DOCSLIB.ORG

Math 660-Lecture 15: Spectral Methods: Fourier Pseduo-Spectral Method

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 660-Lecture 15: Spectral Methods: Fourier Pseduo-Spectral Method Math 660-Lecture 15: Spectral methods: Fourier pseduo-spectral method The topics here will be in the book `Spectral methods in Matlab' (Chap- ter 3); Another reference book is `Spectral methods in chemistry and physics.' 1 Spectral methods v.s. pseudo-spectral methods 1.1 Spectral method A spectral method is to represent the solution of a differential equation in terms of a basis of some vector space and then reduce the differential equations to an ODE system for the coefficients. In this sense, the (contin- uous) Fourier Transform and the Finite Element methods are all spectral methods. In particular, suppose we have some equation T (u) = f(x) where T is some temporal-spatial operator for the differential equation. Then, under the basis we have the error: X X R(x) = T ( ci(t)φi(x)) − f~iφi i i Pick test functions fχjg, which is the dual basis of fφi(x)g (if the space is a Hilbert space like in FEM, these two are identical) and we require hχj;Ri = 0: This then gives a system of equations for cj. • Advantages: Reduce differential operators to multiplications, and solv- ing ODEs might be easier. • Disadvatange: The multiplication in the spatial variable becomes con- volution. Then, we may have a big matrix-vector multiplication. 1.2 Pseudo-spectral method Here, represent the solution in terms of the basis but impose the equa- tion only at discrete points. As we can see, the pseudo-spectral method in some sense is a spectral method in discrete space. There are some typical pseudo-spectral methods 1 • Fourier Pseudo-spectral method with trigonometric basis • Chebyshev pseudo-spectral method with Chebyshev polynomial basis • Collocation methods with polynomial basis (especially for ODEs solvers); (Some people also call the pseudo-spectral method the collocation method.) For discrete points, we often choose χj = δ(x − xj), so that we require R(xj) = 0: The pseudo-spectral method may have efficient algorithms, which makes solving PDEs fast. For example, to compute a(x)ux(x; t), using spectral method where x is continuous, we have an infinite convolution. Even if we truncate, we have a matrix multiplied with a vector. However, if we use the Fourier pseudo-spectral method, we can make use of the FFT and IFFT to change the function back into x space and perform the multiplication in x space. This will save a lot of time. In this course, we will only cover Fourier pseudo-spectral method. You can read the Chebyshev yourself, which is also useful. 2 Basic theories for Discrete Fourier Transform (DFT) Consider N (an even number) points v1; v2; : : : ; vN . We define the DFT as N X −ikxn v^k = e vn; k 2 Z n=1 2π where xn = N n. This means that we regard v as a function on (0; 2π] and xn is a sample point on the interval. • Note that the definition here is a little different from the book. We use this because it's more common. Also in Matlab, the summation is PN −ikxn−1 n=1 e vn, which will not cause a trouble, since we can imagine that we are doing a shifted sequence or vn is associated with xn−1 instead of xn. PN−1 −ikxn • Other people also use the definition as n=0 e vn. Of course, this is equivalent to ours if we regard xN as x0. 2 • Different definitions (shifting the sequence) may result in a uniform factor phase for the DFT. However, as long as we have consistent inverse formula, it is just fine. Properties • v^k =v ^k+mN . In Matlab, the output is for k = 0; 1; 2;:::;N − 1 or k = 0; 1; 2; : : : ; N=2; −N=2 + 1;:::; −1. In other words, fft(v) gives a vector w such that w(1) =v ^0; : : : ; w(N) =v ^N−1. ¯ • If v is real valued, thenv ^k = v^−k, where the bar means complex conjugate. If follows thatv ^N=2 is real. • Define the discrete convolution: N X wn = (u ∗ v)n = ujvn−j j=1 where vn−j = vN+n−j if n − j < 0. Then, w^k =u ^kv^k: • The discrete Parseval equality holds: N=2 X 1 X u v = u^ v^ n n N k k n k=−N=2+1 Let us prove the last two. Proof. For the discrete convolution, we have N N N N X −ikxn X X −ikxj X −ik(xn−xj ) w^k = e ujvn−j = uje e vn−j n=1 j=1 j=1 n=1 By the cyclic symmetry, we have the result. We have N=2 X X X −ikxm X ikxn X X ik(xn−xm) u^kv^k = ( ume )( vne ) = umvn e k=−N=2+1 k m n m;n k P The sum on the right hand side is Nδm;n, so we have N n unvn. 3 Remark 1. Some people want to define N 1 X v^ = e−ikxn v ; k N n n=1 then the Parseval is N=2 2π X X ( ) u v = 2π u^ v^ N n n k k n k=−N=2+1 Inverse DFT The inverse DFT is given by N−1 N=2 1 X 1 X v = eikxn v^ = eikxn v^ : n N k N k k=0 k=−N=2+1 1 PN−1 ikxn−1 (In Matlab convention, we should have vn = N k=0 e v^k, as vn is associated with xn−1.) Proof. First of all, e−ik(xn−xj ) = e−ikxn−j , X X −ikxn X X −ikxn w^k = ujvn−je = uj vn−je n j j n X −ikxj X −ikxn−j X −ikxj = uje vn−je = uje v^k =u ^kv^k: j n j For the second, N−1 N−1 1 X 1 X X 1 X X eikxn v^ = eikxn v e−ikxm = v eik(xn−xm) N k N m N m k=0 k=0 m m k P ik(xn−xm) P ik(n−m)h Clearly, k e = k e . If n = m, the sum is N. If n 6= m, the sum equals zero by the geometric sum since [eik(n−m)h]N = 1. Hence, 1 X X 1 X v eik(xn−xm) = v Nδ = v : N m N m n−m n m k m 4 3 FFT: fast algorithms to compute DFT The most frequently used one is the Cooley-Tukey algorithm (the algo- rithm was independently discovered also by Gauss). The idea is based on the simple fact. Let N be even, then N N=2 N=2 X −ikn 2π X −ikm 2π ik 2π X −ikm 2π une N = u2me (N=2) + e N u2m−1e (N=2) n=1 m=1 m=1 The DFT of an array of size N is reduced to 2 DFT of arrays with size N=2 plus extra N operations. By this way, the whole complexity is O(N log N). 4 Fourier Differentiation For general x, we can define a function N=2 1 X f(x) = eikxv^ ; N k k=−N=2+1 (m) (m) and then we use f (xj) to approximate the derivatives v (xj). 4.1 Algorithm for Fourier differentiation The above idea generates the following Fourier differentiation: • Given v = (v1; : : : ; vN ), computev ^. m • Definew ^k = (ik) v^k, for k = −N=2+1; : : : ; N=2 (or k = 0; : : : ; N=2; −N=2+ 1;:::; −1 in Matlab.) • Compute the inverse DFT (inverse FFT) and get wk, the real part of which is the approximation of the derivative. we take real part because numerical error may introduce some complex val- ues. Corollary 1. For the Fourier differentiation, without taking the real part, the discrete integration by parts by Fourier differentiation holds. This is a corollary of the Paserval equality. 1 PN−1 ikx Exercise: Will it matter if we use f(x) = N k=0 e v^k to compute the derivatives? If it matters, which one is better? Choose a periodic smooth 5 function and code up to check out. Can you explain this? (Hint: Use the Aliasing formula in the next lecture.) However, the k = N=2 mode is a little bit strange. Assume v is a real array. Then, v^N=2 = v^−N=2 = v^N=2: This meansv ^N=2 is real. This mode will contribute derivative iN i N x v^ e 2 j N=2 2 Clearly, this part is imaginary at the given grid points. 6.
Load more

Recommended publications
  • On the Linear Stability of Plane Couette Flow for an Oldroyd-B Fluid and Its
    J. Non-Newtonian Fluid Mech. 127 (2005) 169–190 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation Raz Kupferman Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel Received 14 December 2004; received in revised form 10 February 2005; accepted 7 March 2005 Abstract It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non- modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their “passive" transport by shearing streamlines generates large cross- stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings. © 2005 Elsevier B.V. All rights reserved. Keywords: Couette flow; Oldroyd-B model; Linear stability; Generalized functions; Non-normal operators; Stress diffusion 1. Introduction divided into two main groups: (i) numerical solutions of the boundary value problem defined by the linearized system The linear stability of Couette flow for viscoelastic fluids (e.g. [2,3]) and (ii) stability analysis of numerical methods is a classical problem whose study originated with the pi- for time-dependent flows, with the objective of understand- oneering work of Gorodtsov and Leonov (GL) back in the ing how spurious solutions emerge in computations.
    [Show full text]
  • A Highly Efficient Spectral-Galerkin Method Based on Tensor Product for Fourth-Order Steklov Equation with Boundary Eigenvalue
    An et al. Journal of Inequalities and Applications (2016)2016:211 DOI 10.1186/s13660-016-1158-1 R E S E A R C H Open Access A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue Jing An1,HaiBi1 and Zhendong Luo2* *Correspondence: [email protected] Abstract 2School of Mathematics and Physics, North China Electric Power In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order University, No. 2, Bei Nong Road, Steklov equation with boundary eigenvalue. By making use of the spectral theory of Changping District, Beijing, 102206, compact operators and the error formulas of projective operators, we first obtain the China Full list of author information is error estimates of approximative eigenvalues and eigenfunctions. Then we build a 1 ∩ 2 available at the end of the article suitable set of basis functions included in H0() H () and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results. MSC: 65N35; 65N30 Keywords: fourth-order Steklov equation with boundary eigenvalue; spectral-Galerkin method; error estimates; tensor product 1 Introduction Increasing attention has recently been paid to numerical approximations for Steklov equa- tions with boundary eigenvalue, arising in fluid mechanics, electromagnetism, etc. (see, e.g.,[–]). However, most existing work usually treated the second-order Steklov equa- tions with boundary eigenvalue and there are relatively few articles treating fourth-order ones. The fourth-order Steklov equations with boundary eigenvalue also have been used in both mathematics and physics, for example, the main eigenvalues play a very key role in the positivity-preserving properties for the biharmonic-operator under the border conditions w = w – χwν =on∂ (see []).
    [Show full text]
  • Chebyshev and Fourier Spectral Methods 2000
    Chebyshev and Fourier Spectral Methods Second Edition John P. Boyd University of Michigan Ann Arbor, Michigan 48109-2143 email: [email protected] http://www-personal.engin.umich.edu/jpboyd/ 2000 DOVER Publications, Inc. 31 East 2nd Street Mineola, New York 11501 1 Dedication To Marilyn, Ian, and Emma “A computation is a temptation that should be resisted as long as possible.” — J. P. Boyd, paraphrasing T. S. Eliot i Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series .................................... 20 2.3 Orders of Convergence ............................... 25 2.4 Convergence Order ................................. 27 2.5 Assumption of Equal Errors ...........................
    [Show full text]
  • Isospectral Families of High-Order Systems∗
    Isospectral Families of High-order Systems∗ Peter Lancaster† Department of Mathematics and Statistics University of Calgary Calgary T2N 1N4, Alberta, Canada and Uwe Prells School of Mechanical, Materials, Manufacturing Engineering & Management University of Nottingham Nottingham NG7 2RD United Kingdom November 28, 2006 Keywords: Matrix Polynomials, Isospectral Systems, Structure Preserving Transformations, Palindromic Symmetries. Abstract Earlier work of the authors concerning the generation of isospectral families of second or- der (vibrating) systems is generalized to higher-order systems (with no spectrum at infinity). Results and techniques are developed first for systems without symmetries, then with Hermi- tian symmetry and, finally, with palindromic symmetry. The construction of linearizations which retain such symmetries is discussed. In both cases, the notion of strictly isospectral families of systems is introduced - implying that properties of both the spectrum and the sign-characteristic are preserved. Open questions remain in the case of strictly isospectral families of palindromic systems. Intimate connections between Hermitian and unitary sys- tems are discussed in an Appendix. ∗The authors are grateful to an anonymous reviewer for perceptive comments which led to significant improve- ments in exposition. †The research of Peter Lancaster was partly supported by a grant from the Natural Sciences and Engineering Research Council of Canada 1 1 Introduction ` By a high-order system we mean an n × n matrix polynomial L(λ) = A`λ + ··· + A1λ + A0 n×n with coefficients in C . Such a polynomial with A` 6= 0 is said to have degree `, (sometimes referred to as “order” `) and “high-order” implies that ` ≥ 2. It will be assumed throughout that the leading coefficient A` is nonsingular.
    [Show full text]
  • Introduction to Spectral Methods
    Introduction to spectral methods Eric Gourgoulhon Laboratoire de l’Univers et de ses Th´eories (LUTH) CNRS / Observatoire de Paris Meudon, France Based on a collaboration with Silvano Bonazzola, Philippe Grandcl´ement,Jean-Alain Marck & J´eromeNovak [email protected] http://www.luth.obspm.fr 4th EU Network Meeting, Palma de Mallorca, Sept. 2002 1 Plan 1. Basic principles 2. Legendre and Chebyshev expansions 3. An illustrative example 4. Spectral methods in numerical relativity 2 1 Basic principles 3 Solving a partial differential equation Consider the PDE with boundary condition Lu(x) = s(x); x 2 U ½ IRd (1) Bu(y) = 0; y 2 @U; (2) where L and B are linear differential operators. Question: What is a numerical solution of (1)-(2)? Answer: It is a function u¯ which satisfies (2) and makes the residual R := Lu¯ ¡ s small. 4 What do you mean by “small” ? Answer in the framework of Method of Weighted Residuals (MWR): Search for solutions u¯ in a finite-dimensional sub-space PN of some Hilbert space W (typically a L2 space). Expansion functions = trial functions : basis of PN : (Á0;:::;ÁN ) XN u¯ is expanded in terms of the trial functions: u¯(x) = u˜n Án(x) n=0 Test functions : family of functions (Â0;:::;ÂN ) to define the smallness of the residual R, by means of the Hilbert space scalar product: 8 n 2 f0;:::;Ng; (Ân;R) = 0 5 Various numerical methods Classification according to the trial functions Án: Finite difference: trial functions = overlapping local polynomials of low order Finite element: trial functions = local smooth functions (polynomial
    [Show full text]
  • Validated and Numerically Efficient Chebyshev Spectral Methods For
    Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations Florent Bréhard, Nicolas Brisebarre, Mioara Joldes To cite this version: Florent Bréhard, Nicolas Brisebarre, Mioara Joldes. Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations. ACM Transactions on Mathematical Soft- ware, Association for Computing Machinery, 2018, 44 (4), pp.44:1-44:42. 10.1145/3208103. hal- 01526272v3 HAL Id: hal-01526272 https://hal.archives-ouvertes.fr/hal-01526272v3 Submitted on 5 Jun 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations Florent Bréhard∗ Nicolas Brisebarrey Mioara Joldeşz Abstract In this work we develop a validated numerics method for the solu- tion of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for nu- merically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity anal- ysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map.
    [Show full text]
  • Applying an Iterative Method Numerically to Solve N × N Matrix
    Applying an iterative method numerically to solve n × n rsta.royalsocietypublishing.org matrix Wiener–Hopf equations with exponential factors Research Matthew J. Priddin1, Anastasia V. Kisil2 Article submitted to journal and Lorna J. Ayton1 1DAMTP, University of Cambridge, Cambridge, CB3 Subject Areas: 0WA, UK 2Department of Mathematics, The University wave motion, applied mathematics, of Manchester, Manchester, M13 9PL, UK differential equations Keywords: This paper presents a generalisation of a recent iterative approach to solving a class of 2 2 Wiener–Hopf equations, × matrix Wiener–Hopf equations involving exponential Riemann-Hilbert problem, iterative factors. We extend the method to square matrices of methods, scattering, n-part arbitrary dimension n, as arise in mixed boundary boundaries value problems with n junctions. To demonstrate the method we consider the classical problem of Author for correspondence: scattering a plane wave by a set of collinear plates. The results are compared to other known Matthew Priddin methods. We describe an effective implementation e-mail: [email protected] using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. 1. Introduction The Wiener–Hopf technique [1] provides a powerful tool to approach varied problems such as the solution of integral equations and random processes. Applications may so be found in such diverse fields as aeroacoustics [2], metamaterials [3], geophysics [4], crack propagation [5] and financial mathematics [6]. The mathematical problem underlying the Wiener– Hopf technique may be simply stated: find functions that are analytic in adjacent (or overlapping) regions of a complex manifold that satisfy a prescribed jump condition at their common boundary.
    [Show full text]
  • Msc in Applied Mathematics
    MSc in Applied Mathematics Title: Mathematical and Technological Aspects of Quantum Information Processing Author: Jordi Tura i Brugués Advisor: Sebastià Xambó i Descamps Co-Advisor: Antonio Acín dal Maschio Department: Departament de Matemàtica Aplicada II Academic year: 2010-2011 Universitat Polit`ecnicade Catalunya Facultat de Matem`atiquesi Estad´ıstica Master's Degree Thesis Mathematical and Technological Aspects of Quantum Information Processing Jordi Tura i Brugu´es Advisor: Sebasti`aXamb´oi Descamps Co-Advisor: Antonio Ac´ındal Maschio Departament de Matem`aticaAplicada II Abstract Keywords: Quantum, information, processing, computation, coding, cryptography, entanglement, field theory, entropy, key distribution, protocols. Since its genesis, quantum mechanics has proved to be a very accurate model for predicting the behavior of the world below the nanoscale. However, crucial breakthroughs in technology were needed in order to be able to effectively access and manipulate such small magnitudes. During the last twenty years, the field of quantum information processing has experienced a growing interest, in its many variants, both theoretically and practically. Despite being still at a very basic stage, expectations are high. The uniqueness of quantum phenomena (superposition of states, creation of entanglement, etc.) have no classical analogue and allow novelties such as another paradigm of computation, more secure com- munications, quantum teleportation, quantum dense coding, etc. which are presented and analyzed here. The aim of this Thesis is to present in a unified way the main mathematical methods used in quantum information processing, as well as the state of the art of their corresponding technological implementa- tions. Our contributions are based in making a self-contained presentation; seeking completeness, that is, treating the most relevant fields of research involved, focusing on their relations; and picking the most relevant, insightful references, given the quantity of literature produced in this field.
    [Show full text]
  • Computing Spectral Measures and Spectral Types
    Commun. Math. Phys. 384, 433–501 (2021) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-04072-4 Mathematical Physics Computing Spectral Measures and Spectral Types Matthew J. Colbrook Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK. E-mail: [email protected] Received: 26 February 2020 / Accepted: 16 March 2021 Published online: 11 April 2021 – © The Author(s) 2021 Abstract: Spectral measures arise in numerous applications such as quantum mechan- ics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular contin- uous parts) often characterise relevant physical properties such as the long-time dy- namics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific exam- ples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algo- rithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each col- umn decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on L2(Rd ).
    [Show full text]
  • Analysis of Parametric Models — Linear Methods and Approximations
    Analysis of parametric models — linear methods and approximations∗ Hermann G. Matthies* and Roger Ohayono *Institute of Scientific Computing Technische Universität Braunschweig 38092 Braunschweig, Germany e-mail: [email protected] oLaboratoire de Mécanique des Structures et des Systèmes Couplés Conservatoire National des Arts et Métiers (CNAM) 75141 Paris Cedex 03, France Abstract Parametric models in vector spaces are shown to possess an associated linear map. This linear operator leads directly to reproducing kernel Hilbert spaces and affine- / linear- representations in terms of tensor products. From the associated linear map analogues of covariance or rather correlation operators can be formed. The associated linear map in fact provides a factorisation of the correlation. Its spectral decomposition, and the associated Karhunen-Loève- or proper orthogonal decomposition in a tensor product follow directly. It is shown that all factorisations of a certain class are unitarily equivalent, as well as that every factorisation induces a different representation, and vice versa. A completely equivalent spectral and factorisation analysis can be carried out in kernel space. The relevance of these abstract constructions is shown on a number of mostly familiar examples, thus unifying many such constructions under one the- oretical umbrella. From the factorisation one obtains tensor representations, which may be cascaded, leading to tensors of higher degree. When carried over to a dis- arXiv:1806.01101v2 [math.NA] 17 Jun 2018 cretised level in the form of a model order reduction, such factorisations allow very sparse low-rank approximations which lead to very efficient computations especially in high dimensions. Key words: parametric models, reproducing kernel Hilbert space, correlation, factorisation, spectral decomposition, representation MSC subject classifications: 35B30, 37M99, 41A05, 41A45, 41A63, 60G20, 60G60, 65J99, 93A30 ∗Partly supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB 880 and SPP 1886.
    [Show full text]
  • Spectral Algorithms
    Spectral Algorithms Ravindran Kannan Santosh Vempala August, 2009 ii Summary. Spectral methods refer to the use of eigenvalues, eigenvectors, sin- gular values and singular vectors. They are widely used in Engineering, Ap- plied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to \discrete" as well \continuous" problems. This book describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. In the first part of the book, we present applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A fea- ture of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on \sampling on the fly” from massive matrices. Good estimates of singular values and low rank ap- proximations of the whole matrix can be provably derived from a sample. Our main emphasis in the second part of the book is to present these sampling meth- ods with rigorous error bounds. We also present recent extensions of spectral methods from matrices to tensors and their applications to some combinatorial optimization problems. Contents I Applications 1 1 The Best-Fit Subspace 3 1.1 Singular Value Decomposition . .4 1.2 Algorithms for computing the SVD . .8 1.3 The k-variance problem . .8 1.4 Discussion . 11 2 Mixture Models 13 2.1 Probabilistic separation . 14 2.2 Geometric separation . 14 2.3 Spectral Projection .
    [Show full text]
  • Some Spectral Ideas Applied to Finance and to Self-Similar and Long-Range Dependent Processes
    SOME SPECTRAL IDEAS APPLIED TO FINANCE AND TO SELF-SIMILAR AND LONG-RANGE DEPENDENT PROCESSES A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Anna Srapionyan May 2019 ⃝c 2019 Anna Srapionyan ALL RIGHTS RESERVED SOME SPECTRAL IDEAS APPLIED TO FINANCE AND TO SELF-SIMILAR AND LONG-RANGE DEPENDENT PROCESSES Anna Srapionyan, Ph.D. Cornell University 2019 This dissertation consists of four parts. The aim of the first part is to present original trans- formations on tractable Markov processes (or equivalently, on their semigroup) in order to make the discounted transformed process a martingale, while keeping its tractability. We refer to such procedures as risk-neutral pricing techniques. To achieve our goal, we resort to the concept of intertwining relationships between Markov semigroups that en- ables us, on the one hand to characterize a risk-neutral measure and on the other hand to preserve the tractability and flexibility of the models, two attractive features of models in mathematical finance. To illustrate the usefulness of our approach, we proceed by ap- plying this risk-neutral pricing techniques to some classes of Markov processes that have been advocated in the literature as substantial models. In the second part, we introduce spectral projections correlation functions of a stochastic process which are expressed in terms of the non-orthogonal projections into eigenspaces of the expectation operator of the process and its adjoint. We obtain closed- form expressions of these functions involving eigenvalues, the condition number and/or the angle between the projections, along with their large time asymptotic behavior for three important classes of processes: general Markov processes, Markov processes sub- ordinated in the sense of Bochner and non-Markovian processes which are obtained by time-changing a Markov process with an inverse of a subordinator.
    [Show full text]