems method using a LS method. This preconditioned method is known to not be mathematically correct when the ma- At the Institute for Propulsion Technology of the German trix elements are complexes. Then, we propose a solution Aerospace Center (DLR), the parallel simulation system to use this method using the same storage space on each (Turbo-machinery Research Aerodynamic Com- nodes that other classical GMRES methods. We propose putational Environment) has been developed specifically numerical experimentations using several GRID5000 sites for the calculation of internal turbo-machinery flows. The and we compare the obtained performance, time and itera- finite volume approach with block-structured grids requires tion, with other methods for large matrices from industrial the parallel, iterative solution of large, sparse systems of applications. linear equations. For convergence acceleration of the itera- tion, parallel preconditioners based on multi-level imcom- Serge G. Petiton plete factorizations have been investigated. Numerical and INRIA and LIFL, Universite de Lille performance results of these methods will be presented for [email protected] typical TRACE problems on parallel computer systems, including multi-core architectures. Guy Berg`ere LIFL, CNRS Achim Basermann, Hans-Peter Kersken guy.bergere@lifl.fr German Aerospace Center (DLR) Simulation and Software Technology (SISTEC) [email protected], [email protected] Ye Zhang CNRS/LIFL ye.zhang@lifl.fr CP11 Extending Algorithm Based Fault Tolerance in Or- Pierre-Yves Aquilanti der to Support Error on the Fly LIFL; CNRS pierre-yves.aquilanti@lifl.fr In a previous paper, we showed that Algorithm Based Fault Tolerance can be used to perform high performant and fault tolerant matrix-matrix multiplication. We ex- CP12 tend these ideas to most of linear algebra operations. We Numerical Approaches for Nonlinear Ill-Posed In- believe our approach is systematic and simply relies on a verse Problems ABFT BLAS. We will show result of a prototype ABFT LAPACK. Difficult nonlinear inverse problems arise in a variety of sci- entific applications, and iterative optimization algorithms Julien Langou are often used to efficiently compute solutions to large- Department of Mathematical & Statistical Sciences scale problems. In this work, we investigate Krylov-based University of Colorado Denver solvers and appropriate preconditioners for linear systems [email protected] that arise within nonlinear optimization schemes. Further- more, regularization methods for nonlinear ill-posed prob- lems are required for computing stable solution approxi- CP11 mations. Applications from medical image processing are On the use of Cholesky in the CholeskyQR algo- considered. rithm Julianne Chung In practice, the CholeskyQR algorithm is the fastest known University of Maryland algorithm for the QR factorization of a tall and skinny ma- [email protected] 48 LA09 Abstracts

CP12 control theory is studied. A numerical method is developed Calculating the Numerical Rank of Sparse Matrices for verifying the positive realness of a given proper ratio- nal matrix H(s) for which H(s)+HT (−s) may have purely Algorithms for calculating the numerical rank of sparse imaginary zeros. The proposed method is only based on matrices are compared. The algorithms include (1) rank orthogonal transformations, it is structure-preserving and calculation using Sylvester’s inertia theorem, (2) calcu- has a complexity which is cubic in the state dimension of lating the rank using the sparse QR factorization spqr H(s). Some examples are given to illustrate the perfor- (http://www.cise.ufl.edu/research/sparse/SPQR/) and mance of the proposed method. (3) post-processing the spqr results. The comparison will be based on matrices from Delin Chu the San Jose State University Singular Matrix Database Department of Mathematics, National University of (http://www.math.sjsu.edu/singular/matrices/). Singapore 2 Science Drive 2, Singapore 117543 Leslie V. Foster [email protected] San Jose State University Department of Mathematics Peter Benner [email protected] TU Chemnitz Germany [email protected] CP12 Limited Approximation of Numerical Range of Xinmin Liu Department of Electrical and Computer Engineering Let A be an n × n normal matrix, whose numerical range University of Virginia,USA NR[A]isak–polygon. Adam and Maroulas (On compres- [email protected] sions of normal matrices, Linear Algebra Appl., 341(2002), v ∈ W ⊆ Cn 403-418) established that if a unit vector , with CP13 dimW = k and the point v∗Av ∈ IntNR[A], then NR[A] is circumscribed to NR[P ∗AP ], where P is an n×(k −−1) Dimension of Orbits of Singular Systems under ⊥ n Proportional and Derivative Feedback isometry of {span{v}}W → C . In this paper, we investi- NR A gate an internal approximation of [ ] by an increasing We consider triples of matrices (E, A, B), representing sin- NR C C R∗AR sequence of [ s] of compressed matrices s = s s, gular linear time invariant systems in the form Ex˙(t)= R∗R I ,s , , ···,n −−k with s s = k+s−−1 =12 and addi- Ax(t)+Bu(t), with E,A ∈ Mp×n(C) (not necessarily NR A tionally [ ] is expressed as limit of numerical ranges of squares), and B ∈ Mn×m(C), under proportional and k A –compressions of . derivative feedback. After to obtain a canonical reduced form preserving the structure of the systems where the John B. Maroulas standard subsystem is maximal and using geometric tech- National Technical University niques, we obtain the tangent and normal space to the Department of Mathematics orbits of equivalent triples. This description permit us to [email protected] deduce the dimension of the orbits and analyze structural stability of the triples. Maria Adam University of Central Greece Maria Isabel Garcia-Planas Department of Computer Science and Biomedical Universitat Politecnica de Catalunya Informatics [email protected] [email protected]

CP13 CP12 Analyzing Controllability and Observability by Ex- Accurate Solutions of Structured Linear Systems ploiting the Stratification of Matrix Pairs from Rank-Revealing Decompositions. The qualitative information about nearby linear time- For many structured linear systems (among them, Cauchy invariant (LTI) systems is revealed by the theory of strat- and Vandermonde, either totally positive, or non totally ification. Utilizing the software tool StratiGraph, we show positive) for which there are not accurate solvers, we show how the stratification theory is used to analyze the control- how to compute accurate solutions by using rank-revealing lability and observability characteristics. We demonstrate decompositions (RRD). In this way we take advantage of with two examples, a mechanical system consisting of a the great effort done in the last decades to compute accu- thin uniform platform supported at both ends by springs, rately the SVD using RRD’s. The algorithms used involve and a linearized Boeing 747 model. For both examples, 3 O(n ) flops. nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system Juan M. Molera, Froilan M. Dopico pencils. Some quantitative results are also reported. Department of Mathematics Universidad Carlos III de Madrid Stefan Johansson [email protected], [email protected] Ume˚aUniversity Computing Science [email protected] CP13 Title of the Paper Here. Don’t Use All Caps. Pedher Johansson Ume˚aUniversity In this paper the positive realness problem in circuit and Computing Science and HPC2N LA09 Abstracts 49

[email protected] supported by Grant DGI MTM2007-64477.

Erik Elmroth ALICIA Herrero Department of Computing Science Instituto de Matematica Multidisciplinar Ume˚aUniversity Universidad Politecnica de Valencia, Spain [email protected] [email protected]

Bo T. K˚agstr¨om Francisco J. Ramirez Ume˚aUniversity Instituto Tecnologico de Santo Domingo Computing Science and HPC2N Dominican Republic [email protected] [email protected] Nestor Thome CP13 Instituto de Matematica Multidisciplinar Exposing Structure Algorithms for a Nonsymmet- Universidad Politecnica de Valencia, Valencia, Spain ric Algebraic Riccati Equation Arising in Transport [email protected] Theory.

The nonsymmetric algebraic Riccati equation arising in CP14 transport theory is a special algebraic Riccati equation A Note on the dqds Algorithm with Rutishauser’s whose coefficient matrices having some special structures. Shift By reformulating the Riccati equation, they were found that its arbitrary solution matrix is of a Cauchy-like form We are concerned with the behaviour of the dqds algo- and the solution matrix can be computed from a vector rithm for computing sigular values, when incorporated form Riccati equation instead of the matrix form that. In with Rutishauser’s shift. It is proved that in the itera- this talk, we review some classical-type and Newton-type tive process Rutishauser’s shift is selected finally and the iterative methods for solving the vector form Riccati equa- dqds converges cubically. Furthermore the phenomenon is tion and present some work under investigation . observed that once Rutishauser’s shift is selected ever af- ter it is selected. In this talk we will give an theoretical Linzhang Lu explanation for the phenomenon. School of Mathematical Science Xiamen University, Xiamen, P.R. China Kensuke Aishima [email protected] [email protected] Graduate School of Information Science and Technology, University of Tokyo Kensuke [email protected] CP13 A Parameter Free ADI-like Method for the Numer- Takayasu Matsuo ical Solution of Large Scale Lyapunov Equations Graduate School of Information Science and Technology, the University of Tokyo A parameter free ADI-like method is presented for con- [email protected] structing an approximate numerical solution to a large scale Lyapunov equation. The algorithm uses an approxi- Kazuo Murota mate power method iteration to obtain a basis update and Graduate School of Information Science and Technology, then constructs a re-weighting of this basis update to pro- University of Tokyo vide a factorization update that satisfies ADI-like conver- [email protected] gence properties. Hung Nong Masaaki Sugihara Sandia National Laboratories Graduate School of Information Science and Technology, [email protected] the University of Tokyo m [email protected] Danny C. Sorensen Rice University CP14 [email protected] Extraction Strategies for Nonsymmetric Eigen- value Problems CP13 We consider the extraction phase of subspace methods like Nonnegativity of the Group-Projectors Used to the Jacobi-Davidson method for solving large eigenvalue Obtain the Nonnegativity of Control Singular Sys- problems. In the nonsymmetric case, known approaches do tems not satisfy any optimality property and, in practice, their In POSTA06 we studied the nonnegativity of control sin- behavior may be unsatisfactory, especially when searching gular systems via state-feedbacks by using a characteriza- for the eigenvalues with largest real part. In this talk, we tion of the nonnegativity of the group-projector of a square shall review and compare existing methods, and also pro- matrix E assuming the nonnegativity of the matrix E. In pose some new strategies providing enhanced robustness. POSTA09 we will present a characterization of the nonneg- Julien Callant ativity of the group-projector, slightly simplified. In this Service de M´etrologieNucl´eaire work, we use this last characterization to improve the study Universit´eLibre de Bruxelles (ULB) mentioned at the beginning. This work has been partially [email protected] 50 LA09 Abstracts

Yvan notay Problems with Spectral Transformations Universite Libre de Bruxelles [email protected] This talk concerns algorithms for solving generalized eigen- value problems, with emphasis on efficient iterative solu- tion of the sequence of linear systems that arise when in- CP14 exact subspace iteration and implicitly restarted Arnoldi On Singular Two-Parameter Eigenvalue Problem method are applied to detect interior eigenvalues. We pro- vide new insights into the tuning of preconditioning pro- In the 1960s Atkinson introduced an abstract algebraic set- posed by Spence and his collaborators, and show that the ting for multiparameter eigenvalue problems. He showed cost of solving these linear systems can be further reduced that a nonsingular multiparameter eigenvalue problem is by the combined use of several other strategies. equivalent to the associated system of generalized eigen- value problems. We extend his result to singular two- Fei Xue parameter eigenvalue problems and show that under very University of Maryland mild conditions the finite regular eigenvalues of both prob- [email protected] lems agree. This enables one to solve such problems by computing the common regular eigenvalues of two singular Howard C. Elman generalized eigenvalue problems. University of Maryland, College Park [email protected] Andrej Muhic Institute of Mathematics, Physics and Mechanics Jadranska 19, SI-1000 Ljubljana, Slovenia CP15 [email protected] An Algorithm for Constructing a and the of Arbitrary Linear Bor Plestenjak Operators over Any . Department of Mathematics University of Ljubljana It is well known that the Jordan normal form for a linear [email protected] operator A is defined and is constructed together with the corresponding basis in case when the basic field k is al- gebraically closed or, more generally, if the characteristic CP14 (equivalently, the minimal) polynomial of A is decomposed Tropical Scaling of Polynomial Eigenvalue Problem into a product of linear polynomials. We define the (gener- alized) Jordan normal form (of the second kind) for a linear We introduce a general scaling technique, based on tropical operator A of a linear space L over an arbitrary field k and algebra,to compute the eigenvalues of a matrix polynomial. with an arbitrary characteristic polynomial. For such an We show that under nondegeneracy conditions, the orders operator we give an algorithm which is used to construct of magnitude of the different eigenvalues are given by cer- this form by finding a corresponding canonical basis. As tain “tropical roots’. We use them to construct a scaling, application some other results are obtained, too. and we show by experiments that this improves the accu- racy of the computations, particularly when the data have Samuel H. Dalalyan various orders of magnitude. [email protected] [email protected] Meisam Sharify,St´ephane Gaubert INRIA Saclay Ile-de-France & Centre de Math´ematiquesappliqu´ees,Ecole CP15 Polytechnique A General Subspace Structure Parameterization of [email protected], [email protected] a Pre-Compesator for a Non-Interacting Controller in Robotic Manipulation Systems and Mechanisms

CP14 This paper deals with a total general geometric approach On MRRR-Type Algorithms for the Tridiagonal to the study of robotics manipulators and general mecha- Symmetric Eigenproblem and the Bidiagonal SVD nisms dynamics. In particular a parametrization of a pre- compensator through structures of subspaces for a nonin- The focus of our research is extending Dhillon & Parlett’s teracting controller is presented. Due to the technological algorithm MR3 to compute accurate singular vectors for a development, robotics applications are growing in many . Doing this without spoiling accuracy is industrial sectors and even in such applications as medical a challenge. Among other aspects we had to evaluate how one (micro manipulation of internal tissues or laparoscopy). key parts of MR3 could be varied to improve reliability and In such a framework some typical problems in robotics are efficiency. As main theoretical contribution we regard our mathematically formalized and analyzed. The outcomes investigation of using block factorizations, allowing 2 × 2- are so general that it is possible to speak of structural prop- pivots, in the representation tree. Numerical results will erties in robotic manipulation and mechanisms. The prob- detail the effectiveness of our techniques. lem of designing of force/motion controller is investigated. A generalized linear model is used and a careful analysis Paul R. Willems is made. The presented results consist of proposing a gen- University of Wuppertal, Germany eral geometric structure to the study of mechanisms. In Applied Computer Science Group particular, a Lemma and a Theorem are shown which offer [email protected] a parametrization of a pre-compesator for a task–oriented choice of input subspaces. The existence of these input subspaces are a necessary condition for a structural nonin- CP14 Numerical Solutions of Generalized Eigenvalue LA09 Abstracts 51

teraction property. the matching moment property of the Lanczos, CG and Arnoldi methods, and specify techniques for estimation of Paolo Mercorelli cΛA-1b with A non- Hermitian, including a new algorithm University of Applied Sciences Wolfsburg based on the BiCG method. We show its mathematical [email protected] equivalence to the existing estimates which use a complex generalization of Gauss quadrature, and discuss its numer- ical properties. The proposed estimate will be compared CP15 with existing approaches using analytic arguments and nu- A Sinkhorn-Knopp Fixed Point Problem merical experiments on a practically important problem that arises from the computation of diffraction of light on We consider the fixed media with periodic structure. point problem x =(AT (Ax)(−1) )(−1) (where (−1) is the entry-wise inverse), which arises from the Sinkhorn-Knopp Petr Tichy Algorithm for transforming a positive matrix into a unique Czech Academy of Sciences doubly . We discuss basic properties of [email protected] solutions, including for non-positive and complex matrices, and we consider some special cases, including some inter- Zdenek Strakos esting results involving patterned matrices. This work was Academy of Sciences of the performed in part by four undergraduates as part of an Czech Republic, Prague NSF-sponsored undergraduate research project. [email protected] David M. Strong Department of Mathematics CP15 Pepperdine University [email protected] Jordan Forms For Two Matrices Which Commute This article gives a method of obtaining the Jordan form of CP15 two commuting matrices R and A when R is diagonalizable. An nxn complex matrix R with the property that a positive Some Properties of the K and (s+1) - Potent Ma- power of R gives the identity is examined and examples are trices given to demonstrate the methods developed in the paper. The situations where a A coincides with any of its powers have been studied as well as another class of James R. Weaver matrices which involves an . So, the University of West Florida concept of idempotency can be generalized. This concept [email protected] is widely used in several applications. We introduce a new kind of matrices called K and (s+1)-potent matrices as an extension of both situations mentioned above. A study of CP16 these matrices is developed with some examples. Matching Figures with Sentences in Technical Ab- Leila Lebtahi stracts Universidad Politecnica de Valencia, Spain Figures are key content in technical literature, so the abil- Departamento de Matematica Aplicada ity to quickly access images relevant to a sentence in a [email protected] document’s abstract would provide a valuable resource to scientists. We use two main ideas to realize this goal: a Nestor Thome mixture language model, which can be viewed as a non- Instituto de Matematica Multidisciplinar negative matrix factorization, and a Markov model. We Universidad Politecnica de Valencia, Valencia, Spain introduce a utility metric and report results of applying [email protected] both models to 100 documents from the biomedical litera- ture. CP15 John M. Conroy On Efficient Numerical Approximation of the Scat- IDA Center for Computing Sciences tering Amplitude [email protected]

This talk presents results on efficient and numerically well- Joseph Bockhorst behaved estimation of the scalar value cΛx, where cΛ de- University of Wisconsin - Milwaukee notes the conjugate of c and x solves the lin- Dept. of Electrical Eng. and Comp. Sci. ear system Ax = b, A 2 CNN is a nonsingular complex [email protected] matrix and b and c are complex vectors of length N. In other words, we wish to estimate the scattering amplitude cΛA-1b . In our understanding, various approaches for nu- Dianne P. O’Leary merical approximation of the scattering amplitude can be University of Maryland, College Park viewed as applications of the general mathematical con- Department of Computer Science cept of matching moments model reduction, formulated [email protected] and used in applied mathematics by Vorobyev in his re- markable book [3]. Using the Vorobyev moment problem, Hong Yu matching moments properties of Krylov subspace methods University of Wisconsin - Milwaukee can be described in a very natural and straightforward way, Depts. of Health Sciences and Computer Sci. see [1]. This talk further develops the ideas from [1] into [email protected] efficient estimates of cΛA-1b, see [2]. We briefly outline 52 LA09 Abstracts

CP16 trix Problem Perturbation Theory for the LDU Factorization of Diagonally Dominant Matrices and Its Application Motivated by the classical Newton-Schulz method for find- to Accurate Computation of Singular Values ing the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive semidefitive solution X− of the matrix rational We present a new structured perturbation theory for the ∗ −1 LDU factorization of diagonally dominant matrices that equation X + A X A = I, where I is the allows us to prove that the relative errors in the factors and A is a given matrix. This equation appears in many computed by using an algorithm developed by Q. Ye in applied areas including control theory, dynamical program- 2008 are bounded by O(n4), where n is the dimension of ming, ladder networks, stochastic filtering and statistics. the matrix and  the machine precision. The previous er- We present convergence results and discuss stability prop- ror bounds proved for this algorithm were O(8n), that are erties when the method starts from the available matrix ∗ useless for matrices with dimension as small as n =20in AA . We also present numerical results to compare our double precision. This new error bound proves rigurously proposal with some recently developed inversion-free tech- that the computed LDU factorization can be used to com- niques for solving the same rational matrix equation. pute accurately the SVD of this type of matrices. Marlliny Monsalve Froilan M. Dopico Departamento de Computaci Department of Mathematics ’{o}n. Facultad de Ciencias Universidad Carlos III de Madrid Universidad Central de Venezuela [email protected] marlliny,[email protected]

Plamen Koev Marcos Raydan San Jose State University Departamento de Calculo Cientifico y Estadistica. [email protected] Universidad Simon Bolivar. Caracas-Venezuela [email protected]

CP16 PageRank As a Function of a Matrix Instead of a CP16 Scalar Small-World Model with Asymmet- ric Transition Probabilities The PageRank vector for a stochastic matrix is a rational function of a scalar parameter. Replacing the scalar with Motivated by existing results concerning the case of sym- a matrix yields a PageRank function of a matrix parame- metric transition probabilities, we study the effects of a ter. The trivial relationship between the Markov chain and varying degree of asymmetry in transition probabilities on linear system for PageRank then disappears. We therefore the behavior of the Markov chain small-world model of investigate a few other analogies. These ideas are prelimi- D. Higham. Asymptotic results in this regard are devel- nary and one goal is to solicit feedback on other potential oped following a matrix-theoretic approach. An interest- relationships. We show a few numerical examples. ing outcome is that the behavior of the model is strongly influenced by the intensity of asymmetry, which may find David F. Gleich applications in real-world networks involving imbalanced Stanford interactions. [email protected] Jianhong Xu Paul Constantine Southern Illinois University Carbondale Stanford University Department of Mathematics Institute for Computational and Mathematical [email protected] Engineering [email protected] CP17 A Numerical Method for Quadratic Constrained CP16 Maximization Models for Beta-Wishart Random Matrices The problem of maximizing a quadratic function subject We present our recent progress on developing empirical to an ellipsoidal constraint is considered. The algorithm models for Beta-Wishart random matrices. These mod- produces a nearly optimal solution in a finite number of it- els bridge the gap between the classical real, complex, and erations. In particular, the method can be used to solve the quaternion Wishart random matrices and provide a contin- ill-conditioned problems in which the solution consists of uous treatment of the theory for any positive beta. This two parts from two orthogonal subspaces. Without restric- work generalizes the identity-covariance (Beta-Laguerre) tive assumptions, the solution generated by the method models of Dumitriu and Edelman to arbitrary covariance. satisfies the first and second order necessary conditions for a maximizer of the objective function. Numerical experi- ments clearly demonstrate that the method is successful. Plamen Koev San Jose State University Alireza Esna Ashari [email protected] INRIA- Rocquencourt alireza.esna [email protected]

CP16 Ramine Nikoukhah A Simplified Newton’s Method for a Rational Ma- INRIA - Rocquencourt [email protected] LA09 Abstracts 53

Stephen L. Campbell CP17 North Carolina State Univ Variable Projection Methods for Separable Nonlin- Department of Mathematics ear Least Squares Learning [email protected] We describe variable projection methods (VPM) for neural- network learning. We explain why classical VPM may not CP17 work excellently, and describe how to modify VPM to in- Joint Spectral Characteristics of Matrices: a Conic clude the second-order information of the residual Hessian Programming Approach matrix unlike the standard VPM that uses the Levenberg- Marquardt Hessian. The modification, supported by effi- The joint spectral radius and subradius are a research topic cient stage-wise evaluation of the Hessian that requires es- of growing impact in hybrid systems, combinatorics, nu- sentially no more computation than for the Gauss-Newton merical analysis, etc. We propose a new approach to com- Hessian, allows us to exploit the negative curvature if it pute them. To our knowledge this constitutes the first exists. method to compute the joint spectral subradius. We show how previous results can be interpreted as particular cases Eiji Mizutani of ours. We show the efficiency of our algorithms by ap- Dept. of Industrial Management, National Taiwan plying them to several problems of combinatorics, number University theory and discrete mathematics. of Science and Technology [email protected] Raphael Jungers Massachusetts Institute of Technology James W. Demmel Laboratory for Information and Decision Systems University of California [email protected] Division of Computer Science [email protected] Vladimir Y. Protasov Moscow State University Department of Mechanics and Mathematics CP17 [email protected] Solving Large-Scale Eigenvalue Problems from Op- timization Vincent Blondel Universite catholique de Louvain. We describe parameterized eigenvalue problems arising in [email protected] methods for large-scale trust-region subproblems and regu- larization. We present comparisons of matrix-free methods for solving the eigenproblems. CP17 Minimization of the Kohn-Sham Energy with a Lo- Marielba Rojas calized, Projected Search Direction Technical University of Denmark [email protected] With the constantly increasing power of computers, the realm of experimental chemistry is increasingly being brought into the field of computational mathematics. This CP17 work integrates non-linear programming methods with a Preconditioning for PDE-Constrained Optimiza- projected search direction to directly minimize the Kohn- tion Sham energy. By ensuring that the search direction re- mains sparse, we guarantee that the computationally- Advances in algorithms and hardware have enabled more intensive energy function is only evaluated at sparse it- research on the optimization of functions with constraints erates. Maintaing sparsity, while avoiding local minima, given by partial differential equations. Problems of this is a critical first-step in developing a fully linear-scaling type arise in a variety of applications and pose significant algorithm to minimize the Kohn-Sham energy. challenges to optimization algorithms and numerical meth- ods. In this talk we present preconditioners for problems Marc Millstone in a general setup and also when additional box constraints Courant Institute of Mathematical Sciences are introduced for the control. These constraints add an New York University extra layer of complexity to the optimization method and [email protected] the efficient solution of the linear system is very important. Block-diagonal preconditioners present one possibility to Juan C. Meza solve the arising saddle point problems efficiently. In addi- Lawrence Berkeley National Laboratory tion, we discuss the use of block-triangular preconditioners [email protected] that can be used in a non-standard inner product iterative method. We show that the drawbacks of this method can be easily overcome by choosing appropriate precondition- Michael L. Overton ers for the blocks of the saddle point system. We present New York University an eigenvalue analysis for the preconditioners and illustrate Courant Instit. of Math. Sci. their competitiveness on some examples. [email protected] Martin Stoll Chao Yang University of Oxford Lawrence Berkeley National Lab [email protected] [email protected] Tyrone Rees Oxford University 54 LA09 Abstracts

[email protected] [email protected]

Andy Wathen Cedric Chevalier Oxford University Sandia National Laboratories Computing Laboratory Scalable Algorithms Department [email protected] [email protected]

CP18 CP18 Generalized Laplacians and First Transit Times for Multivariable Toeplitz Matrices Directed Graphs We are interested in formulas for inverse of positive definite We extend results on average commute-times for undi- multivariable Toeplitz matrices T =(tk−l)k,l∈Λ with Λ ⊂ rected graphs to fully-connected directed graphs, corre- N d. Attempts to provide a direct generalization of the sponding to irreducible Markov chains. We show how the Gohberg-Semencul formula have failed so far. In this paper pseudo-inverse of an unsymmetrized generalized Laplacian we use a multivariable Gohberg-Semencul type expression matrix yields the first-transit times and commute times which serves as an upper bound for the inverse. with formulas, almost matching the undirected graph case. The matrices are positive semi-definite, leading to a natural Selcuk Koyuncu, Hugo Woerdeman embedding in Euclidean space preserving commute times. Drexel University These results are illustrated with an application to an anal- [email protected], hugo @ math.drexel.edu ysis of ad-hoc asymmetric wireless networks. Daniel L. Boley CP18 University of Minnesota Structured Distance to Normality in the Banded Department of Computer Science Toeplitz Case [email protected] In this talk we are concerned with the structured and un- Gyan Ranjan, Zhi-Li Zhang structured distances to normality of banded Toeplitz ma- University of Minnesota trices in the Frobenius norm. We present formulas for the Computer Science distance of banded Toeplitz matrices of suitable restricted [email protected], [email protected] bandwidth to the algebraic variety of similarly structured normal matrices. We then investigate the distance to gen- eralized circularity, formulating a conjecture on the dis- CP18 tance to the variety of normal matrices. The talk is based Graph Laplacians and Logarithmic Forest Dis- on joint work with Lothar Reichel. tances Silvia Noschese A new family of distances for graph vertices is proposed. Dipartimento di Matematica ”Guido Castelnuovo” They are a kind of logarithmically transformed forest dis- SAPIENZA Universita‘ di Roma tances. They reduce to the shortest path distance and to [email protected] the resistance distance at the extreme values of the family parameter. Additionally, they satisfy bottleneck additivity: CP18 d(i, j)+d(j, k)=d(i, k) if and only if every path from i to k contains j. The construction of the distances is based Factorization of Block Toeplitz Matrices. Applica- on the matrix forest theorem and the graph bottleneck in- tions to Wavelets and Functional Equations. equality. Block (or slanted) Toeplitz matrices have found a lot of ap- Pavel Chebotarev plications in functional equations, wavelets theory, proba- Institute of Control Sciences bility, combinatorics, etc. In the simplest case, 2-block N × N T p of the Russian Academy of Sciences Toeplitz matrices are -matrices ( s)ij = 2i−j+s−1, s , i, j ∈{ ,...,N} [email protected] =0 1, and 1 for a sequence of complex co- efficients p0,...,pN . We obtain a complete spectral factor- ization of those matrices depending on the sequence pk, de- CP18 scribe the structure of their kernels, root subspaces, and all Parallel Unsymmetric Nested Dissection Ordering common invariant subspaces. These results allow us to sim- for Sparse Direct Solvers plify a formula for the exponent of regularity of wavelets, to obtain a factorization theorem for refinable functions, and Fill-reducing ordering algorithms for unsymmetric prob- to characterize the manifold of smooth refinable functions. lems are often inherently sequential (e.g., COLAMD). For This also solves the problem of continuity of solutions of large problems, a common approach is to apply nested dis- the refinement equations with respect to their coefficients section on the symmetrized matrix. The HUND algorithm, and gives a criterion of convergence of the corresponding based on hypergraph partitioning, was recently proposed cascade algorithms and subdivision schemes. to overcome these drawbacks. We present the first parallel implementation of the HUND algorithm, soon available in Vladimir Y. Protasov the Zoltan toolkit. We show preliminary results for some Moscow State University application matrices. Department of Mechanics and Mathematics [email protected] Erik G. Boman Sandia National Labs, NM Scalable Algorithms Dept. LA09 Abstracts 55

MS1 MS1 Title Not Available at Time of Publication Computing the Phi-function of a Matrix

Abstract not available at time of publication. The phi-functions are certain functions related to the expo- nential: the zeroth phi-function is the exponential itself and x Ernesto Estrada the first one is defined by φ1(x)=(e − 1)/x. They play University of Strathclyde an important role in exponential integrators. We present [email protected] an algorithm to evaluate the action of φk(A) on a vector. The key ingredients in the algorithm are a Krylov-subspace method to reduce the size of the matrix and a time-stepping MS1 method akin to the scaling-and-squaring method for ma- Rational Approximation to Trigonometric Opera- trix exponentials. tors Jitse Niesen We will discuss the approximation of trigonometric opera- University of Leeds tor functions that arise in the numerical solution of wave [email protected] equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions with- Will M. Wright out exponential decay show superlinear convergence be- La Trobe University havior if the number of steps is larger than the norm of [email protected] the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this talk, a rational Krylov subspace method is proposed which converges not MS2 only for finite element or finite difference approximations Ritz Value Localization and Convergence of Non- to differential operators but even for abstract, unbounded Hermitian Eigensolvers operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the oper- Rayleigh-Ritz eigenvalue estimates for Hermitian matrices ator and thus of its spatial discretization. We will discuss obey the well-known interlacing property, which plays a efficient implementations for finite element discretizations crucial role in explaining convergence of the restarted Lanc- and illustrate our analysis with numerical experiments. zos method with exact shifts. Much less is understood about Ritz values of non-Hermitian matrices, and conse- Volker Grimm quently a satisfactory convergence theory for the restarted Friedrich-Alexander Universitaet Erlangen-Nuernberg Arnoldi method has proved elusive. We shall describe [email protected] recent progress on the localization of Ritz values, which provide sufficient (though strict) conditions for restarted Marlis Hochbruck Arnoldi convergence. Heinrich-Heine-University Duesseldorf [email protected] Russell Carden, Mark Embree Computational and Applied Mathematics Rice University MS1 [email protected], [email protected] A New Scaling and Squaring Algorithm for the Ma- trix Exponential MS2 The scaling and squaring method for the matrix exponen- Low Rank Tensor Approximation A −s 2s tial is based on the approximation e ≈ (rm(2 A)) , x where rm(x) is the [m/m] Pad´eapproximant to e and the We present a black-box type algorithm for the approxima- m and s are to be chosen. Several authors have tion of tensors in high dimension d. The algorithm deter- identified a weakness of existing scaling and squaring algo- mines adaptively the positions of entries of the tensor that rithms termed overscaling, in which a value of s much larger have to be computed or read, and using these few entries than necessary is chosen, causing a loss of accuracy in float- it constructs a low rank tensor approximation. For effi- ing point arithmetic. Building on the scaling and squaring ciency reasons the positions are located on fibre-crosses. algorithm of Higham [SIAM J. Matrix Anal. Appl.,26 The minimization problem is solved by Newton’s method (4):1179–1193, 2005], which is used by MATLAB’s expm, which requires the Hessian. It can be evaluated in data- we derive a new algorithm that alleviates the overscaling sparse form in O(d). problem. Lars Grasedyck Nicholas J. Higham Max-Planck-Institute University of Manchester Math in Sciences School of Mathematics [email protected] [email protected] MS2 Awad H. Al-Mohy School of Mathematics The Concept of Augmented Backward Stability for The University of Manchester Some Iterative Algorithms, with Emphasis on the [email protected] Lanczos Process Jim Wilkinson developed the ideas of backward stable nu- merical algorithms. Some crucial algorithms are clearly not backward stable, but provide useful computed results. Several of these are based on orthogonality, but fail miser- 56 LA09 Abstracts

ably to provide this orthogonality. However there exists a Pennsylvania State University truly orthogonal higher dimensional matrix for each com- [email protected] puted one. By using this fact, some of these algorithms are shown to give correct answers for “nearby’ higher di- Ira Livshits mensional problems. We call this “augmented backward Ball State University stability’. [email protected] Christopher C. Paige McGill University, Montreal MS3 [email protected] Towards Nonsymmetric Adaptive Smoothed Ag- gregation for Systems of Pdes

MS2 Smoothed aggregation (SA) applied to nonsymmetric lin- Tensor Network Computations and Analysis ear systems, Ax = b, lacks a minimization principle in coarse-grid correction. Consider applying SA to symmet- A tensor network is a way to build high-order tensors √ √ ric positive definite (SPD) matrices, AtA or AAt, for through structured contractions of many low-order tensors. which minimization principles exist. It is not computation- The manipulation of such objects can lead to sequences of ally efficient to use either matrix directly in coarse-grid cor- QR factorizations and SVDs. Product versions of these rection as they are typically full. The proposed approach well known matrix tools turn out to be important for nu- efficiently approximates these corrections using SA adap- merical reasons. tively to approximate right and left singular vectors of A. Charles Van Loan Linear systems arising from discretizations of nonsymmet- Cornell University ric systems of PDEs are considered. Department of Computer Science Geoffrey D. Sanders [email protected] University of Colorado, Department of Applied Mathematics MS3 geoff[email protected] Algebraic Multigrid Marian Brezina In this talk, we will give an overview of the algebraic multi- U. of Colorado, Boulder grid (AMG) method and some of the research issues asso- Applied Math Department ciated with it. The talk will primarily serve as background [email protected] and motivation for the other presentations in the minisym- posium, filling in on topic areas that will not otherwise be Tom Manteuffel covered. In particular, we will discuss in more detail de- University of Colorado, Department of Applied velopments in AMG theory and on coarsening techniques Mathematics based on compatible relaxation. [email protected] Robert Falgout Steve McCormick Center for Applied Scientific Computing Department of Applied Math Lawrence Livermore National Laboratory CU-Boulder [email protected] [email protected]

MS3 John Ruge Bootstrap AMG in Lattice QCD University of Colorado [email protected] Operators arising in lattice quantum-chromodynamics (LQCD) simulations pose various challenges to current solvers. For physically interesting configurations the per- MS3 formance of Krylov-subspace methods deteriorates quickly Algebraic Multigrid for Massively Parallel Multi- which fueled the search for preconditioners. Due to the na- Core Architectures ture of the background SU(n) gauge field classical algebraic multigrid (AMG) fails that task, but by introduction of Algebraic multigrid has shown to be extremely efficient on adaptive techniques and generalization of AMG principles distributed-memory architectures. However, the new gen- we were able to overcome some of the immanent challenges eration of high-performance computers with an increasing with our Bootstrap approach. number of nodes, which include multiple cores sharing the bus and several caches and an increased core-to-main mem- Karsten Kahl ory performance gap, presents new challenges. Algorithms University of Wuppertal need to have good data locality, few synchronization con- Department of Mathematics flicts, and increased small grain parallelism while at the [email protected] same time improving scalability to deal effectively with millions of cores. This presentation discusses the design Achi Brandt and implementation of algebraic multigrid on such archi- UCLA tectures. [email protected] Ulrike Yang Lawrence Livermore National Laboratory James Brannick [email protected] Department of Mathematics LA09 Abstracts 57

MS4 trix. The talk is based on joint work with Lionello Pasquini Computing the Distance to Bounded Realness via and Lothar Reichel. the Solution of Structured Eigenvalue Problems Silvia Noschese We study control systemsx ˙ = Ax + Bu, y = Cx + Du Dipartimento di Matematica ”Guido Castelnuovo” that arise via discretization and model reduction in elec- SAPIENZA Universita‘ di Roma tric and magnetic field computation. While the original [email protected] system is typically strictly passive (in simple words it does not generate energy) the approximate system often has lost its passivity. The system is then usually made pas- MS4 sive by small perturbations to the system matrices. The Structured Quadratic Matrix Polynomials: Sol- computation of small or smallest perturbations that per- vents and Inverse Problems form this task leads to the task of finding smallest pertur- Q λ bations of Hamiltonian matrices that move eigenvalues off A monic polynomial ( ) of degree 2 can be factorized into a product of two linear matrix polynomi- the imaginary axis. We give explicit formulas for the min- Q λ Iλ−S Iλ−A imal perturbations to move purely imaginary eigenvalues als, say ( )=( )( ). For the inverse problem of of different sign characteristics together and also to move finding a quadratic matrix polynomial with given spectral data it is natural to prescribe a right divisor A and then multiple purely imaginary eigenvalues to specific values in S Q λ the complex plane. We also present a numerical method determine compatible left divisors . In applications, ( ) to compute upper bounds for the minimal perturbations may be Hermitian, real symmetric, hyperbolic, elliptic, or and discuss how to employ these ideas in the context of have mixed real/non-real spectrum, and these structures passivation. add extra constraints to the inverse problem. These issues are explored in this talk. Rafikul Alam Assistant Professor, Department of Mathematics, Francoise Tisseur Indian Institute of Technology Guwahati, INDIA University of Manchester rafi[email protected] Department of Mathematics [email protected] Shreemayee Bora Department of Mathematics, IIT Guwahati, India MS4 [email protected] Computing the Hamiltonian Schur Form by Blocks

Michael Karow We discuss block methods for computing the Hamiltonian Technical University of Berlin Schur form of a . By placing each clus- Germany ter of eigenvalues together in a block, we obtain a more ro- [email protected] bust algorithm. This is joint work with Christian Schroeder and Volker Mehrmann. Volker Mehrmann Inst. f.Mathematik, TU Berlin David S. Watkins [email protected] Washington State University Department of Mathematics [email protected] Julio Moro Universidad Carlos III de Madrid Departamento de Matem´aticas MS5 [email protected] High Performance Algorithms for Materials from Nanocrystals to Crystals

MS4 A challenging problem in materials physics is to develop Structured Distance to Normality and Eigenvalue algorithms that permit prediction of materials properties Sensitivity in the Banded Toeplitz Case at disparate length scales. Most algorithms are specialized for a periodic system such as a crystal, or a localized sys- Matrix nearness problems have been the focus of much re- tem such as a nanocrystal. I will discuss new algorithms to search in linear algebra. In particular, characterizations handle both regimes. Our algorithms are based on imple- of the algebraic variety of normal matrices and distance menting the Kohn-Sham problem in real space and solving measures to this variety have received considerable atten- the corresponding eigenvalue problem without explicit re- tion. Banded Toeplitz matrices arise in many applications sort to standard diagonalization methods. in signal processing, time-series analysis, and numerical methods for the solution of partial differential equations. Jim Chelikowsky In this talk we first describe spectral properties of nor- University of Texas at Austin mal (2k + 1)-banded Toeplitz matrices of order n, with [email protected] k ≤ n/2 . We then present formulas for the distance of (2k + 1)-banded Toeplitz matrices to the algebraic va- riety of structured normal matrices of same bandwidth. MS5 We focus on the perturbation analysis in connection with The Quantum Monte Carlo Method and Linear Al- the distance to normality, devoting special attention to the gebra Problems sensitivity of the spectrum to structure-preserving pertur- bations in the tridiagonal Toeplitz case, and describing an In the Quantum Monte Carlo method we must evaluate interesting example involving a Jordan block. Finally, we determinant ratios for successive matrices in a very long report on an application regarding the determination of a sequence. One approach is to solve a sequence of linear set containing the field of values of a (2k + 1)-banded ma- systems, another to solve generalized eigenvalue problems, 58 LA09 Abstracts

and yet another to approximate the determinants or ratios to a 180x speedup on a CPU-GPU system. Our approaches directly estimating bilinear forms of matrix functions. We are generally applicable to hybrid systems with heteroge- will give a brief overview and then focus on linear solvers, neous parallel processors. particularly on a multilevel preconditioner that can be up- dated very cheaply. Bryan Catanzaro EECS Department Eric De Sturler University of California; Berkeley Virginia Tech [email protected] [email protected] Vasily Volkov, Bor-Yiing Su, Narayanan Sundaram EECS Department MS5 University of California, Berkeley Electronic Structure: Basic Theory and Practical [email protected], [email protected], Methods [email protected] Recent advances in quantitative theory of electronic prop- erties of materials have occurred through the combination Jim Demmel of new theoretical developments, clever algorithms, and in- UC Berkeley creased computational power. This talk will focus on as- [email protected] pects of the basic theory, with emphasis upon the struc- ture of the methods. In particular, I will describe current Kurt Keutzer work to combine density functional theory with many-body EECS Department methods in order to create robust methods to treat broad University of California, Berkeley classes of materials. [email protected] Richard M. Martin Department of Applied Physics MS6 Stanford University Parallel Distributed Reduction to Condensed [email protected] Forms - Scheduling Sequences of Givens Rotations Some algorithms for two-sided reduction of matrices and MS5 matrix pairs to condensed forms generate sequences of On the Convergence of the Self Consistent Field Givens rotations acting on rows or columns i and i + 1 for Iteration for a Class of Nonlinear Eigenvalue Prob- contiguous indices i. This raises some interesting schedul- lems ing problems when using 2D block-cyclic distribution of the matrices. We present static scheduling techniques based Methods based on Density Functional Theory (DFT) re- on partial wavefronts for both rectangular and triangular quire solving a nonlinear eigenvalue problem derived from matrices. The techniques also apply to the application of the Schrodinger equation. At the heart of these codes, accumulated Givens rotations using level 3 BLAS. one typically finds a Self Consistent Field (SCF) iteration, which is the predominant method for solving this under- Lars Karlsson lying nonlinear eigenvalue problem. Despite its popularity Ume˚aUniversity however, there are few convergence results available. I will Computing Science and HPC2N discuss some new and interesting results showing the con- [email protected] vergence behavior for the SCF iteration. Bo T. K˚agstr¨om Juan C. Meza Ume˚aUniversity Lawrence Berkeley National Laboratory Computing Science and HPC2N [email protected] [email protected]

Chao Yang Lawrence Berkeley National Lab MS6 [email protected] Iterative Refinement of Schur Decompositions for Mixed Precision Computation Weiguo Gao Assume that the Schur or spectral decomposition of a ma- Fudan University, China A [email protected] trix is available at a considerably low machine precision but the application demands for higher accuracy. Exam- ples for this setting of current interest include mixed sin- MS6 gle/double precision computations on Cell processors and CPU-GPU Hybrid Eigensolvers for Symmetric GPUs as well as mixed double/quad precision on standard Eigenproblems CPUs. In both cases, the operations performed in high pre- cision are significantly more costly and should therefore be We examine the use of CPU-GPU hybrid systems to ac- limited to the bare minimum. This talk will discuss refine- celerate a few symmetric eigensolvers. We investigate the ment procedures for entire decompositions that allow us in bisection and inverse iteration methods for tridiagonal ma- many cases to have a highly accurate trices by performing redundant computation, optimizing essentially at the cost of the low precision computation. for higher resource utilization and reduced memory traf- Particular attention will be paid to situations where some fic. We then examine a Lanczos method for a particu- of the eigenvalues of A are highly ill-conditioned. lar spectral partitioning problem, finding that using the Cullum-Willoughby test without reorthogonalization leads Daniel Kressner ETH Zurich LA09 Abstracts 59

[email protected] Numerical results using the Matlab toolbox Intlab will be reported. Christian Schr¨oder TU Berlin Andreas J. Frommer [email protected] Bergische Universitaet Wuppertal Fachbereich Mathematik und Naturwissenschaften [email protected] MS6 A Novel Parallel QR Algorithm for Hybrid Dis- Behnam Hashemi tributed Memory HPC Systems Amirkabir University of Technology, Tehran hashemi [email protected] Key techniques used in our novel parallel QR algorithm in- clude multi-window bulge chain chasing and distributed ag- gressive early deflation, which enable level 3 chasing oper- MS7 ations and improved eigenvalue convergence. Mixed MPI- Green’s Function Calculations in Nano Research OpenMP coding techniques are utilized for DM platforms with multithreaded nodes, such as multicore processors. The non-equilibrium Green’s function formalism provides Application and test benchmarks confirm the superb per- a powerful conceptual and computational framework for formance of our parallel QR algorithm. In comparison with treating quantum transport in nanodevices. The Green’s the existing ScaLAPACK code, the new parallel software is function that we are interested in is a function of a bi- one to two orders of magnitude faster for sufficiently large infinite block tridiagonal Hermitian matrix, and only one problems. special diagonal block of the Green’s function is required in applications. Computing this special matrix has been Robert Granat a challenging problem for nano-scientists. In this talk we Dept. of Computing Science and HPC2N present a satisfactory solution to this problem. Ume˚aUniversity, Sweden [email protected] Chun-Hua Guo University of Regina Bo T. K˚agstr¨om Dept of Math & Statistics Ume˚aUniversity [email protected] Computing Science and HPC2N [email protected] Wen-Wei Lin National Chiao Tung University Daniel Kressner [email protected] ETH Zurich [email protected] MS7 Computing Matrix Means MS6 The definition of geometric mean of two positive definite Accelerating the Reduction to Upper Hessenberg matrices is not straightforward. However, imposing some Form through Hybrid GPU-based Computing expected properties leads to the following matrix We present a Hessenberg reduction algorithm for hybrid −1 1/2 multicore+GPU systems that gets 16x performance im- A(A B) , provement over the current LAPACK algorithm running just on multicores (in double precision). This enormous ac- which is defined as the geometric mean of A and B and celeration is due to proper matching of algorithmic require- denoted by A#B. For more than two matrices the situa- ments to architectural strengths of the hybrid components. tion is more complicated. Filling a list of properties that The reduction is an important linear algebra problem, es- a geometric mean should verify does not lead to a unique pecially with its relevance to eigenvalue solvers. The re- matrix. In fact, there is no complete agreement on the def- sults are significant because Hessenberg reduction has not inition of the geometric mean of more than two matrices. yet been accelerated on multicore architectures. We briefly survey some applications and problems related to the matrix geometric means and present some numerical Stanimire Tomov methods to compute them. Innovative Computing Laboratory University of Tennessee; Knoxville Bruno Iannazzo [email protected] Universita‘ di Perugia, Italy [email protected] Jack Dongarra University of Tennessee Knoxville [email protected] MS7 An Effective Matrix Geometric Mean Satisfying the Ando-Li-Mathias Properties MS7 Computing Enclosures for the Matrix There are in literature several proposals to define a proper generalization of the geometric mean to k ≥ 3 positive def- Given an approximation X to the inite matrices. Ando, Li and Mathias listed ten properties A, we present methods based on interval arithmetic which that a ”good” mean should fulfill, and proposed a definition compute an entrywise enclosure for the ‘true’ square root. satisfying them, based on the limit of an iteration process. Our approach is based on a modification of Krawczyk’s We propose another definition, as the limit of a cubically method which reduces the complexity from O(\)toO(\). convergent iteration, having all the desired properties but 60 LA09 Abstracts

much easier to compute. MS8 Do We Understand Gaussian Elimination? Dario Bini, Beatrice Meini Dept. of Mathematics Despite the enormous progress in improving efficiency of University of Pisa, Italy algorithms based on Gaussian elimination for solving large [email protected], [email protected] sparse systems of linear equations, there is still a strong need for analyzing and improvements of the resulting pro- Federico Poloni cedures, in particular, in the field of incomplete decompo- Scuola Normale Superiore sitions. This talk will review some results in this area and [email protected] discuss the possibilities for further developments. A spe- cial attention will be given to the role of the matrix inverse which is implicitly present in the considered algorithms. MS8 Missing Value Estimation Using Matrix Factoriza- Miroslav Tuma tions Academy of Sciences of the Czech Republic Institute of Computer Science Many predictive modeling tasks in data analysis may be [email protected] posed as missing value estimation problems. Recently, such problems have sparked enormous interest in the context of recommender systems, specifically due to the Netflix MS9 million-dollar challenge problem. In this talk, I will sur- A Week in the Life of an Industrial Linear Alge- vey and discuss methods for such missing value estimation braist problems that rely on low-rank matrix factorizations of in- complete data matrices. We will present a survey of all of the linear algebra prob- lems that we have been working on during our years at Inderjit S. Dhillon LSTC including - KKT systems for nonlinear solution for University of Texas at Austin implicit mechanics - methods including low [email protected] rank approximations for subblocks for applications in elec- tromagnetics and acoustics - Iterative methods for CFD and thermal mechanics - Consistent constraint explicit MS8 where we are applying the consistent constraint technol- Krylov Subspace-Based Techniques for Order Re- ogy of implicit to the explicit formulation. duction of Large-Scale RCL Networks Cleve Ashcraft The development of many of the most powerful algorithms Livermore Software Technology in linear algebra was driven by the need to solve challeng- Corporation ing problems arising in new application areas. In this talk, [email protected] we describe how the need for order reduction of large-scale RCL networks arising in VLSI circuit simulation led to the Roger Grimes development of Krylov subspace-based order-reduction of Livermore Software Technology Corporation linear dynamical systems. In particular, we focus on re- [email protected] cent advances in reduction techniques that preserve certain structures of RCL networks. MS9 Roland W. Freund The Role of Numerical Linear Algebra in Large- University of California, Davis Scale Electromagnetic Modeling Department of Mathematics [email protected] We will present a survey of all of the linear algebra prob- lems that we have been working on during our years at LSTC including KKT systems for nonlinear solution for MS8 implicit mechanics, block matrix methods including low Polynomial Approximation Problems and Conver- rank approximations for subblocks for applications in elec- gence of Krylov Subspace Methods tromagnetics and acoustics, iterative methods for CFD and thermal mechanics, and consistent constraint explicit The convergence of Krylov subspace methods for solving where we are applying the consistent constraint technology linear algebraic systems or eigenvalue problems can be de- of implicit to the explicit formulation. scribed or at least bounded by the value of certain polyno- mial approximation problems involving the given matrix. Lie-Quan Lee Unfortunately, such problems are notoriously difficult to SLAC National Accelerator Laboratory analyze, particularly for nonnormal matrices. In this talk [email protected] I will review some of the results in this area, and discuss recent work on the so-called Chebyshev polynomials of ma- trices. MS9 Role of Numerical Linear Algebra in DOE J¨orgLiesen Institute of Mathematics The U.S. Department of Energy is the owner of one of the TU Berlin largest research enterprises in the United States. It has [email protected] more than ten research laboratories scattered throughout the country. These laboratories are well known for their work in physical and biological sciences. Numerical linear algebra plays a crucial role in many of the scientific appli- LA09 Abstracts 61

cations. In this talk the speaker will sample some of these Under the framework of alternative minimization method, applications and highlight the impact of numerical linear we solve each sub-problem and update the regularization algebra. parameter iteratively. Experimental results show that the performance of the proposed method is quite promising. Esmond G. Ng Lawrence Berkeley National Laboratory Haiyong Liao Lawrence Berkeley National Laboratory Hong Kong Baptist University, Hong Kong [email protected] [email protected]

Michael K. Ng MS9 Department of Mathematics, Hong Kong Baptist Linear Algebra Meets the Long Tail University [email protected] In this presentation we discuss large data problems for a marketplace like eBay. eBay is a Long tail market place where buyers and sellers come together to trade for, typ- MS10 ically, one-of-a-kind items. Given this long tail nature of A Fast Algorithm for Updating and Tracking the products for sale building applications like Search, Rec- Dominant Principal Components ommender systems and Classification solutions pose huge challenges. Data dimensions are represented by users, Many important kernel methods in the machine learning keywords, products, items, item properties, or categories. area, i.e., kernel Principal Component Analysis, feature Most of these dimensions extend to tens to hundreds of mil- approximation, denoising, compression, prediction require lions. Matrices representing combinations of these dimen- the computation of the dominant set of eigenvectors of the sions tend to be sparse and are candidates for dimension- symmetrical kernel . Recently, an algorithm ality reduction. Given the scale of the data these compu- for tracking the dominant kernel eigenspace has been pro- tations need to be highly parallelized. Other sets of prob- posed. In this talk,a new algorithm to incrementally com- lems include large volume text data short text (feedback pute the dominant kernel eigenbasis, allowing to track the comments) or long text (product description and product kernel eigenspace dynamically, will be presented. This ap- reviews). We will discuss the problem space and the chal- proach exploits the properties of some structured matrices, lenges and also approaches to address these challenges. like arrowhead, Cauchy and Householder ones. Neel Sundaresan Nicola Mastronardi eBay Research Labs Istituto per le Applicazioni del Calcolo tba National Research Council of Italy, Bari [email protected] MS10 Maximum Entropy Principle for Composition Vec- MS10 tor Method in Phylogeny Tensor Decomposition Methods in Hyperspectral Data Analysis The composition vector (CV) method is an alignment-free method for phylogeny. Since the phylogenetic signals in Nonnegativity constraints on solutions, or approximate so- the biological data are often obscured by noise and bias, lutions, to numerical problems are pervasive throughout denoising is necessary when using the CV method. Re- computational science and engineering [?]. An important cently a number of denoising formulas have been proposed research area in hyperspectral image data analysis is to and found to be successful in phylogenetic analysis of bac- use nonnegativity constraints to identify materials present teria, viruses etc. By using the maximum entropy principle in the object or scene being imaged and to quantify their for denoising and utilizing the structure of the constraint abundance in the mixture. Since spectral unmixing is an matrix to simplify the optimization, we derive several new ill-posed inverse problem, the use of prior information and formulas. With these formulas, we obtain a phylogenetic regularization constraints are essential tools. In this work, tree which identifies correct relationships between differ- we develop analysis methods based on nonnegative ma- ent gerera of Archaea strains in family Halobacteriaceae trix and tensor decompositions that focus on the follow- by using the 16S rRNA dataset; and also a phylogenetic ing goals: material identification, material abundance esti- tree which correctly groups birds and reptiles together, and mation, data compression, segmentation, and compressed then Mammals and Amphibians successively by using the sensing. Applications include satellite and debris tracking 18S rRNA dataset. and analysis for space activities [?]. Raymond Chan Bob Plemmons The Chinese University of Hong Kong Wake Forest University tba Dept. Mathematics, Box 7388 [email protected] MS10 Alternative Minimization Method with Automatic MS11 Selection of Regularization Parameter for Total Full Multigrid and Least Squares for Incompress- Variation Image Restoration ible Resistive Magnetohydrodynamics

Total variation (TV) based image restoration has been in- Magnetohydrodynamics (MHD) is a fluid theory that de- tensively studied recent years for its nice property of edge scribes Plasma Physics by treating the plasma as a fluid of preservation. In this talk, we present an alternative min- charged particles. Hence, the equations that describe the imization method with automatic selection of regulariza- plasma form a nonlinear system that couples Navier-Stokes tion parameter for TV based image restoration problem. 62 LA09 Abstracts

with Maxwells equations. This talk develops a nested- Lawrence Livermore National Laboratory iteration-Newton-FOSLS-AMG approach to solve this type [email protected], [email protected] of system. Most of the work is done on the coarse grid, in- cluding most of the linearizations. We show that at most one Newton step and a few V-cycles are all that is needed MS11 on the finest grid. Here, we describe how the FOSLS Local Fourier Analysis for Systems of PDEs method can be applied to incompressible resistive MHD and how it can be used to solve these MHD problems ef- For many systems of PDEs, multigrid methods are amongst ficiently in a full multigrid approach. An algorithm is de- the most efficient techniques for solving the resulting ma- veloped which uses the a posterior error estimates of the trix systems. This efficiency results from achieving appro- FOSLS formulation to determine how well the system is priate complementarity in the smoothing and coarse-grid being solved and what needs to be done to get the most correction processes. In this talk, we discuss recent work in accuracy per computational cost. In addition, various as- local Fourier analysis of multigrid methods for systems of pects of the algorithm are analyzed, including a timestep- PDEs, including staggered discretizations and overlapping ping analysis to confirm stability of the numerical scheme multiplicative smoothers. The resulting tools are demon- as well as the benefits of an efficiency based adaptive mesh strated for the Stokes, curl-curl, and grad-div systems. refinement method. A 3D steady state and a reduced 2D time-dependent test problem are studied. The latter equa- Scott Maclachlan tions can simulate a large aspect-ratio tokamak. The goal Department of Mathematics is to resolve as much physics from the test problems with Tufts University the least amount of computational work. This talk shows [email protected] that this is achieved in a few dozen work units. (A work unit equals one fine grid residual evaluation.) Kees Oosterlee Delft University of Technology James H. Adler [email protected] University of Colorado at Boulder [email protected] MS12 The Use of Sylvester Equation Solvers for the Com- MS11 putation of Eigenvalues of Integral Operators Algebraic Multigrid for Linear Elasticity Sylvester equations are important in mathematical mod- We are interested in the efficient solution of large systems els of many areas of applied mathematics, such as con- of PDEs arising from elasticity applications. We propose trol theory, stochastic dynamic general equilibrium models, extending the AMG interpolation operator to exactly in- etc. Here we consider its application to the computation terpolate the rotational rigid body modes by adding addi- of spectral elements of integral operators when there are tional degrees of freedom at each node. Our approach is eigenvalues of multiplicity greater than 1. In this case, a an unknown-based approach that builds upon any existing method for the computation of eigenpairs of Fredholm in- AMG interpolation strategy and requires nodal coarsening. tegral operators in a , which combines defect The approach fits easily into the AMG framework and does correction with method, requiring the so- not require any matrix inversions. We demonstrate the ef- lution of an intermediate Sylvester equation, will be used. fectiveness on 2D and 3D elasticity problems. Different solvers for Sylvester equations will be compared. Allison H. Baker Filomena DIAS D’ALMEIDA Center For Applied Scientific Computing Fac. Engenharia Univ. Porto Lawrence Livermore National Laboratory CMUP - Centro Matem´aticaUniv. Porto [email protected] [email protected]

Tzanio V. Kolev, Ulrike Meier Yang Paulo B. VASCONCELOS Center for Applied Scientific Computing Fac. Economia Univ. Porto Lawrence Livermore National Laboratory CMUP - Centro Matem´aticaUniv. Porto [email protected], [email protected] [email protected]

MS11 MS12 AMG for Definite and Semi-definite Maxwell Prob- Triple dqds Algorithm: Eigenvalues of Real Un- lems symmetric Tridiagonals

Recently, there has been a significant interest in auxiliary- The progressive quotient difference algorithm with shifts space methods for linear systems arising in electromagnetic (qds) was presented by Rutishauser as early as 1954. It is diffusion simulations. Based on a novel stable decomposi- equivalent to the shifted LR algorithm written in a special tion of the Nedelec finite element space due to Hiptmair notation for tridiagonal matrices. The much more recent and Xu, we implemented a scalable solver for second order differential qds (dqds) is a sophisticated variant of qds. (semi-)definite Maxwell problems, utilizing internal AMG We will present the 3dqds algorithm which performes im- V-cycles for scalar and vector Poisson-like matrices. In plicitly three dqds steps and such that real arithmetic is this talk we present the linear algebra theory, numerical maintained in the presence of complex eigenvalues. We performance, and discuss some recent developments in the will show some interesting numerical experiments. algorithm. in its implementation. Carla FERREIRA Tzanio V. Kolev, Panayot S. Vassilevski Mathematics Department Center for Applied Scientific Computing University of Minho LA09 Abstracts 63

[email protected] of a square complex matrix. Numerical evidence is pro- vided with famous matrices of the Gallery collection. We Beresford PARLET propose a nonlinear formulation of Schur and Gauss trian- Computer Science Department gularizations in such a way that Newton-like methods can University of California, Berkeley be used to refine the initial approximations to these upper [email protected] triangular forms. Alain LARGILLIER MS12 Laboratoire de Math´ematiques A Framework for Analyzing Nonlinear Eigenprob- Universit´ede Saint-Etienne lems and Parametrized Linear Systems [email protected]

Associated with an n × n matrix polynomial of degree , Mario AHUES P λ λj A P λ x Laboratoire de Math´ematiquesde ( )= j=0 j , are the eigenvalue problem ( ) =0 and the linear system problem P (ω)x = b, where in the l’Universit´ede Saint-Etienne (EA3989) latter case x is to be computed for many values of the pa- [email protected] rameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)z =0orL(ω)z = c that is lin- Muzafar HAMA ear in the parameter λ or ω. This linearization process has Laboratoire de Math´ematiques received much attention in recent years for the eigenvalue Universit´ede Saint-Etienne problem, but it is less well understood for the linear system [email protected] problem. We develop a framework in which more general versions of both problems can be analyzed, based on one- sided factorizations connecting a general nonlinear matrix MS13 function N(λ) to a simpler function M(λ), typically a poly- On the Early History of Matrix Iterations: The nomial of degree 1 or 2. Our analysis relates the solutions Italian Contribution of the original and linearized problems and in the linear sys- tem case indicates how to choose c and recover x from z. In this talk I will review some early work on iterative meth- For the eigenvalue problem this framework includes many ods for solving linear systems by Italian mathematicians special cases studied in the literature, including the vec- during the 1930s, with particular attention to the contri- butions of Lamberto Cesari (1910–1990) and Gianfranco tor spaces of pencils L1(P ) and L2(P ) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of Cimmino (1908–1989). I will also provide some background rational problems. We use the framework to investigate information on Italian applied mathematics and especially the conditioning and stability of the parametrized linear on Mauro Picone’s Istituto Nazionale per le Applicazioni system P (ω)x = b and thereby study the effect of scaling, del Calcolo, where most of this early numerical work took both of the original polynomial and of the pencil L. Our place. Finally, I will illustrate the influence of Cimmino’s results identify situations in which scaling can potentially work on modern and current research. greatly improve the conditioning and stability and our nu- Michele Benzi merical results show that dramatic improvements can be Department˜ofMathematics˜ and Computer Science achieved in practice. Emory University Laurence GRAMMONT [email protected] Universite de Saint-etienne, [email protected] MS13 Cayley, Sylvester, and Early Matrix Theory Francoise Tisseur University of Manchester Last year marked the 150th anniversary of “A Memoir on Department of Mathematics the Theory of Matrices’ by Arthur Cayley (1821–1895)— [email protected] the first paper on matrix algebra. The subject was devel- oped by Cayley and his colleague James Joseph Sylvester Nicholas HIGHAM (1814–1897), who coined the term “matrix’, along with School of Mathematics many others, in particular the Berlin school of Weiertrass, The University of Manchester Kronecker, and Frobenius. I will give an overview of these [email protected] early developments, leading up the time when matrices be- came an accepted tool in applied mathematics. MS12 Nicholas J. Higham Newton-like Methods for Schur and Gauss Trian- University of Manchester gular Forms School of Mathematics [email protected] Ill conditioned eigenproblems may lead to poor numerical approximations of eigenvalues, for example when the data matrix is far from being a normal one or of eigenvectors, for MS13 example when the data matrix has close eigenvalues. How- Hermann Grassmann and the Foundations of Lin- ever, these numerical approximations may serve as initial ear Algebra points of iterative schemes which are able to refine them up to a desired precision. The goal of this paper is to show Hermann Grassmann (1809-1877), one of the mathemat- how the classical Newton-Kantorovich method for differ- ical geniuses of the 19th century, laid the foundations of entiable nonlinear problems and its Fixed Slope variant Linear Algebra with his “Calculus of Extension” of 1844. can be adapted in real arithmetic to refine eigenelements Being notoriously difficult to read and far ahead of its time, 64 LA09 Abstracts

Grassmann’s work was accepted only gradually by other context. mathematicians. On the occasion of Grassmann’s 200th birthday, I will discuss his main mathematical achieve- Heike Fassbender ments as well as some of the influences that led to his TU Braunschweig, Germany creation of Linear Algebra. Institut Computational Mathematics [email protected] J¨orgLiesen Insitute of Mathematics TU Berlin MS14 [email protected] Discrete Empirical Interpolation for Nonlinear Model Reduction

MS13 A dimension reduction method called Discrete Empir- Kron(A,B): A Product of the Times ical Interpolation (DEIM) is proposed and shown to dramatically reduce the computational complexity of A quick study of Zehfuss (1858) and subsequent develop- the Proper Orthogonal Decomposition (POD) method ments suggests that the Kronecker product (KP) should for constructing reduced-order models for unsteady really be referred to as the ”Zehfuss product.” But never and/or parametrized nonlinear partial differential equa- mind, we will fast forward to the 1960s and later and trace tions (PDE). DEIM is a modification of POD that reduces how this all-important matrix operation has worked its way the complexity as well as the dimension of general nonlin- into mainstream numerical linear algebra. Connections ear systems of ordinary differential equations. In particular to fast transforms, PDE discretization, high-dimensional it applies to nonlinear ODE arising from discretization of modeling, and tensor computations will be stressed. Using PDE. history as a guide, we will argue that KP-based problem- solving will become increasing prominent as N approaches Danny C. Sorensen, Saifon Chaturantabut infinity. Rice University [email protected], [email protected] Charles Van Loan Cornell University Department of Computer Science MS14 [email protected] Approaches to Nonlinear Model Order Reduction In the design of electrical circuits, simulation has always MS14 been a crucial factor, e.g., for circuit and system level veri- Lyapunov Equations, Energy Functionals, and fication and for capturing parasitic effects. Shrinking struc- Model Order Reduction ture sizes and increasing packing densities require refined models, causing very high dimensional linear and nonlin- We give an overview on the use of algebraic Gramians in the ear dynamical systems to be treated numerically. Mea- context of model order reduction (MOR). Particularly, we sures have to be taken to enable simulation in adequate emphasize their relation to energy functionals, Lyapunov(- time. Model order reduction for linear and nonlinear prob- type) equations, and balancing transformations. New re- lems is one, industry is interested in. We give an overview sults on using these concepts for balanced truncation MOR of techniques and present questions and approaches from methods for stochastic and bilinear systems will be pre- semiconductor industry. sented. Using Carleman linearization together with bal- anced truncation for bilinear systems, nonlinear reduced- Michael Striebel order models for nonlinear systems can be computed. Nu- Technische Universit¨atChemnitz, Germany merical examples illustrate our results. [email protected]

Peter Benner Joost Rommes TU Chemnitz NXP Semiconductors, The Netherlands Germany [email protected] [email protected]

Tobias Damm MS15 TU Kaiserslautern Nonlinear Eigenvalue Problems in Accelerator Germany Cavity Modeling [email protected] Computational design and optimization of accelerator cav- ities involves solving large-scale numerical eigenvalue prob- MS14 lems. The electromagnetic power loss due to external cou- Model Reduction in Computational Flight Testing pling in the cavities leads to nonlinear terms which depend on the eigenvalues. Due to the complexity of cavity geome- Brute-force use of usual CFD-methods for numerical sim- tries, the problem size of the systems is often very large. ulation in the context of flight testing is not feasible due We will present the formulation and algorithms we em- to large numbers of different simulation settings and high ployed to solve those large-scale nonlinear eigenvalue prob- simulation cost per setting. Models of reduced order for lems from accelerator cavity modeling. the calculation of stationary and instationary aerodynam- ical data of the airplane are desired, which are applicable Lie-Quan Lee for a wide range of flight envelopes. We will report on first SLAC National Accelerator Laboratory steps towards the use of model reduction methods in this [email protected] LA09 Abstracts 65

MS15 Courant Institute, NYU The Computation of Purely Imaginary Eigenvalues USA with Application to the Detection of Hopf Bifurca- [email protected] tions in Large Scale Problems

The computation of purely imaginary eigenvalues with ap- MS16 plication to the detection of Hopf Bifurcations in large scale Minimizing Communication in Linear Algebra problems The determination of Hopf bifurcations of a dy- namical system is often a challenging problem. The goal A lower bound on communication (words of data moved) is to compute the system parameter values for which the needed to perform dense matrix multiplication is known to be Omega(f/sqrt(M)) where f is the number of multiplica- system has purely imaginary eigenvalues. In this talk, we 3 present a method that computes these parameter values tions (n ) and M is the fast memory size. We extend this without computing the imaginary eigenvalues themselves. result to all direct methods of linear algebra (BLAS, LU, The method is based on residual inverse iteration for a QR, eig, svd, etc), to sequential or parallel algorithms, and problem of squared dimension. to dense or sparse matrices. We show large speedups over algorithms in LAPACK and ScaLAPACK. Karl Meerbergen K. U. Leuven James Demmel [email protected] UC Berkeley, USA [email protected]

MS15 Grey Ballard On the Stationary Schr¨odingerEquation Describ- University of California, Berkeley ing Electronic States of Quantum Dots in a Mag- [email protected] netic Field Olga Holtz In this work we derive a pair/quadruple of nonlinear eigen- University of California, Berkeley value problems corresponding to the one–band effective Technische Universitat Berlin Hamiltonian governing the electronic states of a quantum [email protected] dot in a homogenous magnetic field. Exploiting the min- max characterization of its eigenvalues we devise an effi- cient iterative projection methods simultaneously handling Oded Schwartz the pair/quadruple of nonlinear problems and thereby sav- Technische Universitat Berlin ing about 25%/40-50% of the computation time compar- [email protected] ing to the Nonlinear Arnoldi method applied to each of the problems separately. MS16 Heinrich Voss Structured Matrix Computations: Recent Ad- Hamburg Univ of Technology vances and Future Work Institute of Numerical Simulation [email protected] Fast algorithms for strucutred matrices, such as the Toeplitz matrices and Cauchy matrices, have a long and rich history. Recent advances have made many of such al- Marta M. Betcke gorithms both fast and numerically stable. In addition, Univ of Manchester strucutred matrix techniques have been successfully used School of Mathematics to solve new classes of problems that are not structured in [email protected] the classical sense. In this talk, we discuss some recent ad- vances and interesting open questions in structured matrix MS15 computations. A Parallel Additive Schwarz Preconditioned Ming Gu Jacobi-Davidson Algorithm for Polynomial Eigen- University of California, Berkeley value Problems in Quantum Dot Simulation Mathematics Department [email protected] Abstract not available at time of publication.

Weichung Wang MS16 National Taiwan University Department of Mathematics Minimum Sobolev Norm Schemes [email protected] Minimum Sobolev Norm (MSN) schemes are a higher order mesh-free discretization schemes that suppress the Runge MS16 Phenomenon. The main idea is to find interpolatory poly- nomials of order >> N-the number of nodes, with min- Eigenvalue Problems in Sensing and Imaging imum Sobolev Norm. The infinite order case of the in- Nonuniqueness arises from typical sensing and imaging terpolation operator, produces a Golumb Wienberger ker- problems, and can be resolved by seeking sparse solution to nel, that has FMM structure. We present the ideas be- underdetermined linear (integral) equations, usually with hind the MSN scheme, results from convergence, and fast- a Fourier kernel. We connect the task of sparse solution algorithms for their evaluation. to the design of Gaussian quadratures and to a class of Karthik Jayaraman eigenvalue problems with their computational issues. Dept. Elect. & Comp. Engr. Yu Chen U. Cal. Santa Barbara 66 LA09 Abstracts

[email protected] senberg matrix. These laters are solved by using a block updating procedure for the QR decomposition of the Hes- senberg matrix based on Givens rotations. A more effective MS16 alternative was given in which uses the Householder reflec- Fast Inversion of Integral Operators tors. In this paper we will propose a new and a simple implemen- The talk will describe a scheme for rapidly constructing an tation of the block least squares problem. Our approach is approximate inverse to the dense matrices arising from the based on a genralization of Ayachour’s method, given for discretization of 1D integral operators and boundary inte- the GMRES with a single right hand side. Finally we will gral operators in the plane. The scheme is easily derived give some numerical experiments to illustrate the perfor- via hierarchical application of the Woodbury formula for mance of the new implementation. matrix inversion. Its computational cost typically scales linearly with problem size, with a small constant of pro- Hassane Sadok portionality. The use of randomized sampling to facilitate Universit the compression of matrices will be discussed. ’e du Littoral Calais Cedex, France Gunnar Martinsson [email protected] Dept. of Applied Mathematics U. of Colorado, Boulder Lothar Reichel [email protected] Kent State University [email protected] MS17 The Extended Krylov Subspace Method and Or- MS17 thogonal Laurent Polynomials On a Structure-preserving Arnoldi-like Method An orthogonal basis for the extended Krylov subspace K,m A {A−+1v,...,A−1v,v,Av,...,Am−1v} In this talk, an Arnoldi-like method is presented. The ( ) = span method preserves the structures of Hamiltonian or skew- can be generated using short recursion formulas. These Hamiltonian matrices. Different variants are also intro- formulas are derived using properties of Laurent polynomi- duced. The obtained structure-preserving methods are als. The derivation is presented for the general case when m k k needed for reducing the size of large and sparse Hamil- = where is a positive non-zero . The method tonian or skew-Hamiltonian matrices. Numerical experi- is applied to the approximation of matrix functions of the ments are given. form f(A)v. Ahmed Salam Carl Jagels Universite Lille-Nord de France, Calais, France. Hanover College [email protected] [email protected]

MS18 MS17 Recent Developments on the Inertias of Patterns A Newton-Smith-Block Arnoldi Method for Large Scale Algebraic Riccati Equations A number of techniques have been introduced in the liter- ature to show a specific pattern is inertially arbitrary. We In this talk, we present a Newton-Block Arnoldi method for will focus on the problem of finding sets of inertias that solving large continuous algebraic Riccati equations. Such a pattern can attain. To do this, we describe the inertias problems appear in control theory, model reduction, cir- realizable by certain sets of polynomials, the effect the in- cuit simulation and others. At each step of the Newton ertia of a matrix has on its characteristic polynomial, and process, we solve a large Lyapunov matrix equation with provide sufficient conditions for reducible patterns to be a low rank right hand side. These equations are solved by inertially arbitrary. using the block Arnoldi process associated with a precon- ditioner based on the Alternating Direction Implicit (ADI) Michael Cavers iteration method. We give some theoretical results and Dept of Mathematics and Statistics report numerical tests to show the effectiveness of the pro- University of Regina posed approach. [email protected] Khalid Jbilou University Littoral Cote dOpale, Calais, France MS18 [email protected] On Nilpotence Indices of Sign Patterns

We provide classes of n by n sign patterns of nilpotence MS17 indices at least 3, if they are potentially nilpotent. Fur- A New Implementation of the Block GMRES thermore it is shown that if a full sign pattern A of order Method n has nilpotence index k with 2 ≤ k ≤ n − 1, then sign pattern A has nilpotent realizations of nilpotence indices The Block GMRES is a block Krylov solver for solving non k, k +1,...,n. symmetric systems with multiple right hand side. This method is classicaly implemented by first applying the In-Jae Kim Arnoldi iteration as a block orthogonalization process to Minnesota State University, Mankato create a basis of the block Krylov space and then solving a [email protected] block least squares problems, which is equivalent to solv- ing several least squares problems but with the same Hes- LA09 Abstracts 67

MS18 [email protected] Allow Problems Concerning Spectral Properties of Sign Patterns MS19 A sign pattern has entries in {+, −, 0}. For a given prop- Computing the Matrix Sign Function in the Over- erty P, a sign pattern S allows P if there exists at least lap Fermion Model of QCD one real matrix with the sign pattern of S that has prop- erty P. The following problems are surveyed: sign patterns At non-zero chemical potential, the overlap Dirac operator that allow all possible spectra (spectrally arbitrary); sign is obtained as the sign function of a non-hermitian ma- patterns that allow all inertias (inertially arbitrary); and trix. We consider techniques to compute the action of the sign patterns that allow nilpotency (potentially nilpotent sign function on a vector, including the deflation of small patterns). Proof techniques, constructions and examples eigenvalues. One approach uses rational functions and a will be discussed, and open problems identified. multishift version of deflated FOM, the other is based on the Arnoldi-Lanczos recurrence with a new idea to evaluate Dale Olesky the matrix function on the projected system. Department of Computer Science University of Victoria Andreas J. Frommer [email protected] Bergische Universitaet Wuppertal Fachbereich Mathematik und Naturwissenschaften [email protected] MS18 Potentially Nilpotent Zero-Nonzero Patterns over Finite Fields MS19 Title Not Available at Time of Publication The idea of exploring spectrally arbitrary patterns was ini- tially related to inverse eigenvalue and eigenvalue comple- Abstract not available at time of publication. tion problems. For various patterns, one seemingly critical step for determining whether a pattern is spectrally arbi- Scott Maclachian trary is to determine if the pattern is potentially nilpo- Tufts University tent (PN). We introduce an investigation into which zero- [email protected] nonzero patterns are PN over finite fields. Tools from al- gebraic geometry and commutative algebra are developed. MS19 We classify irreducible PN patterns of order three over Zp. Large-scale Eigenvalue Solvers in Computational Materials Kevin N. Vander Meulen Department of Mathematics Eigenvalue problems continue to be a largely unavoidable Redeemer University College bottleneck in materials science applications. We discuss [email protected] current state of the art eigenmethods and revisit Lanc- zos type algorithms which have fallen out of favor because they cannot utilize multiple initial guesses and precondi- Adam Van Tuyl tioning. We show that our recent eigCG technique that Department of Mathematics finds Lanczos quality eigenpairs without keeping the whole Lakehead University Lanczos basis can be used by itself or together with other [email protected] methods such as Jacobi-Davidson in PRIMME to improve eigenvalue convergence. MS19 Andreas Stathopoulos Quantum Monte Carlo Methods: The Use of Lin- College of William & Mary ear Solvers and Evaluating Determinant Ratios Department of Computer Science [email protected] Quantum Monte Carlo is one of the most accurate methods to calculate physical properties of large systems. Simula- tions are currently done on a number of electrons N be- MS20 tween 100 and 1000. These calculations involve evaluating PMOR, an Interpolation Approach a series of determinants of N × N matrices that change (slowly) throughout the simulation. Asymptotically this 3 Recently, the Loewner matrix pencil framework to system evaluation costs O(N ) and is one of the dominating prac- identification from measured data has been developed. In tical bottlenecks in simulating larger systems. In addi- this talk we will propose an extension to this framework tion, functions of these determinants such as gradients and which includes systems depending on parameters. This laplacians must be calculated. We will discuss the nature amounts to treating the transfer function of the system of this problem and explore a series of techniques that are as a rational function in several complex or real variables. designed to decrease the asymptotic cost of these evalua- Reduction of the order of the underlying system follows tions in system of physical interest. naturally, as in the single parameter case.

Bryan Clark Athanasios C. Antoulas Department of Physics Dept. of Elec. and Comp. Eng. University of Illinois at Urbana-Champaign Rice University [email protected] [email protected]

Kapil Ahuja Virginia Tech 68 LA09 Abstracts

MS20 eigenvalue problems. To calculate the eigen- H∈-Optimal Interpolation for Model Reduction of vectors associated to the lowest eigenvalues we propose Parameterized Systems a Subspace Accelerated Inexact Newton (SAIN) method combined with a multigrid preconditioner. This efficient We consider interpolatory model reduction of parameter- iterative solver takes into account the non-linear character ized linear dynamical systems executed in such a way that of the Hamiltonian operator and avoids solving a series of the parametric dependence of the original system is repro- surrogate linearized problems. The method is illustrated duced in the reduced order model. Interpolation is guar- with examples of real applications with hundreds of eigen- anteed both respect to selected complex frequencies and vectors. selected parameter value choices. We develop optimal pa- rameter selection strategies that will produce H∈-optimal Jean-Luc Fattebert reduced order systems that also minimize a least squares Lawrence Livermore National Lab. error measure taken over the parameter range. [email protected] Christopher A. Beattie Virginia Polytechnic Institute and State University MS21 [email protected] Nonlinear Subspace Iteration To compute the electronic structure of materials, the Den- MS20 sity Functional Theory (DFT) technique converts the orig- Interpolation Based Optimal Model Reduction for inal n-particle problem into an effective one-electron sys- LTI-Systems tem, resulting in a coupled one-electron Schr¨odingerequa- tion and a Poisson equation. This coupling is nonlinear and The input-output behaviour of a linear time invariant (LTI) rather complex. A recently proposed method [Y. Zhou et system can be characterized by the transfer function H(s) al., Phys. Rev. E. 74, pp.066704 (2006)] to solve this prob- of the system, a p × m dimensional rational matrix func- lem is to resort to a nonlinear variant of subspace iteration. tion, where m is the number of inputs and p the number The usual (linear) subspace iteration [Bauer, 1960] builds a of outputs of the system. To compute reduced order mod- basis of the subspace associated with the p largest eigenval- els for a large-scale LTI system we may project the state ues of a matrix by a kind of polynomial iteration (filtering) space such that the transfer functions of the reduce order on an initial basis. Convergence is usually enhanced by systems are rational tangential Hermite interpolations of using Chebyshev acceleration. The nonlinear form of this the original transfer function. The interpolation data can approach consists of only updating the Hamiltonian matrix be chosen in such a way that first order necessary condi- at each restart. We will illustrate this approach and show tions for an optimal approximation in the H2,α-, or h2,α- techniques to enhance its robustness by using Homotopy. norm are satisfied, where α is a bound for the poles of the systems. Properties as well as the computation of such re- Youcef Saad duced order models are presented and relations to balanced University of Minnesota truncation methods are discussed. [email protected]

Angelika Bunse-Gerstner, Dorota Kubalinska Universitaet Bremen MS21 Zentrum fuer Technomathematik, Fachbereich 3 Solving Nonlinear Eigenvalue Problems in Elec- [email protected], [email protected] tronic Structure Calculations bremen.de One of the key tasks in electronic structure calculation is to solve a nonlinear eigenvalue in which the matrix Hamil- MS21 tonian depends on the eigenvectors to be computed. This Large Scale Non-linear Eigenvalue Optimization in problem can be solved either as a system of nonlinear equa- Electronic Structure Calculations tions or as a constrained nonlinear optimization problem. We will examine the numerical algorithms used in both We will discuss very large scale non-linear eigenvalue prob- approaches and compare their convergence properties. We lems that arise in Density Functional Theory calculations. will also discuss the use of Broyden updates and trust re- In particular, we are interested in simulations that involve gion techniques in these algorithms. many thousands of atoms and we will illustrate the chal- lenges that arise in extreme scale-out of such problems. We Chao Yang will illustrate the success of powerful non-linear optimiza- Lawrence Berkeley National Lab tion methods and show their scalability to thousands of [email protected] processors. Costas Bekas MS22 IBM Research Use of Semi-Separable Approximate Factorization Zurich Research Laboratory and Direction-Preserving for Constructing Effec- [email protected] tive Preconsitioners For a symmetric positive definite matrix A of size n-by- MS21 n, we developed a fast and backward stable algorithm Subspace Accelerated Inexact Newton Method for to approximate A by a symmetric positive-definite semi- Solving the Kohn-Sham Equations in Large Scale separable Cholesky factorization, accurate to any pre- Condensed Matter Applications. scribed tolerance. Our algorithm ensures that the Schur complements during the Cholesky factorization all remain Discretizing the Kohn-Sham equations of Density Func- positive definite after approximation. In addition, it pre- tional Theory by Finite Differences methods leads to large serves the product, AZ, for a given matrix Z of size n-by-d, LA09 Abstracts 69

where d n. We present numerical results to demonstrate to frequencies or wave numbers. the effectiveness of the preconditioners, and discuss appli- cations in large-scale computations. Jianlin Xia Purdue University Xiaoye S. Li [email protected] Computational Research Division Lawrence Berkeley National Laboratory Ming Gu [email protected] University of California, Berkeley Mathematics Department Ming Gu [email protected] University of California, Berkeley Mathematics Department [email protected] MS22 Butterfly Algorithm and its Applications Panayot Vassilevski Lawrence Livermore National Laboratory We describe the Butterfly algorithm, which is a general ap- [email protected] proach for the rapid evaluation of oscillatory interactions. We also discuss two applications: sparse Fourier transform and discrete Fourier integral operators. In each case, the MS22 butterfly algorithm is combined with special interpolation On the Schur Complements of the Discretized tools to achieve optimal computational complexity. Laplacian Lexing Ying We show that the Schur complements of the matrix ob- University of Texas tained by finite difference discretization of the Laplacian Dept. of Mathematics on the unit square with Dirichlet and Neumann boundary [email protected] conditions exhibit off-diagonal blocks with low rank. This is also true of the finite difference matrix in three dimen- MS23 sions under suitable reordering of the grid. This in turn implies that fast direct solvers could be used to efficiently FORTRAN 77 Implementations of Superfast solve these problems. Solvers Naveen Somasunderam We will describe the current status of FORTRAN 77 im- Department of Electrical Engineering plementations of superfast algorithms for solving positive definite Toeplitz systems of equations. The algorithms in- University of California, Santa Barbara O n 2 n [email protected] clude the original ( log ) method described in [SIMAX 9(1988):61–76] as well as the O(n log3 n) procedure of Stew- art [SIMAX 25(2003):669–693], which has improved stabil- Shiv Chandrasekaran ity properties. U.C. Santa Barbara [email protected] Greg Ammar Northen Illinois University Ming Gu [email protected] University of California, Berkeley Mathematics Department [email protected] MS23 An Implicit Multishift QR-algorithm for Symmet- Patrick DeWilde ric Plus Low Rank Matrices TU Delft, Netherlands [email protected] We present an efficient computation of all the eigenvalues of the Hessenberg form of a hermitian matrix perturbed by a possibly non-symmetric low-rank correction. We propose a MS22 representation based on Givens transformations that allows to perform steps of the implicit multishift QR algorithm us- Robust and Efficient Factorization and Precondi- O \ tioning for Sparse Matrices ing ( ) operations per iteration. A deflation technique is proposed as well. The representation by means of Givens We present a robust approximate factorization and precon- rotations, makes the algorithm numerically stable as con- ditioning method for sparse matrices such as those arising firmed by the extensive numerical tests. from the discretization of some PDEs. Sparse matrix tech- inques are combined with structured factorizations of dense Gianna M. Del Corso matrices. Structural compressions of dense matrix blocks University of Pisa, Department of Computer Science lead to high efficiency. Robustness enhancement is auto- [email protected] matically integrated into the factorization without extra cost. The factor can work as an effective preconditioner Raf Vandebril without strict structural requirement. The cost and storage Katholieke Universiteit Leuven for such a factorization are roughly linear in the matrix size. Department of Computer Science The efficiency, robustness, and effectiveness are discussed [email protected] and demonstrated with examples such as Helmholtz, linear elasticity, and Maxwell equations. Numerical experiments show that this preconditioner is also relatively insensitive MS23 Structured Regularization Operators for 70 LA09 Abstracts

Tikhonov’s Method MS24 Acyclic and Unicyclic Graphs Whose Minimum We consider the solution of linear discrete ill-posed prob- Skew Rank is Equal to the Minimum Skew Rank lems with the aid of Tikhonov regularization. The regu- of a Diametrical Path larized problem is said to be in standard form if the reg- ularization operator is the identity. Tikhonov regulariza- The minimum skew rank of a simple graph G over the field tion problems in standard form are easier to solve than of real numbers is the smallest possible rank among all regularization problems in general form. We discuss the skew-symmetric real matrices whose (i, j)-entry (for i = construction of structured square regularization operators j) is nonzero whenever {i, j} is an edge in G and is zero that allow easy transformation of the regularized problem otherwise. In this paper we give necessary and sufficient to standard form. This talk presents joint work with Qiang conditions for a connected acyclic graph to have minimum Ye. skew rank equal to the minimum skew rank of a diametrical path. We also characterize connected unicyclic graphs with Lothar Reichel this property. Kent State University [email protected] Luz M. DeAlba Drake University Department of Mathematics MS23 [email protected] On Tridiagonal Matrices Unitarily Equivalent to Normal Matrices MS24 In this talk the unitary equivalence transformation of nor- Minimum Rank Problems mal matrices to tridiagonal form is studied. It is well- known that any matrix is unitarily equivalent to a tridi- A graph (directed or undirected, simple or allowing loops) agonal matrix. In case of a normal matrix the resulting can be used to describe a pattern of nonzero entries of a tridiagonal inherits a strong relation between its super- matrix (over the real numbers, or more generally over a and subdiagonal elements. The corresponding elements of field). A minimum rank problem is a problem of determin- the super- and subdiagonal will have the same absolute ing the minimum of the ranks of the matrices described value. In this talk some basic facts about a unitary equiva- by a graph, perhaps requiring additional matrix properties lence transformation of an arbitrary matrix to tridiagonal such as symmetry or positive semidefiniteness. This talk form are firstly studied. Both an iterative manner based will survey recent developments in minimum rank prob- on Krylov sequences as a direct manner via Householder lems and applications, and serve as an introduction to the transformations are deduced. This equivalence transforma- mini-symposium. tion is then reconsidered for the normal case and equality of the absolute value between the super- and subdiago- Leslie Hogben nals is proved. Self-adjointness of the resulting tridiago- Iowa State University nal matrix with regard to a specific scalar product will be [email protected] proved. Flexibility in the reduction will then be exploited and properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary MS24 matrices and their relations with, e.g., complex-symmetric Diagonal Rank and Skew Symmetric Matrices and pseudo-symmetric matrices are presented. It will be shown that the reduction can then be used to compute the A beautiful result of Fiedler’s is that if A is an n × n sym- singular value decomposition of normal matrices making metric matrix such that the rank of D + A is at least n − 1 use of the Takagi factorization. Finally some extra prop- for each D, then A is permutationally sim- erties of the reduction as well as an efficient method for ilar to an irreducible, tridiagonal matrix. In this talk, we computing a unitary complex symmetric decomposition of discuss the analogous problem for skew-symmetric matri- a normal matrix are given. ces. Raf Vandebril Bryan L. Shader, Reshmi Nair, Colin Garnett, Mary Katholieke Universiteit Leuven Allison Department of Computer Science Department of Mathematics [email protected] University of Wyoming [email protected], [email protected], [email protected], [email protected] MS24 The Minimum Rank Problem for Planar Graphs MS25 The minimum rank problem for a given simple, undirected Application of the IDR Theorem to Stationary It- graph G is to determine the minimum rank (or maximum erative Methods and their Performance Evaluation nullity ) over all symmetric matrices whose off-diagonal nonzero pattern corresponds to G. Two graph parame- The appearance of IDR(s) methods based on the IDR The- ters, the path cover number and zero forcing number, have orem strongly influenced some researchers in Krylov sub- played an important role in the determination of the maxi- space (KS) methods. Like some KS methods, classical mum nullity for trees, unicyclic graphs, outerplanar graphs, stationary iterative (SI) methods, e.g, Gauss-Seidel and and rectangular grids. This talk will survey several known SOR, may be seen in a new light too. This change of view results and report on recent progress. builds on the relation between the conventional residual recursion rk+1 = Brk of the SI method and the recursion Wayne Barrett rk+1 = B(rk + γk(rk − rk−1)) of the IDR(s) method. Here Brigham Young University [email protected] LA09 Abstracts 71

B is the iteration matrix, and γk is a scalar. tablished technique for uncertainty quantification (UQ). Its implementation involves a number of challenging linear al- Seiji Fujino gebra tasks, among these the solution of very large linear Computing and Communications Center systems of equations, long sequences of closely related lin- Kyushu University ear and nonlinear systems of equations and large dense [email protected] eigenvalue problems. This talk will lead into the minisym- posium by describing the basic UQ problem setting, intro- ducing the key discretization techniques and also present MS25 some of our own recent work in addressing these tasks. IDR — A Brief Introduction Oliver G. Ernst The Induced Dimension Reduction (IDR) method intro- TU Bergakademie Freiberg duced first by Sonneveld in 1979 has been reconsidered and Fakultaet Mathematik und Informatik generalized by Sonneveld and van Gijzen recently. While [email protected] the original IDR became the forerunner of Bi-CGSTAB, the new IDR(s) is related to the ML(k)BiCGSTAB method of Yeung and Chan and to the nonsymmetric band Lanczos MS26 algorithm. It proved amazingly effective and offers some Parallelization and Preconditioning of Iterative flexibility. This talk intends to give a brief overview of Solvers for Linear SFEM Systems these approaches and their connections. We develop iterative solvers for the solution of linear sys- Martin H. Gutknecht tems stemming from a stochastic Galerkin projection of Seminar for Applied Mathematics stochastic partial differential equations. We demonstrate ETH Zurich the parallelism potential of these solvers and implement [email protected] them with the Trilinos high performance computing en- vironment. We show how these algorithms can mitigate the curse of dimensionality while being very efficient and MS25 attuned to high performance resources. These solvers are ML(n)BiCGStab: Reformulation, Analysis and Im- variations on solvers previously developed by one of the au- plementation thors which have been adapted to current computational resources. With the help of index functions, we rederive the ML(n)BiCGStab algorithm in a more systematic way. Roger Ghanem, Ramakrishna Tipireddy There are n ways to define the ML(n)BiCGStab residual University of Southern California vector. Each different definition will lead to a different Aerospace and Mechanical Engineering and Civil ML(n)BiCGStab algorithm. We demonstrate this by de- Engineering riving a second algorithm which requires less storage. We [email protected], [email protected] also analyze the breakdown situations. Implementation is- sues are also addressed. We discuss the choices of the pa- Maarten Arnst rameters in ML(n)BiCGStab and their effects on the per- University of Southern California formance. [email protected] Man-Chung Yeung University of Wyoming MS26 [email protected] Iterative Solvers for Stochastic Galerkin Finite El- ement Discretizations MS25 A linear stochastic partial differential equation can be con- An IDR-variant that Efficiently Exploits Bi- verted by the stochastic Galerkin method into a high di- orthogonality Relations mensional algebraic system. Such system, characterized by The IDR theorem provides a way to construct a sequence of a tensor product structure, can be solved efficiently by a nested subspaces, of which the smallest subspace is zero di- multigrid solver. This yields a convergence rate indepen- mensional. IDR-based algorithms construct residuals that dent of the discretization parameters. Non-optimal meth- are in these nested subspaces. The translation from theo- ods, based on block splitting approaches or on a hierarchy rem to algorithm is not unique. In the talk we will present in stochastic space, may require however less computing a variant of IDR(s) that imposes bi-orthogonalization con- time. A comparison of several solvers will be presented. ditions on the iteration vectors. The resulting algorithm is Eveline Rosseel accurate and efficient. Dept. Computer Science Martin B. van Gijzen, Peter Sonneveld K.U. Leuven Delft Inst. of Appl. Mathematics [email protected] Delft University of Technology [email protected], [email protected] Stefan Vandewalle Scientific Computing Research Group KULeuven MS26 [email protected] Linear Algebra and Stochastic Finite Element Computations MS26 Stochastic Finite Element methods for solving partial dif- Iterative Solvers in Stochastic Galerkin Finite El- ferential equations with random data have become an es- ement Computations: Preconditioners and Multi- 72 LA09 Abstracts

level Techniques Chris Siefert Sandia National Laboratories The discretization of linear stochastic partial differential [email protected] equations (SPDEs) by means of the stochastic Galerkin fi- nite element method [1] results in general in a very large and coupled but structured linear system of equations. MS27 Our task is the design of efficient iterative solvers for Smoothed Aggregation AMG Solvers for Least- these Galerkin equations. The stochastic diffusion equa- Squares Finite Element Systems tion serves as our model problem. For the primal formu- lation of this SPDE we review the recently proposed Kro- Least-squares finite element methods are an attractive class necker product preconditioner [2] for the Galerkin matrix of methods for the numerical solution of certain div-curl which - in contrast to the popular mean-based precondi- systems. We consider a discretization where we approxi- tioner - makes use of the entire information contained in the mate H(curl, Ω) ∪ H(div, Ω) conformally with respect to Galerkin matrix. Furthermore, we extend the idea of Kro- the divergence operator (e.g. using face elements) and use necker product preconditioning to the discretized mixed a discrete approximation of the curl operator. We derive a formulation of the stochastic diffusion equation and present multigrid method from the previous work of Bochev, Hu, numerical results of test problems where the stochastic dif- Siefert, Tuminaro, Xu and Zhu (2007), using specialized fusion coefficient is a lognormal random field. Finally we prolongators to transform the system into a nodal Laplace- report on our attempts to utilize multilevel techniques for like problem. the Galerkin equations. In contrast to previous works, where many researchers have applied multilevel methods Chris Siefert exclusively to the deterministic finite element spaces, we Sandia National Laboratories focus on multilevel decompositions for the stochastic vari- [email protected] ational space and combined deterministic/stochastic mul- tilevel approaches for the global Galerkin system. Pavel Bochev Sandia National Laboratories Elisabeth Ullmann Computational Math and Algorithms TU Bergakademie Freiberg [email protected] [email protected] Kara Peterson MS27 Sandia National Laboratories [email protected] Adaptive Least Squares Algebraic Multigrid for Complex-valued Systems MS27 We present an algebraic multilevel method for solving HPD linear systems. The proposed approach adaptively com- A Generalization of Energy Minimization Ideas to putes a set of testfunctions and then uses them in a least Algebraic Multigrid for Nonsymmetric Systems squares fit to define interpolation. In this way, the algo- We consider the generalization of energy minimization al- rithm iteratively improves the multilevel hierarchy to bet- gebraic multigrid concepts to nonsymmetric systems. A ter approximate the lowest modes of the MG relaxation new method is proposed where construction of prolonga- (solver) until suitable performance is achieved. We test tion (and restriction) is based on a few iterations of GM- the proposed approach for a variety of applications in both RES. These iterations essentially minimize the sum of the the Classical AMG and Smoothed Aggregation settings, AT A norm of the grid transfer basis functions. This mini- showing promising results. mization occurs within some Krylov space while satisfying James Brannick a set of constraints. Numerical comparison are given on a Department of Mathematics set of highly convective nonsymmetric problems. Pennsylvania State University Ray S. Tuminaro [email protected] Sandia National Laboratories Computational Mathematics and Algorithms MS27 [email protected] Smoothed Aggregation Multigrid on Multicore Ar- chitectures MS28 Multicore computer architectures present new challenges Teaching Introductory Linear Algebra Incorporat- to linear solvers, including algebraic multigrid multigrid. ing MATLAB The advent of many cores has effectively increased the im- The first wave of technology supporting mathematics in- balance between node compute power and off-node com- struction surged during the 90s. We took the opportunity munication throughput. We report our experiences using to offer a laboratory experience to accompany our general nonstandard smoothed aggregation multigrid on multicore introductory linear algebra course. A fundamental objec- architectures to address the computation/communication tive was to support linear algebra concepts with graphics imbalance. We consider multigrid variants with domain for visualization, design experiments to engage students on decomposition smoothing, where domains are allowed to a variety of levels, and explore connections. A discussion of cover more than one core in a compute node and present tactics, examples, student reactions, changes encountered, parallel numerical results. and challenges involved with the current second wave of Jonathan J. Hu technologies that are emerging will be included. Sandia National Laboratories David Hill Livermore, CA 94551 Temple University [email protected] LA09 Abstracts 73

[email protected] MS29 Interpolation with Semi-separable Matrices: the Ratio-of-Polynomials Case MS28 A Second Course in Linear Algebra for Science and An interesting problem connected to semi-separable ma- Engineering trices and operators is the issue of interpolation. Given a general (upper) matrix or operator, one may wish to This talk will describe a second course in linear algebra that find reduced approximate representations for it of the semi- has evolved over the past ten years. The course is aimed at separable type. This issue leads to all sort of generaliza- students in engineering and in the physical and computer tions of classical interpolation problems, with very similar sciences. Main new topics: (1) geometry in vector spaces, approximation properties. An interesting case is when the (2) matrix games and linear programming, and (3) finite- representation is of the type ’ratio of generalized polyno- state Markov chains. Enrollment has grown over the past mials’. In the generalized case the polynomials are series of five years and now ranges around 200 students annually. shifted , their ratio are essentially ratios of (non- uniformly) banded matrices. The theory provides for a David Lay generalization to matrices of the classical L¨ownerinter- University of Maryland polation theory as originally developed by Antoulas, Ball, [email protected] Kang and Willems, has interesting connections with con- trol theory and is also computationally attractive. MS28 Patrick Dewilde When MATLAB Gives ”Wrong” Answers; How to Institute for Advanced Study of TUM Get Students in Numerical Linear Algebra Classes tba Excited About Finite-precision Computations

One of the main differences between teaching the standard MS29 linear algebra course and a course in numerical linear alge- Structured Matrix Methods for the Construction bra is that in the latter course all computations are done of Interpolatory Subdivision Masks using finite precision arithmetic. One way to illustrate the importance of this difference is to look at examples where In this talk we present a general approach for the construc- computational software packages such as MATLAB appear tion of interpolatory subdivision masks based on polyno- to be giving wrong answers. In this talk we examine four mials and structured matrix computations. This is a joint or five such scenarios. In each case we look at examples work with Costanza Conti from the University of Florence and explain how and why MATLAB arrives at its answers. and Lucia Romani from the University of Milano-Bicocca. In our final example we examine a MATLAB program that clearly produces an impossible answer. In this case, when the author of the program tried to debug it by printing out Luca Gemignani intermediate results, the value of the computed solution Univeristy of Pisa (Italy) changed. What is going on? Is MATLAB exhibiting some Department of Mathematics sort of Heisenberg effect? All will be explained at the talk. [email protected]

Steven J. Leon MS29 University of Massachusetts Dartmouth Efficient Computations with Tensor Structured [email protected] Matrices

In this talk we present a new method for basic matrix op- MS28 erations with multilevel tensor structured matrices, It is Key Points in a Linear Algebra Course based on a special low-parametric and stable representa- tion for tensors and keeps the complexity linear in the I will discuss my experience in teaching and writing about number of levels. Some of these results are presented in linear algebra. I will focus on three points in the course, the following papers: one at the start and one at the heart and one at the very end (often barely reached): I.V.Oseledets and E.E.Tyrtyshnikov, Breaking the 1. I now give an early lecture to point the way – matrix curse od dimensionality, or how to use SVD multiplication and words like independence are introduced in many dimensions, Research Report 09-03, by specific examples. I believe we learn language this way, Kowloon Tong, Hong Kong: ICM HKBU, 2009 by hearing words, making mistakes, and eventually getting (www.math.hkbu.edu.hk/ICM/pdf/09-03.pdf). them right. I.V.Oseledets, 2. The ”Four Fundamental Subspaces” give a structure D.V.Savostyanov, and E.E.Tyrtyshnikov, Linear al- to the subject. Students catch on to the big picture – gebra for tensor problems, Research Report 08-02, dimensions and orthogonality. Kowloon Tong, Hong Kong: ICM HKBU, 2008 3. A highlight at the end is the SVD. How to introduce (www.math.hkbu.edu.hk/ICM/pdf/08-02.pdf). this idea? It depends on A(AA)=(AA)A. It involves positive definiteness. And it gives perfect bases for the Eugene Tyrtyshnikov four subspaces. Institute of Numerical Mathematics, Russian Academy of Sci. Gil Strang [email protected] MIT [email protected] MS29 Multivariable Orthogonal Polynomials and Struc- 74 LA09 Abstracts

tured Matrix Computations University of Waterloo Dept of Combinatorics & Optimization The recurrence coefficients of polynomials in one variable [email protected] with respect to a discrete inner product can be computed by solving a structured inverse eigenvalue problem. In this talk we will investigate how this inverse eigenvalue problem MS30 is modified in case of multivariable orthogonal polynomials. Explicit Sensor Network Localiza- We will also indicate how this inverse eigenvalue problem tion using Semidefinite Programming and Clique can be solved only using orthogonal similarity transforma- Reductions tions. Some numerical experiments will show the validity of this approach. The sensor network localization (SNL) problem in embed- ding dimension r, consists of locating the positions of ad Marc Van Barel hoc wireless sensors, given only the distances between sen- Katholieke Universiteit Leuven sors that are within radio range and the positions of a Department of Computer Science subset of the sensors (called anchors). There are advan- [email protected] tages for formulating this problem as a Euclidean completion (EDMC) problem, and ignoring the dis- tinction between anchors and sensors. Current solution MS30 techniques relax this problem to a weighted, nearest, (posi- Low-rank Matrix Recovery From Noisy Observa- tive) semidefinite programming, SDP, completion problem, tions Using Nuclear Norm Minimization by using the linear mapping between EDMs and SDP ma- trices. The relaxation consists in ignoring the rank r for the The field of Compressed Sensing has demonstrated that SDP matrices. The resulting SDP is solved using primal- sparse signals can be recovered robustly from incomplete dual interior point solvers, yielding an expensive and inex- sets of measurements. It is natural to ask what other ob- act solution. Moreover, the relaxation is ill-conditioned in jects or structures might also be recovered from incom- two ways. First, it is implicitly highly degenerate in the plete, limited, and noisy measurements. It has been shown sense that the feasible set is restricted to a low dimensional recently that a similar framework can be developed for re- face of the SDP cone. This means that the Slater constraint covering low-rank matrices from limited observations, by qualification fails. Second, nonuniqueness of the optimal minimizing the nuclear norm of the matrix. In this talk, solution results in large sensitivity to small perturbations we extend this framework to include cases where the obser- in the data. The degeneracy in the SDP arises from cliques vations are noisy and the matrix is not perfectly low-rank, in the graph of the SNL problem. These cliques implicitly and provide bounds on the recovery error for several classes restrict the dimension of the face containing the feasible of measurements. Furthermore, we discuss conditions un- SDP matrices. In this paper, we take advantage of the der which the matrix recovered from noisy measurements absence of the Slater constraint qualification and derive a will have the correct rank and singular spaces. technique for the SNL problem, with exact data, that ex- Maryam Fazel plicitly solves the corresponding rank restricted SDP prob- University of Washington lem. No SDP solvers are used. We are able to efficiently Electrical Engineering solve this NP-hard problem with high probability, by find- [email protected] ing a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly find- MS30 ing the intersection of subspaces that represent the faces Nuclear Norm Minimization for the Maximum of the SDP cone that correspond to cliques of the SNL Clique and Biclique Problems problem.

We consider the problems of finding a maximum clique Nathan Krislock in a graph and finding a maximum-edge biclique in a bi- University of Waterloo partite graph. Both problems are NP-hard. We write both [email protected] problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be Henry Wolkowicz regarded as a generalization of compressive sensing, has University of Waterloo recently been shown to be an effective way to solve rank Dept of Combinatorics and Optimization optimization problems. In the special cases that the input [email protected] graph has a planted clique or biclique (i.e., a single large clique or biclique plus diversionary edges), our algorithm successfully provides an exact solution to the original in- MS30 stance. For each problem, we provide two analyses of when Fast and Near–Optimal Matrix Completion via our algorithm succeeds. In the first analysis, the diversion- Randomized Basis Pursuit ary edges are placed by an adversary. In the second, they are placed at random. In the case of random edges for the Motivated by the philosophy and phenomenal success of planted clique problem, we obtain the same bound as Alon, compressed sensing, the problem of reconstructing a ma- Krivelevich and Sudakov as well as Feige and Krauthgamer, trix from a sampling of its entries has attracted much at- but we use di erent techniques. tention recently. Such a problem can be viewed as an information–theoretic variant of the well–studied matrix Brendan Ames completion problem, and the main objective is to design University of Waterloo an efficient algorithm that can reconstruct a matrix by in- [email protected] specting only a small number of its entries. Although this is an impossible task in general, it has heen recently shown r Stephen A. Vavasis that under a so–called incoherence assumption, a rank n × n matrix can be reconstructed using semidefinite pro- LA09 Abstracts 75

gramming (SDP) after one inspects O(nr log6 n) of its en- IDRstab can outperform both IDR(s) and BiCGstab( ). tries. In this paper we propose an alternative approach that is much more efficient and can reconstruct a larger Gerard L. Sleijpen class of matrices by inspecting a significantly smaller num- Mathematical Institute ber of the entries. Specifically, we first introduce a class of Utrecht University so–called stable matrices and show that it includes all those [email protected] that satisfy the incoherence assumption. Then, we propose a randomized basis pursuit (RBP) algorithm and show that Martin B. van Gijzen it can reconstruct a stable rank rn×n matrix after inspect- Delft University of Technology ing O(nr log n) of its entries. Our sampling bound is only Delft, The Netherlands a logarithmic factor away from the information–theoretic [email protected] limit and is essentially optimal. Moreover, the runtime of the RBP algorithm is bounded by O(nr2 log n+n2r), which compares very favorably with the Ω(n4r2 log12 n) runtime MS31 of the SDP–based algorithm. Perhaps more importantly, On the Convergence Behaviour of IDR(s) our algorithm will provide an exact reconstruction of the input matrix in polynomial time. By contrast, the SDP– During the development of the prototype IDR(s) method based algorithm can only provide an approximate one in [Sonneveld and van Gijzen, SIAM J. Sci. Comput. 31, polynomial time. 1035-1062 (2008)], several ”intelligent” choices for the s auxiliary vectors were tried, with poor results. Only a Zhisu Zhu random choice for these vectors appears to be suitable. Stanford University In numerical experiments the convergence plots show, for [email protected] increasing s, a kind of ”convergence” to full GMRES. A statistical explanation of this behaviour will be given in Anthony Man-Cho So relation to the random choice of the auxiliary vectors. The Chinese University of Hong Kong, Hong Kong [email protected] Peter Sonneveld Delft Inst. of Appl. Mathematics Delft University of Technology Yinyu Ye [email protected] Stanford University [email protected] MS31 MS31 Eigenvalue Perspectives of the IDR Family BiCGStab2 and GPBiCG Variants of the IDR(s) We investigate the natural correspondence between Krylov Method methods for the approximate solution of linear systems of equations and the approximation of eigenvalues tailored to The the IDR family. This is work in progress. hybrid BiCG methods such as BiCGSTAB, BiCGStab2, GPBiCG, and BiCGstab( ) are well-known efficient solvers Jens-Peter M. Zemke for linear system with nonsymmetric matrices. Recently, Inst. of Numerical Simulation IDR(s) has been proposed, and it has been reported to be TU Hamburg-Harburg more effective than the hybrid BiCG methods. Moreover, [email protected] IDR(s) variants incorporating BiCGstab( ) strategies have been designed, which converge even faster than the orig- s s Martin Gutknecht inal IDR( ) method. Here, we propose IDR( ) variants Seminar for Applied Mathematics incorporating strategies of BiCGStab2 and GPBiCG. ETH Zurich Kuniyoshi Abe [email protected] Faculty of Economics and Information Gifu Shotoku University MS32 [email protected] Model Reduction of Switched Dynamical Systems

Gerard L.G. Sleijpen A hybrid dynamical system is a system described by Mathematical Institute both differential equations (continuous flows) and differ- Utrecht University ence equations (discrete transitions). It has the benefit of [email protected] allowing more flexible modeling of dynamic phenomena, in- cluding physical systems with impact such as the bouncing ball, switched systems such as the thermostat, and even MS31 the internet congestion as examples. Hybrid dynamical Exploiting BiCGstab( ) Strategies to Induce Di- systems pose a challenge since almost all reduction meth- mension Reduction ods cannot be directly applied. Here we show some recent s developments in the area of model reduction of switched IDR( ) and BiCGstab( ) are among the most efficient iter- dynamical systems. ative methods for solving nonsymmetric linear systems of equations. Here, we derive a new method called IDRstab Younes Chahlaoui combining the strengths of IDR(s) and BiCGstab( ). To CICADA derive IDRstab we extend previous work, where we con- The University of Manchester, UK sidered Bi-CGSTAB as an IDR method. We analyze the [email protected] structure of IDR in detail and introduce the new concept of a Sonneveld space. Numerical experiments show that 76 LA09 Abstracts

MS32 Markus Stammberger A Structure-Preserving Method for Positive Real- Hamburg Univ of Technology ness Problem Inst. of Numerical Simulation [email protected] In this talk the positive realness problem in circuit and control theory is studied. A numerical method is developed for verifying the positive realness of a given proper ratio- MS33 nal matrix H(s) for which H(s)+HT (−s) may have purely Scalable Tensor Factorizations with Missing Data imaginary zeros. The proposed method is only based on orthogonal transformations, it is structure-preserving and Handling missing data is a major challenge in many dis- has a complexity which is cubic in the state dimension of ciplines. Missing data problem has been addressed in H(s). Some examples are given to illustrate the perfor- the context of matrix factorizations and these approaches mance of the proposed method. have been extended to tensor factorizations. We are inter- ested in fitting tensor models, in particular, the CANDE- Delin Chu COMP/PARAFAC (CP) model to data with missing en- Department of Mathematics, National University of tries. We propose the use of a gradient-based optimization Singapore algorithm (CP-WOPT) in order to have methods scalable 2 Science Drive 2, Singapore 117543 to large tensors. Using numerical experiments we show [email protected] that the algorithm we propose is accurate and faster than an alternative technique based on nonlinear least squares Peter Benner approach. TU Chemnitz Germany Evrim Acar, Tamara G. Kolda, Daniel M. Dunlavy [email protected] Sandia National Laboratories [email protected], [email protected], [email protected] Xinmin Liu Department of Electrical and Computer Engineering University of Virginia,USA MS33 [email protected] Scalable Implicit Tensor Approximation

We describe some algorithms for producing a best multi- MS32 linear rank r approximation of a symmetric tensor, useful Thick-Restart Krylov Subspace Techniques for Or- for working with higher moments and cumulants of a large der Reduction of Large-Scale Linear Dynamical p-dimensional random vector. Since the approximated d- Systems way tensor has pd entries, for large p it is critical never to explicitly represent the entire tensor. Working implicitly In recent years, Krylov subspace techniques have proven and approximately where appropriate, this approach en- to be powerful tools for order reduction of large-scale lin- ables the use of higher order factor models on large-scale ear dynamical systems. The most widely-used algorithms data. employ explicit projection of the data matrices of the dy- namical systems, using orthogonal bases of the Krylov sub- Jason Morton spaces. For truly large-scale systems, the generation and Stanford University storage of such bases becomes prohibitive. In this talk, we [email protected] explore the use of thick-restart Krylov subspace techniques to reduce the computational costs of explicit projection. MS33 Roland W. Freund Decomposing a Third-Order Tensor in Rank- University of California, Davis (L,L,1) Terms by Means of Simultaneous Matrix Department of Mathematics Diagonalization [email protected] The decompositions of a third-order tensor in block compo- nents are new multilinear algebra tools, that generalize the MS32 Parallel Factor (PARAFAC) decomposition. We focus on AMLS for an Unsymmetric Eigenproblem Govern- one of these recently introduced decompositions, the Block- ing Free Vibrations of Fluid-Solid Structures Component-Decomposition in rank-(L,L,1) terms, referred to as BCD-(L,L,1). In previous work, we have proposed an Automated Multi-Level Substructuring (AMLS) has been Alternating Least Squares (ALS) algorithm for the com- developed to reduce the computational demands of fre- putation of the BCD-(L,L,1) and a sufficient bound for quency response analysis and is very efficient for huge sym- which uniqueness of this decomposition is guaranteed. In metric and definite eigenvalue problems. This contribution this talk, we show that the BCD-(L,L,1) can be equiva- is concerned with an adapted version of AMLS for an un- lently reformulated as a problem of Simultaneous Diago- symmetric eigenvalue problem governing free vibrations of nalization (SD) of a set of matrices. This reformulation fluid solid structures. Although we take advantage of an has two major advantages. First, the resulting SD-based equivalent Hermitian eigenproblem of doubled dimension algorithm reveals to be more accurate and less sensitive our method needs essentially the same computational work to ill-conditioned data than the standard ALS algorithm. as the original AMLS algorithm. Second, the SD-based reformulation of the BCD-(L,L,1) in- Heinrich Voss volves itself a new sufficient uniqueness bound. The latter Hamburg Univ of Technology bound is more relaxed than the one previously derived, in Institute of Numerical Simulation the sense that the number of rank-(L,L,1) terms that can [email protected] be extracted from the tensor is significantly greater, under LA09 Abstracts 77

a few conditions on the dimensions. ear solver will be investigated on large 2D and 3D models. Dimitri Nion K.U. Leuven Azzam Haidar [email protected] INPT-ENSEEIHT-IRIT, France [email protected] Lieven De Lathauwer Katholieke Universiteit Leuven Luc Giraud [email protected] INRIA Toulouse, France [email protected] MS33 On the Convergence of Lower Rank Approxima- H. Benhadjali, S. Operto tions to Super-symmetric Tensors UMR G´eosciencesAzur [email protected], [email protected] We consider a type of rank-r approximation to super- vlfr.fr symmetric tensors of even order (of size N × N ×···×N), in which a set of r orthonormal basis vectors and an r × r × ··· × r core tensor are sought (with r

Matthias Bollhoefer Yogi Erlangga TU Braunschweig Dept. of Earch Sciences [email protected] Univ. of British Columbia [email protected] MS34 Parallel Algebraic Hybrid Linear Solver for Fre- Kees Vuik quency Domain Acoustic Wave Modeling T. U. Delft [email protected] In this talk, we will present a parallel algebraic non- overlapping domain decomposition methods for the so- lution of the Helmholtz equations involved in frequency- MS35 domain full-waveform inversion. The preconditioning ap- Structures and Dimension of Linearizations of Sin- proach is based on the shift Laplacian technique. The nu- gular Matrix Polynomials merical behaviour and the parallel performance of the lin- Recently, the authors have found strong linearizations 78 LA09 Abstracts

which are valid for both regular and singular square matrix MS35 polynomials and that allow us to recover the whole eigen- Smith Forms of Structured Matrix Polynomials structure of the polynomial. Nonetheless, the problem of finding structured linearizations of structured singular ma- A much-used computational approach to polynomial eigen- trix polynomials remains open. In this talk, we study this problems starts with a linearization of the underlying ma- problem for T -palindromic polynomials. In particular, we trix polynomial P , such as the companion form, and then show how to construct T -palindromic linearizations of T - applies a general-purpose method like the QZ algorithm to palindromic polynomials with odd degree. In the case of this linearization. But when P is structured, it can be ad- even degree we provide necessary and sufficient conditions vantageous to use a linearization with the “same’ structure for the existence of T -palindromic linearizations, most of as P , if one can be found. In this talk we discuss the scope which have lower dimension than the usual one. of this structured linearization strategy for the classes of “alternating’ and “palindromic’ matrix polynomials, using Fernando De Ter´an, Froil´anM. Dopico the Smith form as the central tool. Departamento de Matem´aticas Universidad Carlos III de Madrid Steve Mackey [email protected], [email protected] Western Michigan University [email protected] D. Steven Mackey Department of Mathematics Niloufer Mackey Western Michigan University Western Michigan University [email protected] Dept of Mathematics [email protected]

MS35 Christian Mehl Invariant Pairs for Polynomial and Nonlinear Technische Universitaet Berlin Eigenvalue Problems Institut f¨urMathematik [email protected] We consider matrix eigenvalue problems that are polyno- mial or genuinely nonlinear in the eigenvalue parameter. One of the most fundamental differences to the linear case Volker Mehrmann is that distinct eigenvalues may have linearly dependent Inst. f.Mathematik, TU Berlin eigenvectors or even share the same eigenvector. This can [email protected] be a severe hindrance in the development of general numer- ical schemes for computing several eigenvalues of a poly- MS36 nomial or nonlinear eigenvalue problem, either simultane- ously or subsequently. The purpose of this talk is to show Fast and Accurate Computations with Some that the concept of invariant pairs offers a way of repre- Classes of Quasiseparable Matrices senting eigenvalues and eigenvectors that is insensitive to In the last decade many algorithms have been developed this phenomenon. We will demonstrate the use of this con- to perform fast computations with quasiseparable matri- cept with a number of numerical examples. This is partly ces by working with the quasiseparable generators. In gen- joint work with Timo Betcke, University of Reading. eral, the accuracy and stability of these algorithms is not Daniel Kressner guaranteed. We present in this talk some subsets of qua- ETH Zurich siseparable matrices that allow us to perform both fast and [email protected] accurate computations if a proper parametrization is used. These subsets include totally nonnegative quasiseparable matrices and zero diagonal symmetric and skew-symmetric MS35 quasiseparable matrices. Diagonalizing Quadratic Matrix Polynomials Froil´anM. Dopico A quadratic eigenvalue problem (Mλ2 + Dλ + K)x =0 Department of Mathematics with M nonsingular is said to be diagonalizable if its lin- Universidad Carlos III de Madrid earization [email protected] Tom Bella     University of Rhode Island DM −K 0 λ + Department of Mathematics M 0 0 M [email protected]

Vadim Olshevsky Dept. of Mathematics is diagonalizable by equivalence or congruence transfor- U. of Connecticut, Storrs. mations, as appropriate. We characterize all admissible [email protected] canonical forms. Also, we identify isomorphisms between the sets of transformations and subsets of the centralizer of the Jordan form. This is a report on collaborative work MS36 with Ion Zaballa. A Survey of Some Recent Results for Quasisepara- Peter Lancaster Department of Mathematics and Statistics University of Calgary, Canada [email protected] LA09 Abstracts 79

ble Matrices and their Applications [email protected]

Abstract not available at time of publication. Misha E. Kilmer Vadim Olshevsky Tufts University Dept. of Mathematics [email protected] U. of Connecticut, Storrs. [email protected] MS37 Non-smooth Solutions to Least Squares Problems MS36 In an attempt to overcome the ill-posedness or ill- Wavelet Matrices as Green’s Matrices with New conditioning of inverse problems, regularization methods Factorizations are implemented by introducing assumptions on the solu- Olshevsky and Zhlobich have identified a special class tion. Common regularization methods include total vari- of five-diagonal matrices with a remarkable factorization. ation, L-curve, Generalized Cross Validation (GCV), and These matrices are 2 by 2 block bidiagonal. They are prod- the discrepancy principle. It is generally accepted that ucts of 2 by 2 block diagonal matrices in which the blocks all of these approaches except total variation unnecessarily smooth solutions, mainly because the regularization oper- of the second factor are shifted by a row and column. We L2 identify these factors in the case of the famous Daubechies- ator is in an norm. Alternatively, statistical approaches 4 wavelet filters. These are block Toeplitz and the new to ill-posed problems typically involve specifying a priori factorization shows how easily they could be made time- information about the parameters in the form of Bayesian varying (and finite length) without losing their orthogonal- inference. These approaches can be more accurate than ity. typical regularization methods because the regularization term is weighted with a matrix rather than a constant. Gilbert Strang The drawback is that the matrix weight requires informa- tion that is typically not available or is expensive to cal- Massachusetts Institute of Technology 2 [email protected] culate. Ø The χ method developed by the author and colleagues can be viewed as a regularization method that uses statistical information to find matrices to weight the MS36 regularization term. We will demonstrate that unique and 2 A Quasiseparable Matrices Approach to CMV and simple L solutions found by this method do not unec- Fiedler Matrices essarily smooth solutions when the regularization term is accurately weighted with a diagonal matrix. Several new classes of structured matrices have appeared recently in the scientific literature. Among them there are Jodi Mead so-called CMV and Fiedler matrices which are found to be Boise State University related to polynomials orthogonal on the unit circle and Department of Mathematics Horner polynomials respectively. Both matrices are five [email protected] diagonal and have a similar structure, although they have appeared under completely different circumstances. We es- tablish a link between these matrices by showing that they MS37 both belong to the wider class of twisted Hessenberg-Order- Using Confidence Ellipsoids to Choose the Regu- One quasiseparable matrices. We also obtain a general de- larization Parameter for Ill-Posed Problems scription of five-diagonal matrices in terms of recurrence relations satisfied by polynomials they are related to. Confidence ellipsoids for discretized ill-posed problems are elongated in directions corresponding to small singular val- Pavel Zhlobich, Vadim Olshevsky ues. Intersecting a confidence ellipsoid with another ellip- University of Connecticut soid guaranteed to contain the solution gives better local- Department of Mathematics ization of that solution with the same confidence level. We [email protected], [email protected] present a regularization method with a one-to-one corre- spondence between values of the regularization parameter Gilbert Strang and convex combinations of the two ellipsoids. The pa- Massachusetts Institute of Technology rameter value is optimized by minimizing the size of the [email protected] convex combination. Bert W. Rust MS37 National Institute of Standards and Technnology [email protected] Modified, Regularized Total Least Norm Approach to Signal Restoration MS37 Total Least Norm (TLN) is a common choice for model- ing blind deconvolution problems. In this talk, we present Regularization in Polynomial Approximations, a modified, regularized TLN model for signal deblurring Resolution of the Runge Phenomenon problems where both the blurring operator and the blurred The polynomial interpolation based on a uniform grid signal contain noise. We introduce an alternating method yields the well-known Runge phenomenon. The maximum that uses this model to obtain better restoration not only point-wise error is unbounded for functions with complex of the signal, but also of the blurring operator. roots in the Runge zone. In this work, we first investigate Malena I. Espanol the Runge phenomenon with finite precision operations. Graduate Aeronautical Laboratories Then a truncation method based on the truncated singu- California Institute of Technology lar value decomposition is proposed to resolve the Runge 80 LA09 Abstracts

phenomenon. The method consists of two stages: In the practical point of view, the implementation and conver- first stage a statistical filtering matrix is applied to the in- gence of the inexact RPHSS method are also discussed in terpolation matrix as preconditioner. In the second stage a detail. Finally, a number of numerical experiments are used pseudo inverse of the interpolation matrix is formed using to show that RPHSS method is efficient and comparable a truncated singular value decomposition. We investigate to standard HSS iteration method. the structure of the singular vectors and singular values and determine a truncation point based on the oscillatory Jun-Feng Yin behavior of the signular vectors and decay behavior of the Department of Mathematics singular values. We then show with numerical examples Tongji University, Shanghai, P.R. China that exponential decay of the approximation error can be [email protected] achieved if an appropriate smoothness parameter that de- pends on the interpolated function is chosen. MS38 Wolfgang Stefan On Parameterized Uzawa Methods for Saddle- University at Buffalo, SUNY Point Problems wstefan@buffalo.edu Abstract not available at time of publication. Jae-Hun Jung Bing Zheng Department of Mathematics School of Mathematics and Statistics SUNY at Buffalo Lanzhou University, Lanzhou, P.R. China jaehun@buffalo.edu [email protected]

MS38 MS39 Block Preconditioners for Generalized Saddle- Coupled Tensor and Matrix Factorizations Point Matrices Matrix and tensor factorizations have proved to be pow- Abstract not available at time of publication. erful tools in data analysis. There are limited, though, to Zhong-Zhi Bai data that can be represented as a single object. We propose Institute of Computational Mathematics to consider problems where the data has multiple aspects, Chinese Academy of Sciences, Beijing, China represented by multiple matrices and tensors with shared [email protected] modes. Our goal is to analyze these data simultaneously via coupled decompositions. Specifically, we propose all-at- once optimization techniques, which are more effective in MS38 our experiments than alternating methods. In this talk, we A Geometric View of Krylov Subspace Methods on describe the problem, present several examples, and show Singular Systems computational results.

We show why one can apply CG to consistent systems Evrim Acar, Tamara G. Kolda Ax = b where A is symmetric and positive semi-definite, Sandia National Laboratories by decomposing CG into the R(A) and N (A) components, [email protected], [email protected] which shows that the method is essentially equivalent to CG applied to a positive diagonal system in R(A). Next, Danny Dunlavy we analyze GMRES for systems where A is nonsymmetric Computer Science and Informatics Department and singular by decomposing GMRES into the R(A) and Sandia National Laboratories R(A)⊥ components, providing a geometric interpretation [email protected] of the convergence conditions given by Brown and Walker. MS39 Ken Hayami A Fast and Efficient Algorithm for Low-rank Ap- National Institute of Informatics proximation of a Matrix and Generalization to Ten- [email protected] sor Abstract not available at time of publication. Masaaki Sugihara Graduate School of Information Science and Technology, Nam H. Nguyen the University of Tokyo Department of Electrical and Computer Engineering m [email protected] Johns Hopkins University [email protected] MS38 On Restrictively Preconditioned HSS Iteration MS39 Methods for Non-Hermitian Positive-Definite Lin- Commuting Birth-and-Death Processes ear Systems We use methods from combinatorics and algebraic statis- A restrictively preconditioned Hermitian/skew-Hermitian tics to study analogues of birth-and-death processes that splitting (RPHSS) iteration method is presented for the have as their state space a finite subset of the m- solution of the non-Hermitian and positive-definite system dimensional lattice and for which the m matrices that of linear equations. Theoretical analyses on the conver- record the transition probabilities in each of the lattice di- gence condition for RPHSS method , the optimal param- rections commute pairwise. One reason such processes are eters and the choice of preconditioners are given. From of interest is that the transition matrix is straightforward LA09 Abstracts 81

to diagonalize, and hence it is easy to compute n step tran- the application of photonic crystal waveguides and there is sition probabilities. The set of commuting birth-and-death a strong need for the fast solution of the problems. For solv- processes decomposes as a union of toric varieties, with the ing the problems efficiently, we extend the Sakurai-Sugiura main component being the closure of all processes whose method which has been proposed for simply connected re- nearest neighbor transition probabilities are positive. We gion. We also analyze the error of the computed eigenval- exhibit an explicit monomial parametrization for this main ues. component, and we explore the boundary components us- ing primary decomposition. Takafumi Miyata, Lei Du, Tomohiro Sogabe, Yusaku Yamamoto, Shao-Liang Zhang Bernd Sturmfels Nagoya University, Japan University of California , Berkeley [email protected], lei- [email protected] [email protected], [email protected], [email protected], [email protected] Steve Evans, Caroline Uhler u.ac.jp Department of Statistics UC Berkeley [email protected], [email protected] MS40 Efficient Subspace Alignment Technique for the Self-Consistent Field Iterations MS39 Finitely Generated Cumulants: Developments and The subspace alignment is a crucial preprocessing step for Applications the self-consistent field iteration to solve nonlinear eigen- value problems arising from electronic structure calcula- This continues work on algebraic aspects of cumulants de- tions. Its computational kernel involves the matrix po- veloped in Pistone and Wynn (1999, Statistica Sinica) and lar decomposition. The popular scaled Newton method (2000, J. Symb. Comp). The definition of the finitely gen- requires explicit matrix inversion, which is dominated by erated cumulant (FGC) property is that, in the univariate communication in distributed and multi-core computing. or multivariate case, if K(s) is the cumulant generating In this talk, we present a new algo- function for a distribution then the vector of first partial rithm based on the QR-decomposition (without pivoting) derivatives with respect to s, namely K’, and the matrix and a communication optimal implementation of the sub- of second order (partial) derivatives, K’, satisfy an implicit space alignment. polynomial equation f(K’,K’) = 0. This generalizes the Morris class, in which K’ is a quadratic function of K’. The Yuji Nakatsukasa FGC property is a way of introducing polynomial algebra, University of California, Davis and symbolic methods, into continuous distribution theory Applied Math and a surprisingly large class of distributions satisfy the [email protected] FGC property. Applications include the asymptotic theory of maximum likelihood, saddle-point approximations, mul- Zhaojun Bai tivariate dependency for non-Gaussian random variables, University of California mixture models and information geometry. [email protected]

Henry Wynn Francois Gygi London School of Economics University of California, Davis [email protected] Lawrence Livermore National Laboratory [email protected] MS40 A Key for the Choice of the Subspaces in Restarted MS40 Krylov Methods A Hierarchical Parallel Method for Solving Nonlin- ear Eigenvalue Problems The restarted Krylov subspace methods allow computing some eigenpairs of large sparse matrices. The size of the In this talk, we present a parallel method for finding eigen- subspace in these methods is chosen empirically. A poor values in a given domain and corresponding eigenvectors of choice of this size could lead to the non-convergence of the nonlinear eigenvalue problems. In our method, the origi- methods. We propose a technique, based on the projection nal problem is converted to a smaller generalized eigenvalue of the problem on several subspaces instead of a single one, problem, which is obtained numerically by solving a set of to remedy to this problem. Our approach is validated by linear equations. These linear equations are independent its application on IRA method. and can be solved in parallel. Nahid Emad Tetsuya Sakurai University of Versailles, PRiSM Laboratory Department of Computer Science [email protected] University of Tsukuba [email protected] MS40 Junko Asakura A Numerical Method Based on the Residue The- Research and Development Division orem for Computing Eigenvalues in Multiply Con- Square Enix Co. Ltd. nected Region [email protected] In this talk, we consider the solution of eigenvalues in a fi- nite and multiply connected region. Such problems arise in Hiroto Tadano Department of Computer Science 82 LA09 Abstracts

University of Tsukuba with eigenvalues with prespecified algebraic multiplicity. [email protected] Emre Mengi Tsutomu Ikegami Dept. of Mathematics, UC San Diego Information Technology Research Institute [email protected] AIST [email protected] MS41 Characterization and Construction of the Nearest Kinji Kimura via Coalescence of Pseudospectra Department of Applied Mathematics Kyoto University Let w(A) be the distance from A to the set of defective [email protected] matrices. and let c(A) be the supremum of all  for which the of A has n distinct components. It is known that w(A) ≥ c(A), with equality holding for the MS41 2-norm. We show that w(A)=c(A) for the Frobenius Fast Algorithms for Approximating the Pseu- norm too, and that the minimal distance is attained by a dospectral Abscissa and Radius defective matrix in all cases. The results depend on the  n×n geometry of the pseudospectrum near points where coales- The -pseudospectral abscissa and radius of an matrix cence of the components occurs. are respectively the maximum real part and the maximal modulus of points in its -pseudospectrum. Existing tech- Michael L. Overton niques compute these quantities accurately but the cost is New York University multiple SVDs of order n. We present a novel approach Courant Instit. of Math. Sci. based on computing only the spectral abscissa or radius or [email protected] a sequence of matrices, generating a monotonic sequence of lower bounds which, in many but not all cases, converges to the pseudospectral abscissa or radius. MS42 Hessenberg Quasiseparable Matrices and Polyno- Nicola Guglielmi mials University of L’Aquila Italy The classification of tridiagonal matrices in terms of poly- [email protected] nomials orthogonal on a real interval, as well as the classi- fication of unitary Hessenberg matrices in terms of Szeg¨o Michael L. Overton polynomials (orthogonal on the unit circle) are well–known New York University results, and form the basis for many fast algorithms involv- Courant Instit. of Math. Sci. ing these classes. In this talk, we give complete classifica- [email protected] tions of matrices having Hessenberg–quasiseparable struc- ture in terms of the polynomials related in the same way as the above examples. Several subclasses are also classified, MS41 including Hessenberg–semiseparable matrices, and the mo- Structured Pseudospectra of Hamiltonian Matrices tivating special cases of tridiagonal and unitary Hessenberg matrices. We consider the variation of the spectrum of Hamiltonian matrices under Hamiltonian perturbations. The first part Tom Bella of the talk deals with the associated structured pseudospec- University of Rhode Island tra. We show how to compute these sets and give some Department of Mathematics examples. In the second part we discuss the robustness [email protected] of linear stability. In particular we determine the smallest norm of a perturbation that makes the perturbed Hamil- tonian matrix unstable. MS42 Characteristic Polynomials, Eigenvalues Michael Karow and Eigenspaces of Quasiseparable of Order One Technical University of Berlin Matrices Germany [email protected] We discuss spectral properties of quasiseparable of order one matrices. This class of matrices contains at least three well-known classes: diagonal plus semiseparable matrices, MS41 tridiagonal matrices, unitary Hessenberg matrices. We de- Multiple Eigenvalues with Prespecified Multiplici- rive different recurrence relations for characteristic polyno- ties and Pseudospectra mials of principal leading submatrices of a quasiseparable matrix. Some basic algorithms to compute eigenvalues are Suppose z is a point in the complex plane where two com- presented. We obtain conditions when an eigenvalue of a ponents of the epsln-pseudospectrum of A coalesce. Alam quasiseparable matrix is simple. For the case of a multiple and Bora deduced the existence of matrices within epsln- eigenvalue the structure of the corresponding eigenspace is neighborhood of A with z as a multiple eigenvalue. There- described. Illustrative examples are presented. fore smallest epsln such that two components of the epsln- pseudospectrum coalesce is the distance to the nearest ma- Yuli Eidelman trix with a multiple eigenvalue. Here we establish the con- Tel Aviv University nection between the pseudospectra and nearest matrices Department of Mathematics [email protected] LA09 Abstracts 83

Israel Gohberg, Iulian Haimovici putational sense, to the template update algorithm pro- Department of Mathematics posed by Matthew et. al (The Template Update Problem). Tel-Aviv University The update strategy proposed in Matthew et. al. requires [email protected], [email protected] computation of the principal components of the augmented image matrix at every iteration, since the PCs correspond Vadim Olshevsky to the left singular vectors of image matrix, we suggest us- University of Connecticut ing URV updates as an alternative to computing PCs ab Department of Mathematics initio at every iteration. [email protected] Jesse L. Barlow Penn State University MS42 Dept of Computer Science & Eng A Fast Euclidean Algorithm for Quasiseparable [email protected] Polynomials via Non-casual Factorizations Anupama Chandrasekhar We have elaborated that there exist a one to one corre- Pennsylvania State University spondence of Classical Euclid algorithm with GKO, HO [email protected] and OS algorithms when we express the Euclid algorithm in one way of the matrix form. It is noticeable that Be- zoutian plays a major role in here by preserving different MS43 displacement structures according to the above three algo- Analysis of Model Uncertainty Using Optimization rithms. All these result drive us to design a fast Euclid Iterates algorithm for polynomial represent in general bases and the result can be specialized for cases of Quasiseparable, Throughout the course of an optimization run, multiple Orthogonal and Szego polynomials. points are generated and evaluated. In this talk, we will describe how this data can be used to evaluate parameter Sirani Perera sensitivities and uncertainties while providing additional University of Connecticut insight into the model. We will also describe how the opti- Department of Mathematics mization iterate set can be supplemented to provide more [email protected] statistically significant results and how these results might be computed during the course of the optimization to im- Vadim Olshevsky prove the search results. co-author [email protected] Genetha Gray Sandia National Laboratories [email protected] MS42 Parameterization and Stability of Methods for Katie Fowler Quasiseparable Matrices Clarkson University [email protected] For the standard parameterization of a quasiseparable ma- trix, a small perturbation of the parameters ( i.e. small Matthew Grace structured error) do not correspond to a small perturba- Sandia National Labs tion of the matrix (i.e. small normwise error). As a conse- [email protected] quence, when working with a quasiseparable matrix, some additional stability assumption on the parameters is re- quired to guarantee normwise backward stability. This talk MS43 describes alternate parameterizations based on plane rota- Multisplitting for Regularized Least Squares: A tions for which small structured error implies small norm- Multiple Right Hand Side Problem for Image wise error. Restoration and Reconstruction

Michael Stewart Least squares problems are one of the most often used nu- Georgia State University merical formulationsin engineering. Many such problems Department of Mathematics lead to ill-posed systems of equations for which a solution [email protected] may be found by introducing regularization. Here we ex- tend the use of multisplitting least squares, as originally introduced by Renaut (1998) for well-posed least squares MS43 problems, to Tikhonov regularized large scale least squares A More Efficient Version of the Lucas-Kanade problems. Regularization at both the global and subprob- Template Tracking Algorithm Using the Ulv De- lem level is considered, hence providing a means for mul- composition tiple parameter regularization of large scale problems. We find out that in solving the local splitting, each local prob- Template tracking refers to the problem of tracking an lem turn out to be a linear system with multiple right object through a video sequence using a template. The hand sides with updates. Utilizing the characteristics of template is the image of the tracked object, usually ex- the problem itself enables us to apply more efficient al- tracted from the first frame. The aspect of template track- gorithm to obtain the solution. Basic convergence results ing that we are interested in, is the problem of updating follow immediately from the original formulation. Numeri- the template. Template updates are necessary, especially cal validation is presented for some simple one dimensional in long video sequences, because the appearance of the ob- signal restoration simulations. ject changes significantly and the initial template is soon obsolete. In this work we suggest improvements, in a com- Hongbin Guo 84 LA09 Abstracts

Arizona State University origins of these algorithms, emphasizing research themes hb [email protected] that continue to have central importance.

Rosemary A. Renaut Dianne P. O’Leary Arizona State University University of Maryland, College Park Department of Mathematics and Statistics Department of Computer Science [email protected] [email protected]

Youzuo Lin MS44 Arizona State University Antecedents of the LR and QR Algorithms [email protected] or [email protected] The seminal idea in LR and QR is the reversal of factors to form a new matrix. In 1882 Darboux introduced a MS43 transform that takes a 2nd order linear operator (ODE) Accelerating the EM Algorithm and writes it as the composition of two 1st order linear operators PQ and forms a new 2nd order operator QP The EM algorithm is widely used for numerically approx- as the desired transform. If you discretize properly you imating maximum-likelihood estimates in the context of get an instance of the basic LR transform applied to a missing information. This talk will focus on the EM al- tridiagonal matrix. In the 20th century a few physicists gorithm applied to estimating unknown parameters in a (Weyl,Schroedinger, Infeld) applied the transform to ob- (finite) mixture density, i.e., a probability density function tain eigenfunctions for operators that occur in applications. (PDF) associated with a statistical population that is a In that work the transform was applied a small number mixture of subpopulations, using “unlabeled” observations of times, the goal was not to find eigenvalues, and shifts, on the mixture. In the particular case when the subpopu- though present, were incidental and not a tool to accelerate lation PDFs are from common parametric families, the EM convergence. algorithm becomes a fixed-point iteration that has a num- ber of appealing properties. However, the convergence of Beresford N. Parlett the iterates is only linear and may be unacceptably slow if University of California the subpopulations in the mixture are not “well-separated” Department of Mathematics in a certain sense. In this talk, we will review the EM al- [email protected] gorithm for mixture densities, discuss a certain method for accelerating convergence of the iterates, and report on nu- merical experiments. MS44 The Dual Flow Between Linear Algebra and Opti- Homer F. Walker mization Worcester Polytechnic Institute [email protected] Optimization and linear algebra have been closely con- nected for more than 60 years, ever since Courant’s 1943 in- troduction of the quadratic penalty function and Dantzig’s MS44 1947 creation of the simplex method for linear program- Development and History of Sparse Direct Meth- ming. A multitude of linear algebraic subproblems appear ods in optimization methods, which often impose special struc- ture on the associated matrices. This talk will highlight a Direct methods for the solution of large sparse systems selection of symmetrically productive ties between linear were used by the linear programming community from the algebra and optimization, ranging from classical to ultra- 1950s but it was not until the 1960s that they were applied modern. to a wider range of applications including the solution of stiff ODEs and power systems. We sketch these early ori- Margaret H. Wright gins and highlight the key points in the development of New York University sparse direct techniques and software. We also examine Courant Institute of Mathematical Sciences the history of the direct v iterative debate and its current [email protected] resolution.

Iain Duff MS45 Rutherford Appleton Laboratory, Oxfordshire, UK OX11 Regularizing Iterations for ODF Reconstruction 0QX and CERFACS, Toulouse, France We use regularizing iterations to reconstruct local crys- iain.duff@stfc.ac.uk tallographic orientations from X-ray diffraction measure- ments, and we demonstrate that right preconditioning is necessary to provide satisfactory reconstructions. Our MS44 right preconditioner is not a traditional one that acceler- Some History of Conjugate Gradients and Other ates convergence; its purpose is to modify the smoothness Krylov Subspace Methods properties of the reconstruction. We also show that a new stopping criterion, based on the information available in In the late 1940’s and early 1950’s, newly available comput- the residual vector, provides a robust choice of the number ing machines generated intense interest in solving “large’ of iterations. systems of linear equations. Among the algorithms devel- oped were several related methods, all of which generated Per Christian Hansen bases for Krylov subspaces and used the bases to minimize DTU Informatics or orthogonally project a measure of error. The best known Technical University of Denmark of these algorithms is conjugate gradients. We discuss the [email protected] LA09 Abstracts 85

Henning Osholm Sørensen [email protected] Materials Research Division Ris{ Iveta Hnetynkova o} DTU Charles University, Prague [email protected] [email protected]

MS45 Martin Plesinger Technical University Liberec A Generalized Hybrid Regularization Approach for [email protected] Partial Fourier MRI Reconstruction

Accelerated MR imaging techniques require regularization MS46 to ensure robust solutions which must be computed effi- ciently in a manner consistent with the speed of data acqui- Randomized Algorithms in Linear Algebra: From sition. One difficulty is the efficient determination of one or Approximating the Singular Value Decomposition more regularization parameters. We develop a generaliza- to Solving Regression Problems tion of the LSQR-Hybrid approach for the partial-Fourier The introduction of randomization in the design and anal- reconstruction problem that allows us to efficiently select ysis of algorithms for matrix computations (such as matrix two parameters via information generated during various multiplication, least-squares regression, the Singular Value runs of the LSQR-Hybrid algorithm. Phantom and in-vivo Decomposition (SVD), etc.) over the last decade provided results will be presented. a new paradigm and a complementary perspective to tradi- Misha E. Kilmer tional numerical linear algebra approaches. These novel ap- Department of Mathematics proaches were motivated by technological developments in Tufts University many areas of scientific research that permit the automatic [email protected] generation of large data sets, which are often modeled as matrices. In this talk we will outline how such approaches can be used to approximate problems ranging from ma- W. Scott Hoge trix multiplication and the Singular Value Decomposition Brigham and Womens Hospital (SVD) of matrices to approximately solving least-squares Harvard Medical School, Boston, MA problems and systems of linear equations. [email protected] Petros Drineas Computer Science Department MS45 Rennselaer Polytechnic Institute Golub-Kahan Hybrid Regularization in Image Pro- [email protected] cessing

Ill-posed problems arise in many image processing applica- MS46 tions, including microscopy, medicine and astronomy. It- Fast Factorization Of Low-Rank Matrices via Ran- erative methods are typically recommended for these large domized Sampling scale problems, but they can be difficult to use in practice. Lanczos based hybrid methods have been proposed to slow The talk will describe a set of techniques based on ran- the introduction of noise in the iterates. In this talk we domized sampling that dramatically accelerate several key discuss the behavior of Lanczos based hybrid methods for matrix computations, including many needed for solving large scale problems in image processing. partial differential equations and for extracting information from large data-sets (such as those arising from genomics, James G. Nagy the link structure of the World Wide Web, etc). The algo- Mathematics and Computer Science rithms are applicable to any matrix that can in principle Emory University be approximated by a low-rank matrix, are as accurate [email protected] as deterministic methods, and are for practical purposes 100% reliable (the risk of ”failure” can easily be rendered Julianne Chung less than e.g. 1e − 12). Several numerical examples will be University of Maryland presented in which the randomized methods are applied to [email protected] solve problems arising in potential theory, acoustic scatter- ing, image processing, and other application areas. MS45 Per-Gunnar Martinsson Golub-Kahan Iterative Bidiagonalization and Re- University of Colorado at Boulder vealing Noise in the Data Applied Mathematics Department [email protected] Consider an ill-posed problem with a noisy right-hand side (observation vector), where the size of the (white) noise is unknown. We show how the information from the Golub- MS46 Kahan iterative bidiagonalization can be used for revealing The Design and Analysis of Randomized Algo- the unknown level of the noise. Such information can be rithms for High-dimensional Data: New Insights useful in construction of stopping criteria in solving large from Multilinear Algebra and Statistics ill-posed problems. In recent years, the spectral analysis of appropriately de- Zdenek Strakos fined kernels has emerged as a principled way to ex- Institute of Computer Science tract the low-dimensional structure often prevalent in high- Academy of Sciences of the Czech Republic dimensional data. From selecting so-called landmark train- 86 LA09 Abstracts

ing examples in supervised machine learning, to solving MS47 large least-squares problems from classical statistics, ran- Near Threshold Graphs domized algorithms provide an appealing means of dimen- sionality reduction, helping to overcome the computational A conjecture of Grone and Merris states that for any graph limitations currently faced by practitioners with massive G, its Laplacian spectrum, Λ(G), is majorized by its conju- datasets. In this talk we describe how new insights from gate degree sequence, D∗(G). That conjecture prompts an multilinear algebra and statistics can be brought to bear investigation of the relationship between Λ(G) and D∗(G), on the design and analysis of such algorithms, in order and Merris has characterized the graphs G for which the to provide intuitive insight as well as concise formalisms. Λ(G) and D∗(G) are equal. In this talk, we pro- We discuss in particular the role of determinants and com- vide a constructive characterization of the graphs G for pound matrices as it arises in a variety of recent settings which Λ(G) and D∗(G) share all but two elements. of interest to the theoretical computer science community, and illustrate the practical implications of our results by Steve Kirkland way of example applications drawn from signal and image Hamilton Institute processing. (Joint work with Mohamed-Ali Belabbas and National University of Ireland other SISL members.) [email protected] Patrick J. Wolfe Harvard University MS47 Department of Statistics Scrambling Index of Primitive Matrices [email protected] The scrambling index of a primitive matrix A is the small- est positive integer k such that Ak(At)k = J, where At MS47 denotes the transpose of A and J denotes all-ones matrix. Tournaments, Landau’s Theorem, Rado’s Theo- We prove two upper bounds on the scrambling index of rem, and Partial Tournaments primitive matrices. The first bound is in terms of Boolean rank, and the second bound is in terms of the diameter and Landau’s classical theorem gives necessary and sufficient girth of the of A. conditions for the existence of a tournament with a pre- scribed score sequence. The theorem has been strength- Jian Shen ened to give the existence of a tournament with special Texas State University properties. Rado’s theorem gives necessary and sufficient San Marcos, TX conditions for the existence of an independent transversal [email protected] of a family of subsets of a set on which a matroid is de- fined. Rado’s theorem can be used to give a nice proof of Mahmud Akelbek Landau’s theorem, indeed to give conditions for a partial Texas State University tournament to be completed to a tournament with a pre- [email protected] scribed score sequence. In this talk, we shall discuss these topics. Sandra Fital Weber State University Richard A. Brualdi sfi[email protected] University of Wisconsin Department of Mathematics [email protected] MS48 Fixed Points Theorems for Nonnegative Tensors MS47 and Newton Methods Hadamard Diagonalizable Graphs We discuss two theorems for nonnegative nonsymmetric tensors. First, the existence of a unique positive singular The general question of interest is to characterize the undi- value and the corresponding singular vectors. Second, the rected graphs G such that the associated G rescaling of nonnegative tensors to balanced tensors, i.e. with can be diagonalized by some . the tensor version of Sinkhorn’s diagonal scaling to dou- During this presentation, I will survey many interesting bly stochastic matrices. In both cases the values of these and fundamental properties about these graphs along with solutions can be found efficiently by using the fix point the- a partial characterization of the cographs that can be di- orem. We then discuss the Newton method to speed up the agonalized by a Hadamard matrix. computations for these fixed points. Shaun M. Fallat Shmuel Friedland University of Regina University of Illinois at Chicago Canada [email protected] [email protected]

Steve Kirkland MS48 University of Regina Trust-region Method for the Best Multilinear Rank Dept of Math and Stats Approximation of Tensors [email protected] We work on reliable and efficient algorithms for the best Sasmita Barik low multilinear rank approximation of higher-order ten- University of Regina sors. This approximation is used for dimensionality re- [email protected] duction and signal subspace estimation. In this talk, we express it as the solution of a minimization problem on a LA09 Abstracts 87

quotient manifold. Applying the Riemannian trust-region sion of low-dimensional data. scheme with the truncated conjugate gradient method for the trust-region subproblems, superlinear convergence is Eugene Tyrtyshnikov achieved. We comment on the advantages of the algorithm, Institute of Numerical Mathematics, Russian Academy of discuss the issue of local optima and show some applica- Sci. tions. [email protected]

Mariya Ishteva Ivan Oseledets K.U.Leuven, ESAT/SCD Institute of Numerical Mathematics Leuven, Belgium Russian Academy of Sciences, Moscow [email protected] [email protected]

Lieven De Lathauwer Katholieke Universiteit Leuven MS49 [email protected] Numerical Computation with Semiseparable Ma- trices: Distributed Algorithms for Quantum Trans- P.-A. Absil port with Atomistic Basis Sets (with Ragu Balakr- Department of Mathematical Engineering ishnan) Universit { } Two of the most demanding computational problems in ’ e catholique de Louvain quantum transport analysis involve mathematical oper- [email protected] ations with semiseparable matrices. First, we consider determining the diagonal of a semiseparable matrix that Sabine Van Huffel is the inverse of a block-tridiagonal matrix. The second K.U.Leuven, Belgium operation corresponds to computing matrix products in- sabine.vanhuff[email protected] volving semiseparable matrices. We present a parallel in- version algorithm for block-tridiagonal matrices as well as distributed approaches for computing matrix products MS48 based upon an underlying compact representation for the Multi-multilinear Methods: Computing with Sums semiseparable matrices. of Products of Sums of Products Stephen Cauley Representing a multivariate function as a sum of prod- Purdue University ucts of one-variable functions bypasses the curse of dimen- Electrical and Computer Engineering sionality. However, for the wavefunction of a quantum- [email protected] mechanical system consisting of K non-interacting subsys- tems, one can show that the number of terms needed grows exponentially in K. A representation with sums of prod- MS49 ucts of sums of products can handle this case, and poten- Semiseparable Matrices and Fast Transforms for tially provide a size-consistent representation for general Orthogonal Polynomials and Associated Functions systems. A new connection between classical orthogonal polynomi- Martin J. Mohlenkamp als or the corresponding associated functions and semisep- Ohio University arable matrices will be developed. It will be shown how [email protected] this can be exploited to obtain fast FFT-like algorithms for these functions. The results will be demonstrated with two recent methods to compute discrete Fourier transforms MS48 2 on the sphere S and on the rotation group SO(3). Effective Ranks of Tensors and New Decomposi- tions in Higher Dimensions Jens Keiner Universitaet zu Luebeck Decompositions and approximations of d-dimensional ten- Institut f¨urMathematik sors are crucial either in structure recovery problems and [email protected] merely for a compact representation of tensors. How- ever, the well-known decompositions have serious draw- backs: the Tucker decompositions suffer from exponen- MS49 tial dependence on the dimensionality d while fixed-rank Hierarchical Matrix Preconditioners canonical approximations are not stable. In this talk we present new decompositions that are stable and have Hierarchical (H-) matrices provide a powerful technique to the same number of representation parameters as canon- compute and store approximations to dense matrices in a ical decompositions for the same tensor. Moreover, the data-sparse format. The basic idea is the appoximation new format possesses nice stability properties of the SVD of matrix data in hierarchically structured subblocks by (as opposed to the canonical format) and is convenient low rank representations. The usual matrix operations can for basic operations with tensors (see: I.V.Oseledets and be computed approximately with almost linear complexity. E.E.Tyrtyshnikov, Breaking the curse od dimensionality, We use such an h-arithmetic to set up preconditioners for or how to use SVD in many dimensions, Research Re- the iterative solution of sparse linear systems as they arise port 09-03, Kowloon Tong, Hong Kong: ICM HKBU, 2009 in the finite element discretization of PDEs. (www.math.hkbu.edu.hk/ICM/pdf/09-03.pdf). Last not the least, the new decomposition provides a useful method Sabine Le Borne (comparable and even superior to wavelets) for compres- Tennessee Technological University Department of Mathematics [email protected] 88 LA09 Abstracts

MS49 ods (robustness) and iterative solvers (computational cost), Root-finding, Structured Matrices and Additive while alleviating their drawbacks (memory requirements, Preprocessing lack of robustness).

The classical problem of polynomial root-finding was re- Olaf Schenk cently attacked highly successfully based on effective algo- Department of Computer Science, University of Basel rithms for approximation of eigenvalues of rank structured Switzerland matrices. We advance this approach with some techniques [email protected] that employ the displacement structure of the associated matrices and their additive preprocessing. MS51 Victor Pan A Parallel Version of GPBiCR Method Suitable for CUNY Distributed Parallel Computing Lehman v y [email protected] Abstract not available at time of publication. Tong-Xiang Gu MS50 Laboratory of Computational Physics Hybrid Techniques in the Solution of Large Scale Institute of Appl. Physics and Comp. Math., Beijing, Problems China [email protected] The most challenging problems for numerical linear algebra arguably arise from the discretization of partial differential equations from three-dimensional modelling. Often these MS51 problems are intractable to either direct methods (because Structure Exploited Algorithms for Nonsymmet- of fill-in) and direct methods (because of non-convergence). ric Algebraic Riccati Equation Arising in Transport In this talk we will review some recent work by researchers Theory at CERFACS and ENSEEIHT on the combined use of di- rect and iterative methods for solving very large linear sys- The nonsymmetric algebraic Riccati equation arising in tems of equations arising in three-dimensional modelling. transport theory is a special algebraic Riccati equation whose coefficient matrices having some special structures. Iain Duff By reformulating the Riccati equation, they were found Rutherford Appleton Laboratory, Oxfordshire, UK OX11 that its arbitrary solution matrix is of a Cauchy-like form 0QX and the solution matrix can be computed from a vector and CERFACS, Toulouse, France form Riccati equation instead of the matrix form that. In iain.duff@stfc.ac.uk this talk, we review some classical-type and Newton-type iterative methods for solving the vector form Riccati equa- tion and present some work under investigation . MS50 The Impact of Adaptive Solver Libraries on Future Linzhang Lu Many-cores School of Mathematical Science Xiamen University, Xiamen, P.R. China Solving linear systems takes many forms. Matrix proper- [email protected] [email protected] ties (numerical and physical) and computational properties (program and architecture) affect performance delivered. The number of combinations is huge and growing over time. MS51 Adaptive libraries are essential to provide rational paths On Hybrid Preconditioning Methods for Large through this thicket no individual can grasp the optimiza- Sparse Saddle-Point Problems tion issues. A parameter-space exhaustively-trained selec- tion system can make fast run-time selections for users. We Based on the block triangular product approximation to an describe an adaptive Spike-Pardiso (PSpike) combination energy matrix, a class of hybrid preconditioning methods poly-algorithm providing excellent performance compared is designed for accelerating the MINRES method for solv- to other direct and iterative solvers. ing the saddle point problems. The quasi-optimal values for the parameters involved in the new preconditioners are David Kuck estimated, so that the numerical conditioning and the spec- Intel tral property of the saddle-point matrix of the linear system Parallel and Distributed Solutions Division can be substantially improved. Several practical hybrid [email protected] preconditioners and the corresponding preconditioning it- erative methods are constructed and studied, too.

MS50 Zeng-Qi Wang On the PSPIKE Parallel Sparse Linear System Shanghai Jiaotong University Solver [email protected]

The availability of large-scale computing platforms com- prised of thousands of multicore processors motivates the MS51 need for highly scalable sparse linear system solvers for A Symmetric Homotopy and Hybrid Method for symmetric indefinite matrices. The solvers must opti- Solving Mixed Trigonometric Polynomial Systems mize parallel performance, processor performance, as well as memory requirements, while being robust across broad Abstract not available at time of publication. classes of applications. We will present a new parallel solver Bo Yu that combines the desirable characteristics of direct meth- LA09 Abstracts 89

School of Mathematical Science MS53 Dalian University of Technology, Dalian, P.R. China Stochastic Binormalization of Symmetric Matrices [email protected] A A is binormalized if the norm of each row (and column) is the same. Many matrices can be bi- MS52 normalized by symmetric diagonal scaling, and the result- Subset Selection: Deterministic vs. Randomized ing matrix frequently has a smaller . In 2004, Livne and Golub introduced an algorithm to find Subset selection methods try identify those columns of a such a diagonal scaling matrix D. The algorithm must ac- matrix that are ”most” linearly independent. Many deter- cess the elements of A individually. We first answer three ministic subset selection methods are based on a QR de- open questions from their paper concerning the existence composition with column pivoting. In contrast, random- and uniqueness of a binormalizing D. Then we introduce ized methods consist of two stages, where the first stage a stochastic algorithm to find D while accessing A only samples a smaller set of columns according to a proba- through matrix-vector products. Finally, we introduce a bility distribution, and the second stage performs subset limited-memory quasi-Newton method that incorporates selection on these columns. We analyze and compare the stochastic binormalization. performance of deterministic and randomized methods for subset selection. Andrew Bradley Stanford University Ilse Ipsen [email protected] North Carolina State University Department of Mathematics [email protected] MS53 Linear Algebra Issues in SQP Methods for Nonlin- ear Optimization MS52 Randomized Algorithms for Matrices and Large- We consider some linear algebraic issues associated with Scale Data Applications the formulation and analysis of sequential quadratic pro- gramming (SQP) methods for large-scale nonlinearly con- Randomization can be a powerful resource in the design of strained optimization. Recent developments in methods algorithms for linear algebra problems. Although much of for mixed-integer nonlinear programming and optimiza- this work has roots in convex analysis and theoretical com- tion subject to differential equation constraints has led to puter science, and thus has relied on techniques very differ- a heightened interest in methods that may be “hot started’ ent than those traditionally used in numerical linear alge- from a good approximate solution. In this context we fo- bra, recent research has begun to “bridge the gap” between cus on SQP methods that are best able to use “black-box’ the numerical linear algebra and theoretical computer sci- linear algebra software. Such methods provide an effective ence perspectives on these matrix problems. This has led way of exploiting recent advances in linear algebra software to practical numerical implementations of algorithms for for multicore and GPU-based computer architectures. very traditional matrix problems arising in scientific com- puting and numerical linear algebra, and it has led to novel Philip E. Gill, Elizabeth Wong algorithms for new matrix problems arising in large-scale University of California, San Diego scientific and Internet data applications. Several examples Department of Mathematics of this paradigm will be described. [email protected], [email protected] Michael Mahoney Stanford University MS53 Applied and Computational Mathematics An Active-set Convex QP Solver Based on Regu- [email protected] larized KKT Systems

Implementations of the simplex method depend on “basis MS52 repair’ to steer around near-singular basis matrices, and A Near-optimal Performance Analysis for Ran- KKT-based QP solvers must deal with near-singular KKT domized Matrix Approximation systems. However, few sparse-matrix packages have the required rank-revealing features. For convex QP, we ex- Simple randomized algorithms are exceptionally efficient plore the idea of avoiding singular KKT systems by apply- at identifying the numerical range of a matrix, which is ing primal and dual regularization to the QP problem. A a basic building block for constructing approximations of simplified single-phase active-set algorithm can then be de- large matrices. This talk describes a unified analysis of veloped. Warm starts are straightforward from any given the most common algorithms. The argument decouples active set, and the range of applicable KKT solvers ex- the linear algebra from the probability, which permits the pands. QPBLUR is a prototype QP solver that makes use application of powerful results from theory. of the block-LU KKT updates in QPBLU (Hanh Huynh’s The new methods lead to sharper and more transparent PhD dissertation, 2008) but employs regularization and the conclusions than previous analyses. simplified active-set algorithm. The aim is to provide a new QP subproblem solver for SNOPT for problems with Joel Tropp many degrees of freedom. Numerical results confirm the Applied and Computational Mathematics robustness of the single-phase regularized QP approach. Caltech [email protected] Christopher Maes Stanford University [email protected] 90 LA09 Abstracts

Michael A. Saunders [email protected] Systems Optimization Laboratory (SOL) Dept of Management Sci and Eng, Stanford [email protected] MS54 Krylov Subspace Methods for Linear Systems with Kronecker Product Structure MS53 Linear Algebra Computations in Sparse Optimiza- The numerical solution of linear systems with certain Kro- tion necker product structures is considered. Such structures arise, for example, from the finite element discretization of ”Sparse optimization” is a term often used to describe opti- a linear PDE on a d-dimensional hypercube. A standard mization problem in which we seek an approximate solution Krylov subspace method applied to such a linear system that is sparse, in the sense of having relatively few nonze- suffers from the curse of dimensionality and has a compu- ros in the vector of unknowns. The iterative optimization tational cost that grows exponentially with d. The key to methods proposed for these problems make use of many breaking the curse is to note that the solution can often tools from numerical linear algebra, including methods for be very well approximated by a vector of low tensor rank. linear equations, least squares, and singular value decom- We propose and analyse a new class of methods, so called positions. This talk considers several sparse optimization tensor Krylov subspace methods, which exploit this fact problems of current interest, including compressed sensing and attain a computational cost that grows linearly with and matrix completion, and outlines several algorithms for d. each problem. The role of linear algebra methods in these algorithms is highlighted, and the talk considers whether Daniel Kressner alternative techniques may be more effective. ETH Zurich [email protected] Stephen J. Wright University of Wisconsin Christine Tobler Dept. of Computer Sciences Seminar for Applied Mathematics [email protected] ETH Zurich [email protected] MS54 Randomized Preprocessing in Tensor Problems MS54 P-factor Analysis for Nonlinear Tensor Mappings We develop and analyze the techniques of randomized pre- processing and show both theoretically and experimentally This work presents the main concept and results of the that it facilitates some most fundamental computations in p-regulariry theory (also known as p-factor tensor analy- numerical linear algebra such as the solution of general and sis of nonlinear mappings). This approach is based on the structured linear systems of equations and matrix eigen- construction of a tensor type operator (p-factor operator). solving. In the case of structured inputs the computational The main result of this theory gives a detailed description improvement is dramatic. The approach seems to be also of the structure of the zero set of an irregular nonlinear promising for multi-linear algebraic computations. mapping F(x). Applications include a new numerical p- factor method for solving essentially nonlinear problems Victor Pan and p-order optimality conditions for nonlinear optimiza- CUNY tion problems. Lehman v y [email protected] Alexey Tretyakov University of Podlasie, Poland Computer Center of RAS, Moscow MS54 [email protected] Quasi-Newton-Grassamann and Krylov-type Methods for Tensors Approximation MS55 In this talk we will describe two different methods for com- Preconditioners for 3D Modeling Problems Based puting low multilinear rank approximations of a given ten- on Generalized Schur Interpolation sor. The underlying structure of the tensor approximation problem involves a product of Grassmann manifolds, on We consider 3D modeling problems that lead to hierarchi- which the related objective function is defined on. The first cally multibanded matrices. As prototype example we use method, explicitly utilizing this fact, is a quasi-Newton al- the matrix obtained for the Poisson equation on a regular gorithm that operates on product of Grassmann manifolds. 3D grid with a 27 point stencil. Using lexicographic or- The quasi-Newton methods are based on BFGS and limited dering the matrix obtained is block tridiagonal, whereby memory BFGS updates. We will give a general description each sub-block is again block tridiagonal and each sub- of the algorithms and discuss optimality of the BFGS up- sub-block is a tridiagonal matrix. In the past no good pre- date on Grassmannians. In the second method, for tensor conditioner for such (often occurring) matrices has been approximation, we will generalize Krylov methods to ten- determined, to the best of our knowledge. We propose sors. For large and sparse matrix problems Krylov based a new method based on hierarchical Schur complementa- methods are, in many cases, the only practically useful tion, whereby each Schur complement is approximated by methods. As sparse tensors occur frequently in informa- a low complexity representation, this time using an original tion sciences, tensor-Krylov based methods will enable the Schur matrix interpolation method. computation of low rank Tucker models of huge tensors. Patrick Dewilde Berkant Savas Institute for Advanced Study, Technische Universit¨at University of Texas at Austin LA09 Abstracts 91

M¨unchen systems theory. and CAS Section, EEMCS, TU Delft [email protected] Vadim Olshevsky Dept. of Mathematics U. of Connecticut, Storrs. MS55 [email protected] Communication Complexity for Parallel and Se- quential Eigenvalue/Singular Value Algorithms

The flops complexity of an algorithm does not always deter- mine how fast the algorithm will run in practice; communi- cation costs associated to algorithms can make a world of difference. Roughly speaking, one must take into account all movement of data between fast memory, where actual computations occur, and slow memory, where the data is stored for later access. Ideally, desirable algorithms mini- mize all three following key components: flops, bandwidth (proportional to the total number of data words moved), and latency (proportional to the total number of messages passed to and from layers of memory). Recently, there has been a lot of work on finding O(n3) algorithms which achieve optimal communication costs (see the work of Dem- mel, Grigori, Hoemmen, and Langou, and also that of Bal- lard, Demmel, Holtz, and Schwartz). This talk will give a communication-optimizing algorithm (in the big-Oh sense) for computing eigenvalue, generalized eigenvalue, and sin- gular value decompositions. The algorithm is based on an earlier algorithm by Bai, Demmel, and Gu, and it achieves communication optimality by using a randomized rank- revealing decomposition introduced by Demmel, Dumitriu, and Holtz. In addition to being O(n3) and big-Oh optimal, this algorithm has the benefit that, with high probability, it is stable. This is joint work with Grey Ballard and James Demmel. Ioana Dumitriu University of Washington, Seattle [email protected]

MS55 A Fast Implicit Qr for a Class of Structured Matrices

The talk presents a fast adaptation of the implicit QR eigenvalue algorithm for certain classes of rank structured matrices including companion matrices. This is a joint work with D. Bini, P. Boito, Y. Eidelman and I. Gohberg.

Luca Gemignani Univeristy of Pisa (Italy) Department of Mathematics [email protected]

MS55 Signal Flow Graph Approach to Inversion of Qua- siseparable Vandermonde Matrices

We use the language of signal flow graph representation of digital filter structures to solve three purely mathemat- ical problems, including fast inversion of certain polyno- mialVandermonde matrices, deriving an analogue of the Horner and Clenshaw rules for polynomial evaluation in a (H,m)quasiseparable basis, and computation of eigenvec- tors of (H,m) quasiseparable classes of matrices. While algebraic derivations are possible, using elementary op- erations (specifically, flow reversal) on signal flow graphs provides a unified derivation, and reveals connections with