On Computing Inverse Entries of a Sparse Matrix in an Out-Of-Core Environment Patrick R
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On Multigrid Methods for Solving Electromagnetic Scattering Problems
On Multigrid Methods for Solving Electromagnetic Scattering Problems Dissertation zur Erlangung des akademischen Grades eines Doktor der Ingenieurwissenschaften (Dr.-Ing.) der Technischen Fakultat¨ der Christian-Albrechts-Universitat¨ zu Kiel vorgelegt von Simona Gheorghe 2005 1. Gutachter: Prof. Dr.-Ing. L. Klinkenbusch 2. Gutachter: Prof. Dr. U. van Rienen Datum der mundliche¨ Prufung:¨ 20. Jan. 2006 Contents 1 Introductory remarks 3 1.1 General introduction . 3 1.2 Maxwell’s equations . 6 1.3 Boundary conditions . 7 1.3.1 Sommerfeld’s radiation condition . 9 1.4 Scattering problem (Model Problem I) . 10 1.5 Discontinuity in a parallel-plate waveguide (Model Problem II) . 11 1.6 Absorbing-boundary conditions . 12 1.6.1 Global radiation conditions . 13 1.6.2 Local radiation conditions . 18 1.7 Summary . 19 2 Coupling of FEM-BEM 21 2.1 Introduction . 21 2.2 Finite element formulation . 21 2.2.1 Discretization . 26 2.3 Boundary-element formulation . 28 3 4 CONTENTS 2.4 Coupling . 32 3 Iterative solvers for sparse matrices 35 3.1 Introduction . 35 3.2 Classical iterative methods . 36 3.3 Krylov subspace methods . 37 3.3.1 General projection methods . 37 3.3.2 Krylov subspace methods . 39 3.4 Preconditioning . 40 3.4.1 Matrix-based preconditioners . 41 3.4.2 Operator-based preconditioners . 42 3.5 Multigrid . 43 3.5.1 Full Multigrid . 47 4 Numerical results 49 4.1 Coupling between FEM and local/global boundary conditions . 49 4.1.1 Model problem I . 50 4.1.2 Model problem II . 63 4.2 Multigrid . 64 4.2.1 Theoretical considerations regarding the classical multi- grid behavior in the case of an indefinite problem . -
Sparse Matrices and Iterative Methods
Iterative Methods Sparsity Sparse Matrices and Iterative Methods K. Cooper1 1Department of Mathematics Washington State University 2018 Cooper Washington State University Introduction Iterative Methods Sparsity Iterative Methods Consider the problem of solving Ax = b, where A is n × n. Why would we use an iterative method? I Avoid direct decomposition (LU, QR, Cholesky) I Replace with iterated matrix multiplication 3 I LU is O(n ) flops. 2 I . matrix-vector multiplication is O(n )... I so if we can get convergence in e.g. log(n), iteration might be faster. Cooper Washington State University Introduction Iterative Methods Sparsity Jacobi, GS, SOR Some old methods: I Jacobi is easily parallelized. I . but converges extremely slowly. I Gauss-Seidel/SOR converge faster. I . but cannot be effectively parallelized. I Only Jacobi really takes advantage of sparsity. Cooper Washington State University Introduction Iterative Methods Sparsity Sparsity When a matrix is sparse (many more zero entries than nonzero), then typically the number of nonzero entries is O(n), so matrix-vector multiplication becomes an O(n) operation. This makes iterative methods very attractive. It does not help direct solves as much because of the problem of fill-in, but we note that there are specialized solvers to minimize fill-in. Cooper Washington State University Introduction Iterative Methods Sparsity Krylov Subspace Methods A class of methods that converge in n iterations (in exact arithmetic). We hope that they arrive at a solution that is “close enough” in fewer iterations. Often these work much better than the classic methods. They are more readily parallelized, and take full advantage of sparsity. -
Scalable Stochastic Kriging with Markovian Covariances
Scalable Stochastic Kriging with Markovian Covariances Liang Ding and Xiaowei Zhang∗ Department of Industrial Engineering and Decision Analytics The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Abstract Stochastic kriging is a popular technique for simulation metamodeling due to its flexibility and analytical tractability. Its computational bottleneck is the inversion of a covariance matrix, which takes O(n3) time in general and becomes prohibitive for large n, where n is the number of design points. Moreover, the covariance matrix is often ill-conditioned for large n, and thus the inversion is prone to numerical instability, resulting in erroneous parameter estimation and prediction. These two numerical issues preclude the use of stochastic kriging at a large scale. This paper presents a novel approach to address them. We construct a class of covariance functions, called Markovian covariance functions (MCFs), which have two properties: (i) the associated covariance matrices can be inverted analytically, and (ii) the inverse matrices are sparse. With the use of MCFs, the inversion-related computational time is reduced to O(n2) in general, and can be further reduced by orders of magnitude with additional assumptions on the simulation errors and design points. The analytical invertibility also enhance the numerical stability dramatically. The key in our approach is that we identify a general functional form of covariance functions that can induce sparsity in the corresponding inverse matrices. We also establish a connection between MCFs and linear ordinary differential equations. Such a connection provides a flexible, principled approach to constructing a wide class of MCFs. Extensive numerical experiments demonstrate that stochastic kriging with MCFs can handle large-scale problems in an both computationally efficient and numerically stable manner. -
Chapter 7 Iterative Methods for Large Sparse Linear Systems
Chapter 7 Iterative methods for large sparse linear systems In this chapter we revisit the problem of solving linear systems of equations, but now in the context of large sparse systems. The price to pay for the direct methods based on matrix factorization is that the factors of a sparse matrix may not be sparse, so that for large sparse systems the memory cost make direct methods too expensive, in memory and in execution time. Instead we introduce iterative methods, for which matrix sparsity is exploited to develop fast algorithms with a low memory footprint. 7.1 Sparse matrix algebra Large sparse matrices We say that the matrix A Rn is large if n is large, and that A is sparse if most of the elements are2 zero. If a matrix is not sparse, we say that the matrix is dense. Whereas for a dense matrix the number of nonzero elements is (n2), for a sparse matrix it is only (n), which has obvious implicationsO for the memory footprint and efficiencyO for algorithms that exploit the sparsity of a matrix. AdiagonalmatrixisasparsematrixA =(aij), for which aij =0for all i = j,andadiagonalmatrixcanbegeneralizedtoabanded matrix, 6 for which there exists a number p,thebandwidth,suchthataij =0forall i<j p or i>j+ p.Forexample,atridiagonal matrix A is a banded − 59 CHAPTER 7. ITERATIVE METHODS FOR LARGE SPARSE 60 LINEAR SYSTEMS matrix with p =1, xx0000 xxx000 20 xxx003 A = , (7.1) 600xxx07 6 7 6000xxx7 6 7 60000xx7 6 7 where x represents a nonzero4 element. 5 Compressed row storage The compressed row storage (CRS) format is a data structure for efficient represention of a sparse matrix by three arrays, containing the nonzero values, the respective column indices, and the extents of the rows. -
A Parallel Solver for Graph Laplacians
A Parallel Solver for Graph Laplacians Tristan Konolige Jed Brown University of Colorado Boulder University of Colorado Boulder [email protected] ABSTRACT low energy components while maintaining coarse grid sparsity. We Problems from graph drawing, spectral clustering, network ow also develop a parallel algorithm for nding and eliminating low and graph partitioning can all be expressed in terms of graph Lapla- degree vertices, a technique introduced for a sequential multigrid cian matrices. ere are a variety of practical approaches to solving algorithm by Livne and Brandt [22], that is important for irregular these problems in serial. However, as problem sizes increase and graphs. single core speeds stagnate, parallelism is essential to solve such While matrices can be distributed using a vertex partition (a problems quickly. We present an unsmoothed aggregation multi- 1D/row distribution) for PDE problems, this leads to unacceptable grid method for solving graph Laplacians in a distributed memory load imbalance for irregular graphs. We represent matrices using a seing. We introduce new parallel aggregation and low degree elim- 2D distribution (a partition of the edges) that maintains load balance ination algorithms targeted specically at irregular degree graphs. for parallel scaling [11]. Many algorithms that are practical for 1D ese algorithms are expressed in terms of sparse matrix-vector distributions are not feasible for more general distributions. Our products using generalized sum and product operations. is for- new parallel algorithms are distribution agnostic. mulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which 2 BACKGROUND is necessary for parallel scalability. -
MUMPS Users' Guide
MUltifrontal Massively Parallel Solver (MUMPS 5.4.1) Users’ guide * August 3, 2021 Abstract This document describes the Fortran 95 and C user interfaces to MUMPS5.4.1, a software package to solve sparse systems of linear equations, with many features. For some classes of problems, the complexity of the factorization and the memory footprint of MUMPS can be reduced thanks to the Block Low-Rank (BLR) feature. We describe in detail the data structures, parameters, calling sequences, and error diagnostics. Basic example programs using MUMPS are also provided. Users are allowed to save to disk MUMPS internal data before or after any of the main steps of MUMPS (analysis, factorization, solve) and then restart. *Information on how to obtain updated copies of MUMPS can be obtained from the Web page http://mumps-solver.org/ 1 Contents 1 Introduction 5 2 Notes for users of other versions of MUMPS 6 2.1 ChangeLog........................................7 2.2 Binary compatibility....................................7 2.3 Upgrading between minor releases............................7 2.4 Upgrading from MUMPS 5.3.5 to MUMPS 5.4.1 ...................8 2.5 Upgrading from MUMPS 5.2.1 to MUMPS 5.3.5 ...................8 2.6 Upgrading from MUMPS 5.1.2 to MUMPS 5.2.1 ...................8 2.7 Upgrading from MUMPS 5.0.2 to MUMPS 5.1.2 ...................9 2.7.1 Changes on installation issues...........................9 2.7.2 Selective 64-bit integer feature..........................9 2.8 Upgrading from MUMPS 4.10.0 to MUMPS 5.0.2 .................. 10 2.8.1 Interface with the Metis and ParMetis orderings................ -
Solving Linear Systems: Iterative Methods and Sparse Systems
Solving Linear Systems: Iterative Methods and Sparse Systems COS 323 Last time • Linear system: Ax = b • Singular and ill-conditioned systems • Gaussian Elimination: A general purpose method – Naïve Gauss (no pivoting) – Gauss with partial and full pivoting – Asymptotic analysis: O(n3) • Triangular systems and LU decomposition • Special matrices and algorithms: – Symmetric positive definite: Cholesky decomposition – Tridiagonal matrices • Singularity detection and condition numbers Today: Methods for large and sparse systems • Rank-one updating with Sherman-Morrison • Iterative refinement • Fixed-point and stationary methods – Introduction – Iterative refinement as a stationary method – Gauss-Seidel and Jacobi methods – Successive over-relaxation (SOR) • Solving a system as an optimization problem • Representing sparse systems Problems with large systems • Gaussian elimination, LU decomposition (factoring step) take O(n3) • Expensive for big systems! • Can get by more easily with special matrices – Cholesky decomposition: for symmetric positive definite A; still O(n3) but halves storage and operations – Band-diagonal: O(n) storage and operations • What if A is big? (And not diagonal?) Special Example: Cyclic Tridiagonal • Interesting extension: cyclic tridiagonal • Could derive yet another special case algorithm, but there’s a better way Updating Inverse • Suppose we have some fast way of finding A-1 for some matrix A • Now A changes in a special way: A* = A + uvT for some n×1 vectors u and v • Goal: find a fast way of computing (A*)-1 -
High Performance Selected Inversion Methods for Sparse Matrices
High performance selected inversion methods for sparse matrices Direct and stochastic approaches to selected inversion Doctoral Dissertation submitted to the Faculty of Informatics of the Università della Svizzera italiana in partial fulfillment of the requirements for the degree of Doctor of Philosophy presented by Fabio Verbosio under the supervision of Olaf Schenk February 2019 Dissertation Committee Illia Horenko Università della Svizzera italiana, Switzerland Igor Pivkin Università della Svizzera italiana, Switzerland Matthias Bollhöfer Technische Universität Braunschweig, Germany Laura Grigori INRIA Paris, France Dissertation accepted on 25 February 2019 Research Advisor PhD Program Director Olaf Schenk Walter Binder i I certify that except where due acknowledgement has been given, the work presented in this thesis is that of the author alone; the work has not been sub- mitted previously, in whole or in part, to qualify for any other academic award; and the content of the thesis is the result of work which has been carried out since the official commencement date of the approved research program. Fabio Verbosio Lugano, 25 February 2019 ii To my whole family. In its broadest sense. iii iv Le conoscenze matematiche sono proposizioni costruite dal nostro intelletto in modo da funzionare sempre come vere, o perché sono innate o perché la matematica è stata inventata prima delle altre scienze. E la biblioteca è stata costruita da una mente umana che pensa in modo matematico, perché senza matematica non fai labirinti. Umberto Eco, “Il nome della rosa” v vi Abstract The explicit evaluation of selected entries of the inverse of a given sparse ma- trix is an important process in various application fields and is gaining visibility in recent years. -
A Parallel Gauss-Seidel Algorithm for Sparse Power Systems Matrices
AParallel Gauss-Seidel Algorithm for Sparse Power Systems Matrices D. P. Ko ester, S. Ranka, and G. C. Fox Scho ol of Computer and Information Science and The Northeast Parallel Architectures Center (NPAC) Syracuse University Syracuse, NY 13244-4100 [email protected], [email protected], [email protected] A ComdensedVersion of this Paper was presentedatSuperComputing `94 NPACTechnical Rep ort | SCCS 630 4 April 1994 Abstract We describ e the implementation and p erformance of an ecient parallel Gauss-Seidel algorithm that has b een develop ed for irregular, sparse matrices from electrical p ower systems applications. Although, Gauss-Seidel algorithms are inherently sequential, by p erforming sp ecialized orderings on sparse matrices, it is p ossible to eliminate much of the data dep endencies caused by precedence in the calculations. A two-part matrix ordering technique has b een develop ed | rst to partition the matrix into blo ck-diagonal-b ordered form using diakoptic techniques and then to multi-color the data in the last diagonal blo ck using graph coloring techniques. The ordered matrices often have extensive parallelism, while maintaining the strict precedence relationships in the Gauss-Seidel algorithm. We present timing results for a parallel Gauss-Seidel solver implemented on the Thinking Machines CM-5 distributed memory multi-pro cessor. The algorithm presented here requires active message remote pro cedure calls in order to minimize communications overhead and obtain go o d relative sp eedup. The paradigm used with active messages greatly simpli ed the implementationof this sparse matrix algorithm. 1 Intro duction Wehavedevelop ed an ecient parallel Gauss-Seidel algorithm for irregular, sparse matrices from electrical p ower systems applications. -
Arxiv:2105.11220V1 [Cs.DC] 24 May 2021 the Motion of Particles Is Described by a Set of Convection-Diffusion-Reaction Equations
Towards Parallel CFD computation for the ADAPT framework Imad Kissamiy∗, Christophe Ceriny, Fayssal Benkhaldoun∗, and Gilles Scarella∗ Universit´ede Paris 13 y LIPN, ∗ LAGA, 99, avenue Jean-Baptiste Cl´ement, 93430 Villetaneuse, France [email protected] Abstract. In order to run Computational Fluid Dynamics (CFD) codes on large scale infrastructures, parallel computing has to be used because of the computational intensive nature of the problems. In this paper we investigate the ADAPT platform where we couple flow Partial Differ- ential Equations and a Poisson equation. This leads to a linear system which we solve using direct methods. The implementation deals with the MUMPS parallel multi-frontal direct solver and mesh partitioning meth- ods using METIS to improve the performance of the framework. We also investigate, in this paper, how the mesh partitioning methods are able to optimize the mesh cell distribution for the ADAPT solver. The ex- perience gained in this paper facilitates the move to the 3D version of ADAPT and the move to a Service Oriented view of ADAPT as future work. Keywords: Unstructured mesh, Mesh partitioning, Parallel direct solver, Multi-frontal method, MUMPS, METIS, Multi-physics, Multi-scale and Multilevel Algorithms. 1 Introduction and context of the work In the last recent decades CFD (Computational Fluid Dynamics) used to play an important role in industrial designs, environmental impact assessments and academic studies. The aim of our work is to fully parallelize an unstructured adaptive code for the simulation of 3D streamer propagation in cold plasmas. As a matter of fact, the initial sequential version of the Streamer code needs up to one month for running typical benchmarks. -
Decomposition Methods for Sparse Matrix Nearness Problems∗
SIAM J. MATRIX ANAL. APPL. c 2015 Society for Industrial and Applied Mathematics Vol. 36, No. 4, pp. 1691–1717 DECOMPOSITION METHODS FOR SPARSE MATRIX NEARNESS PROBLEMS∗ YIFAN SUN† AND LIEVEN VANDENBERGHE† Abstract. We discuss three types of sparse matrix nearness problems: given a sparse symmetric matrix, find the matrix with the same sparsity pattern that is closest to it in Frobenius norm and (1) is positive semidefinite, (2) has a positive semidefinite completion, or (3) has a Euclidean distance matrix completion. Several proximal splitting and decomposition algorithms for these problems are presented and their performance is compared on a set of test problems. A key feature of the methods is that they involve a series of projections on small dense positive semidefinite or Euclidean distance matrix cones, corresponding to the cliques in a triangulation of the sparsity graph. The methods discussed include the dual block coordinate ascent algorithm (or Dykstra’s method), the dual projected gradient and accelerated projected gradient algorithms, and a primal and a dual application of the Douglas–Rachford splitting algorithm. Key words. positive semidefinite completion, Euclidean distance matrices, chordal graphs, convex optimization, first order methods AMS subject classifications. 65F50, 90C22, 90C25 DOI. 10.1137/15M1011020 1. Introduction. We discuss matrix optimization problems of the form minimize X − C2 (1.1) F subject to X ∈S, where the variable X is a symmetric p × p matrix with a given sparsity pattern E, and C is a given symmetric matrix with the same sparsity pattern as X.Wefocus on the following three choices for the constraint set S. -
Multifrontal Massively Parallel Solver (MUMPS 4.8.4) Users' Guide
MUltifrontal Massively Parallel Solver (MUMPS 4.8.4) Users’ guide ∗ December 15, 2008 Abstract This document describes the Fortran 90 and C user interface to MUMPS 4.8.4 We describe in detail the data structures, parameters, calling sequences, and error diagnostics. Example programs using MUMPS are also given. A preliminary out-of-core functionality (with limited support) is included in this version. ∗Information on how to obtain updated copies of MUMPS can be obtained from the Web pages http://mumps.enseeiht.fr/ and http://graal.ens-lyon.fr/MUMPS/ 1 Contents 1 Introduction 4 2 Main functionalities of MUMPS 4.8.4 5 2.1 Inputmatrixstructure . ....... 5 2.2 Preprocessing ................................... .... 5 2.3 Post-processingfacilities . .......... 6 2.4 Solvingthetransposedsystem. ......... 6 2.5 Reduce/condense a problem on an interface (Schur complement and reduced/condensed right-handside) .................................... 7 2.6 Arithmeticversions .... ............. ............. ...... 8 2.7 Theworkinghostprocessor . ....... 8 2.8 Sequentialversion..... ............. ............. ...... 8 2.9 Sharedmemoryversion . ..... 8 2.10 Out-of-corefacility . ........ 8 3 Sequence in which routines are called 8 4 Input and output parameters 11 4.1 Versionnumber ................................... 11 4.2 Control of the three main phases: Analysis, Factorization,Solve ............. 11 4.3 Controlofparallelism . ....... 12 4.4 Matrixtype ...................................... 13 4.5 Centralized assembled matrix input: ICNTL(5)=0 and ICNTL(18)=0 .......... 13 4.6 Element matrix input: ICNTL(5)=1 and ICNTL(18)=0 . ............. 14 4.7 Distributed assembled matrix input: ICNTL(5)=0 and ICNTL(18)=0 .......... 14 4.8 Scaling.........................................6 . 15 4.9 Givenordering:ICNTL(7)=1 . ....... 15 4.10 Schur complement with reduced (or condensed) right-hand side: ICNTL(19) and ICNTL(26) .......................................