A Tuned Preconditioner for Inexact Inverse Iteration Applied to Hermitian Eigenvalue Problems
|
|
- Scott Riley
- 5 years ago
- Views:
Transcription
1 A Tuned Preconditioner for Applied to Eigenvalue Problems Department of Mathematical Sciences University of Bath, United Kingdom IWASEP VI May 22-25, 2006 Pennsylvania State University, University Park Joint work with: Alastair Spence
2
3
4 Problem and Inverse Find an eigenvalue and eigenvector of a positive definite A: Ax = λx, Inverse : A large, sparse. (A σi)y = x Inverse iteration with preconditioned iterative
5
6 for i = 1 to i max do choose τ (i), σ (i) solve (A σ (i) I)y (i) x (i) τ (i), Rescale x (i+1) = y(i) y (i), Update λ (i+1) = x (i+1)t Ax (i+1), possibly: update the shift σ (i) Test: eigenvalue residual r (i+1) = (A λ (i+1) I)x (i+1). end for
7 Error indicator Error indicator (Orthogonal decomposition for symmetric A, Parlett) Eigenvalue residual x (i) = cos θ (i) x 1 + sin θ (i) x (i), x(i) x 1. sin θ (i) λ 2 λ (i) r (i) sin θ (i) λ n λ 1
8 rates of inexact inverse iteration Decreasing tolerance τ (i) = C r (i) = O(sin θ (i) ) 1 For decreasing tolerance τ (i) C r (i) = O(sin θ (i) ) the inexact method recovers the rate of convergence achieved by exact. 2 Fixed shift σ: linear convergence. 3 Rayleigh quotient shift σ (i) = ρ(x (i) ) = x(i)t Ax (i) x (i)t x : cubic (i) convergence for A = A. Fixed tolerance τ (i) = τ 1 Rayleigh quotient shift: quadratic convergence
9 MINRES (A σi)y = x when A is symmetric Solving a linear system (A σi)y = x standard MINRES for y 0 = 0: x (A σi)y k 2 where κ is the condition number of A σi. Number of inner iterations: then k 1 + κ ( ) k 1 κ 1 x. κ + 1 { log 2 + log x } τ x (A σi)y k τ.
10 Unpreconditioned with MINRES rates for with MINRES for simple eigenvalue If A is positive definite and has a simple eigenvalue then x (i) (A σi)y (i) k 2 2 λ 1 λ n λ 1 σ ( )k 2 κ1 1 Qx (i) 2. κ where Q is the orthogonal projection onto span{x 2,..., x n } and κ 1 is the reduced condition number κ 1 = max i=2,...,n λ i σ min i=2,...,n λ i σ. Number of inner for each i for x (i) (A σ (i) I)y (i) τ (i) ( ) k (i) 2 + κ 1 log 2 λ 1 λ n + log Qx(i) 2 λ 1 σ τ (i)
11 Unpreconditioned with MINRES rates for with MINRES for simple eigenvalue If A is positive definite and has a simple eigenvalue then x (i) (A σi)y (i) k 2 2 λ 1 λ n λ 1 σ ( )k 2 κ1 1 sin θ (i). κ where Q is the orthogonal projection onto span{x 2,..., x n } and κ 1 is the reduced condition number κ 1 = max i=2,...,n λ i σ min i=2,...,n λ i σ. Number of inner for each i for x (i) (A σ (i) I)y (i) τ (i) ( k (i) 2 + κ 1 log 2 λ 1 λ n + log sin ) θ(i) λ 1 σ τ (i)
12
13 Incomplete Cholesky A = LL T + E symmetric of (A σi)y (i) = x (i) : Remarks L 1 (A σi)l T ỹ (i) = L 1 x (i), 1 changes number of inner iterations k (i) 2 + κ 1 (log 2 λ 1 λ n + log 2 k (i) increases with i for τ (i) = C r (i). y (i) = L T ỹ (i) L 1 ) λ 1 σ τ (i)
14 Incomplete Cholesky A = LL T + E symmetric of (A σi)y (i) = x (i) : Remarks L 1 (A σi)l T ỹ (i) = L 1 x (i), 1 changes number of inner iterations k (i) 2 + κ 1 (log 2 λ 1 λ n + log 2 k (i) increases with i for τ (i) = C r (i). y (i) = L T ỹ (i) L 1 ) λ 1 σ τ (i)
15 Derivation Aims 1 modify L L L 1 (A σi)l T ỹ (i) = L 1 x (i), 2 minor extra computation cost for L y (i) = L T ỹ (i) 3 nice right hand side L 1 x (i) (same behaviour as unpreconditioned, e.g. for fixed shifts k (i) does not increase with i)
16 Derivation Aims 1 modify L L L 1 (A σi)l T ỹ (i) = L 1 x (i), 2 minor extra computation cost for L y (i) = L T ỹ (i) 3 nice right hand side L 1 x (i) (same behaviour as unpreconditioned, e.g. for fixed shifts k (i) does not increase with i)
17 Choice of L Condition MINRES indicates that L 1 x (i) should be close to eigenvector of L 1 (A σi)l T Holds if Justification of LL T x (i) = Ax (i) If x (i) = x 1 then LL T x 1 = λ 1 x 1 LL T x (i) = Ax (i) L 1 (A σi)l T L 1 x 1 = λ 1 σ λ 1 L 1 x 1 L 1 (A σi)l T L 1 x (i) = λ 1 σ L 1 x (i) + C r (i) λ 1
18 Choice of L Condition MINRES indicates that L 1 x (i) should be close to eigenvector of L 1 (A σi)l T Holds if Justification of LL T x (i) = Ax (i) If x (i) = x 1 then LL T x 1 = λ 1 x 1 LL T x (i) = Ax (i) L 1 (A σi)l T L 1 x 1 = λ 1 σ λ 1 L 1 x 1 L 1 (A σi)l T L 1 x (i) = λ 1 σ L 1 x (i) + C r (i) λ 1
19 How do we achieve LL T x (i) = Ax (i)? Theorem Let x (i) current eigenvector approximation, e (i) = Ax (i) LL T x (i) (known) and L chosen such that L = L + α (i) e (i) (L 1 e (i) ) T with α (i) root of quadratic function we get LL T x (i) = Ax (i). Implementation 1 Note: LL T = LL T 1 + e (i)t x (i) e(i) e (i)t 2 L is a rank-one update of L.
20 How do we achieve LL T x (i) = Ax (i)? Theorem Let x (i) current eigenvector approximation, e (i) = Ax (i) LL T x (i) (known) and L chosen such that L = L + α (i) e (i) (L 1 e (i) ) T with α (i) root of quadratic function we get LL T x (i) = Ax (i). Implementation 1 Note: LL T = LL T 1 + e (i)t x (i) e(i) e (i)t 2 L is a rank-one update of L.
21 Implementation General positive definite For MINRES implementation only the evaluation of P 1 is necessary Sherman-Morrison formula where z (i) = P 1 Ax (i). P = P + γ (i) e (i) e (i)t P 1 = P 1 (z(i) x (i) )(z (i) x (i) ) T (z (i) x (i) ) T Ax (i)
22 rates The 1 outer convergence rate is retained 2 cheap inner are provided ( sin θ k (i) (i) ) C 1 + C 2 log λ 1 σ τ (i) 3 only a single extra back substitution with P = LL T per outer iteration needed
23 Example SPD matrix from the Matrix Market library (nos5: 3 story building with attached tower) seek eigenvalue near fixed shift σ = 100 A LL T, incomplete Cholesky factorisation (drop tol. = 0.1) compare standard and
24 Fixed shift with standard incomplete Cholesky total number of inner iterations using standard : 2026 total number of inner iterations using : 779
25 Comparison of LL T with LL T Spectral properties of preconditioned matrix Let Theorem If σ / Λ(A) then µ, ξ 0 and L 1 (A σi)l T w = µw L 1 (A σi)l T ŵ = ξŵ min µ Λ(L 1 (A σi)l T ) where γ = 1/(e T x) and v = L 1 e. µ ξ ξ γv v,
26 Comparison of LL T with LL T Spectral properties of preconditioned matrix Let L 1 (A σi)l T w = µw L 1 (A σi)l T ŵ = ξŵ Interlacing property Rewrite second equation Dt = ξ(i + γzz T )t where L 1 (A σi)l T = QDQ T, z = Q T v, (I + αvv T )Qt = ŵ. Interlacing property If γ > 0 eigenvalues are moved towards the origin. If γ < 0 eigenvalues are moved away from the origin.
27 Comparison of LL T with LL T Spectral properties of preconditioned matrix Let L 1 (A σi)l T w = µw L 1 (A σi)l T ŵ = ξŵ Interlacing property Rewrite second equation Dt = ξ(i + γzz T )t where L 1 (A σi)l T = QDQ T, z = Q T v, (I + αvv T )Qt = ŵ. Interlacing property If γ > 0 eigenvalues are moved towards the origin. If γ < 0 eigenvalues are moved away from the origin.
28 Comparison of LL T with LL T Interlacing property µ and ξ interlace each other depending on the sign of γ Clustering properties are preserved reduced condition number κ 1 L κ1 L κ1 L (1 + γvt v)
29 Comparison of LL T with LL T Interlacing property µ and ξ interlace each other depending on the sign of γ Clustering properties are preserved reduced condition number κ 1 L κ1 L κ1 L (1 + γvt v)
30 Changing the right hand side Approach by Simoncini/Eldén [3] Instead of solving L 1 (A σ (i) I)L T ỹ (i) = L 1 x (i), change the right hand side L 1 (A σ (i) I)L T ỹ (i) = L T x (i), y (i) = L T ỹ (i) y (i) = L T ỹ (i)
31 Comparison Tuned and Simoncini & Eldén Example nos5.mtx from Matrix Market. Solves to fixed tolerance τ = Rayleigh quotient shift. Quadratic convergence for both methods. Simoncini & Eldén Tuned Drop Tolerances Outer total
32
33 example Ax = λm x with bcsstk08 (Structural engineering) Figure: Fixed Shift Figure: Rayleigh Quotient Shift
34 example for the eigenproblem Ax = λm x with bcsstk08 (Structural engineering) Figure: Fixed Shift Figure: Rayleigh Quotient Shift
35 M. A. Freitag and A. Spence, rates for inexact inverse iteration with application to preconditioned iterative, Submitted to BIT., A for inexact inverse iteration applied to eigenvalue problems, Submitted to IMA J. Numer. Anal. V. Simoncini and L. Eldén, Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT, 42 (2002), pp
A Tuned Preconditioner for Inexact Inverse Iteration for Generalised Eigenvalue Problems
A Tuned Preconditioner for for Generalised Eigenvalue Problems Department of Mathematical Sciences University of Bath, United Kingdom IWASEP VI May 22-25, 2006 Pennsylvania State University, University
More informationLecture 3: Inexact inverse iteration with preconditioning
Lecture 3: Department of Mathematical Sciences CLAPDE, Durham, July 2008 Joint work with M. Freitag (Bath), and M. Robbé & M. Sadkane (Brest) 1 Introduction 2 Preconditioned GMRES for Inverse Power Method
More informationPreconditioned inverse iteration and shift-invert Arnoldi method
Preconditioned inverse iteration and shift-invert Arnoldi method Melina Freitag Department of Mathematical Sciences University of Bath CSC Seminar Max-Planck-Institute for Dynamics of Complex Technical
More informationInexact inverse iteration with preconditioning
Department of Mathematical Sciences Computational Methods with Applications Harrachov, Czech Republic 24th August 2007 (joint work with M. Robbé and M. Sadkane (Brest)) 1 Introduction 2 Preconditioned
More informationInexact Inverse Iteration for Symmetric Matrices
Inexact Inverse Iteration for Symmetric Matrices Jörg Berns-Müller Ivan G. Graham Alastair Spence Abstract In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av
More informationInexact Inverse Iteration for Symmetric Matrices
Inexact Inverse Iteration for Symmetric Matrices Jörg Berns-Müller Ivan G. Graham Alastair Spence Abstract In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av
More informationInexact inverse iteration for symmetric matrices
Linear Algebra and its Applications 46 (2006) 389 43 www.elsevier.com/locate/laa Inexact inverse iteration for symmetric matrices Jörg Berns-Müller a, Ivan G. Graham b, Alastair Spence b, a Fachbereich
More informationRAYLEIGH QUOTIENT ITERATION AND SIMPLIFIED JACOBI-DAVIDSON WITH PRECONDITIONED ITERATIVE SOLVES FOR GENERALISED EIGENVALUE PROBLEMS
RAYLEIGH QUOTIENT ITERATION AND SIMPLIFIED JACOBI-DAVIDSON WITH PRECONDITIONED ITERATIVE SOLVES FOR GENERALISED EIGENVALUE PROBLEMS MELINA A. FREITAG, ALASTAIR SPENCE, AND EERO VAINIKKO Abstract. The computation
More informationc 2006 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 28, No. 4, pp. 1069 1082 c 2006 Society for Industrial and Applied Mathematics INEXACT INVERSE ITERATION WITH VARIABLE SHIFT FOR NONSYMMETRIC GENERALIZED EIGENVALUE PROBLEMS
More informationInner-outer Iterative Methods for Eigenvalue Problems - Convergence and Preconditioning
Inner-outer Iterative Methods for Eigenvalue Problems - Convergence and Preconditioning submitted by Melina Annerose Freitag for the degree of Doctor of Philosophy of the University of Bath Department
More informationRecent advances in approximation using Krylov subspaces. V. Simoncini. Dipartimento di Matematica, Università di Bologna.
Recent advances in approximation using Krylov subspaces V. Simoncini Dipartimento di Matematica, Università di Bologna and CIRSA, Ravenna, Italy valeria@dm.unibo.it 1 The framework It is given an operator
More informationAN INEXACT INVERSE ITERATION FOR COMPUTING THE SMALLEST EIGENVALUE OF AN IRREDUCIBLE M-MATRIX. 1. Introduction. We consider the eigenvalue problem
AN INEXACT INVERSE ITERATION FOR COMPUTING THE SMALLEST EIGENVALUE OF AN IRREDUCIBLE M-MATRIX MICHIEL E. HOCHSTENBACH, WEN-WEI LIN, AND CHING-SUNG LIU Abstract. In this paper, we present an inexact inverse
More informationLinear algebra issues in Interior Point methods for bound-constrained least-squares problems
Linear algebra issues in Interior Point methods for bound-constrained least-squares problems Stefania Bellavia Dipartimento di Energetica S. Stecco Università degli Studi di Firenze Joint work with Jacek
More informationABSTRACT NUMERICAL SOLUTION OF EIGENVALUE PROBLEMS WITH SPECTRAL TRANSFORMATIONS
ABSTRACT Title of dissertation: NUMERICAL SOLUTION OF EIGENVALUE PROBLEMS WITH SPECTRAL TRANSFORMATIONS Fei Xue, Doctor of Philosophy, 2009 Dissertation directed by: Professor Howard C. Elman Department
More informationConjugate Gradient Method
Conjugate Gradient Method direct and indirect methods positive definite linear systems Krylov sequence spectral analysis of Krylov sequence preconditioning Prof. S. Boyd, EE364b, Stanford University Three
More informationAlternative correction equations in the Jacobi-Davidson method
Chapter 2 Alternative correction equations in the Jacobi-Davidson method Menno Genseberger and Gerard Sleijpen Abstract The correction equation in the Jacobi-Davidson method is effective in a subspace
More informationITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 11 : JACOBI DAVIDSON METHOD
ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND EIGENPROBLEMS CHAPTER 11 : JACOBI DAVIDSON METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation
More informationIs there life after the Lanczos method? What is LOBPCG?
1 Is there life after the Lanczos method? What is LOBPCG? Andrew V. Knyazev Department of Mathematics and Center for Computational Mathematics University of Colorado at Denver SIAM ALA Meeting, July 17,
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large
More information1. Introduction. In this paper we consider the large and sparse eigenvalue problem. Ax = λx (1.1) T (λ)x = 0 (1.2)
A NEW JUSTIFICATION OF THE JACOBI DAVIDSON METHOD FOR LARGE EIGENPROBLEMS HEINRICH VOSS Abstract. The Jacobi Davidson method is known to converge at least quadratically if the correction equation is solved
More informationA Model-Trust-Region Framework for Symmetric Generalized Eigenvalue Problems
A Model-Trust-Region Framework for Symmetric Generalized Eigenvalue Problems C. G. Baker P.-A. Absil K. A. Gallivan Technical Report FSU-SCS-2005-096 Submitted June 7, 2005 Abstract A general inner-outer
More informationModel order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers
MAX PLANCK INSTITUTE International Conference on Communications, Computing and Control Applications March 3-5, 2011, Hammamet, Tunisia. Model order reduction of large-scale dynamical systems with Jacobi-Davidson
More informationDavidson Method CHAPTER 3 : JACOBI DAVIDSON METHOD
Davidson Method CHAPTER 3 : JACOBI DAVIDSON METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology The Davidson method is a popular technique to compute a few of the smallest (or largest)
More informationSolution of eigenvalue problems. Subspace iteration, The symmetric Lanczos algorithm. Harmonic Ritz values, Jacobi-Davidson s method
Solution of eigenvalue problems Introduction motivation Projection methods for eigenvalue problems Subspace iteration, The symmetric Lanczos algorithm Nonsymmetric Lanczos procedure; Implicit restarts
More informationControlling inner iterations in the Jacobi Davidson method
Controlling inner iterations in the Jacobi Davidson method Michiel E. Hochstenbach and Yvan Notay Service de Métrologie Nucléaire, Université Libre de Bruxelles (C.P. 165/84), 50, Av. F.D. Roosevelt, B-1050
More informationEigenvalue Problems CHAPTER 1 : PRELIMINARIES
Eigenvalue Problems CHAPTER 1 : PRELIMINARIES Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Preliminaries Eigenvalue problems 2012 1 / 14
More informationThe Nullspace free eigenvalue problem and the inexact Shift and invert Lanczos method. V. Simoncini. Dipartimento di Matematica, Università di Bologna
The Nullspace free eigenvalue problem and the inexact Shift and invert Lanczos method V. Simoncini Dipartimento di Matematica, Università di Bologna and CIRSA, Ravenna, Italy valeria@dm.unibo.it 1 The
More informationMICHIEL E. HOCHSTENBACH
VARIATIONS ON HARMONIC RAYLEIGH RITZ FOR STANDARD AND GENERALIZED EIGENPROBLEMS MICHIEL E. HOCHSTENBACH Abstract. We present several variations on the harmonic Rayleigh Ritz method. First, we introduce
More informationA robust multilevel approximate inverse preconditioner for symmetric positive definite matrices
DICEA DEPARTMENT OF CIVIL, ENVIRONMENTAL AND ARCHITECTURAL ENGINEERING PhD SCHOOL CIVIL AND ENVIRONMENTAL ENGINEERING SCIENCES XXX CYCLE A robust multilevel approximate inverse preconditioner for symmetric
More informationCONTROLLING INNER ITERATIONS IN THE JACOBI DAVIDSON METHOD
CONTROLLING INNER ITERATIONS IN THE JACOBI DAVIDSON METHOD MICHIEL E. HOCHSTENBACH AND YVAN NOTAY Abstract. The Jacobi Davidson method is an eigenvalue solver which uses an inner-outer scheme. In the outer
More informationA Chebyshev-based two-stage iterative method as an alternative to the direct solution of linear systems
A Chebyshev-based two-stage iterative method as an alternative to the direct solution of linear systems Mario Arioli m.arioli@rl.ac.uk CCLRC-Rutherford Appleton Laboratory with Daniel Ruiz (E.N.S.E.E.I.H.T)
More informationPolynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 28, 2011 T.M. Huang (Taiwan Normal Univ.)
More information7.2 Steepest Descent and Preconditioning
7.2 Steepest Descent and Preconditioning Descent methods are a broad class of iterative methods for finding solutions of the linear system Ax = b for symmetric positive definite matrix A R n n. Consider
More informationCONTROLLING INNER ITERATIONS IN THE JACOBI DAVIDSON METHOD
CONTROLLING INNER ITERATIONS IN THE JACOBI DAVIDSON METHOD MICHIEL E. HOCHSTENBACH AND YVAN NOTAY Abstract. The Jacobi Davidson method is an eigenvalue solver which uses the iterative (and in general inaccurate)
More informationTHEORETICAL AND NUMERICAL COMPARISON OF PROJECTION METHODS DERIVED FROM DEFLATION, DOMAIN DECOMPOSITION AND MULTIGRID METHODS
THEORETICAL AND NUMERICAL COMPARISON OF PROJECTION METHODS DERIVED FROM DEFLATION, DOMAIN DECOMPOSITION AND MULTIGRID METHODS J.M. TANG, R. NABBEN, C. VUIK, AND Y.A. ERLANGGA Abstract. For various applications,
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 1, pp. 1-11, 8. Copyright 8,. ISSN 168-961. MAJORIZATION BOUNDS FOR RITZ VALUES OF HERMITIAN MATRICES CHRISTOPHER C. PAIGE AND IVO PANAYOTOV Abstract.
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 18 Outline
More informationSolving Regularized Total Least Squares Problems
Solving Regularized Total Least Squares Problems Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Numerical Simulation Joint work with Jörg Lampe TUHH Heinrich Voss Total
More informationNotes on PCG for Sparse Linear Systems
Notes on PCG for Sparse Linear Systems Luca Bergamaschi Department of Civil Environmental and Architectural Engineering University of Padova e-mail luca.bergamaschi@unipd.it webpage www.dmsa.unipd.it/
More informationComputational Methods. Eigenvalues and Singular Values
Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations
More informationA Note on Inverse Iteration
A Note on Inverse Iteration Klaus Neymeyr Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, 18051 Rostock, Germany; SUMMARY Inverse iteration, if applied to a symmetric positive definite
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol.??, No.?, pp.?????? c 2009 Society for Industrial and Applied Mathematics GRADIENT FLOW APPROACH TO GEOMETRIC CONVERGENCE ANALYSIS OF PRECONDITIONED EIGENSOLVERS ANDREW V.
More informationMatrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland
Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationLow Rank Approximation Lecture 7. Daniel Kressner Chair for Numerical Algorithms and HPC Institute of Mathematics, EPFL
Low Rank Approximation Lecture 7 Daniel Kressner Chair for Numerical Algorithms and HPC Institute of Mathematics, EPFL daniel.kressner@epfl.ch 1 Alternating least-squares / linear scheme General setting:
More informationA Jacobi Davidson type method for the product eigenvalue problem
A Jacobi Davidson type method for the product eigenvalue problem Michiel E Hochstenbach Abstract We propose a Jacobi Davidson type method to compute selected eigenpairs of the product eigenvalue problem
More information6.4 Krylov Subspaces and Conjugate Gradients
6.4 Krylov Subspaces and Conjugate Gradients Our original equation is Ax = b. The preconditioned equation is P Ax = P b. When we write P, we never intend that an inverse will be explicitly computed. P
More informationLecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector
More informationHOMOGENEOUS JACOBI DAVIDSON. 1. Introduction. We study a homogeneous Jacobi Davidson variant for the polynomial eigenproblem
HOMOGENEOUS JACOBI DAVIDSON MICHIEL E. HOCHSTENBACH AND YVAN NOTAY Abstract. We study a homogeneous variant of the Jacobi Davidson method for the generalized and polynomial eigenvalue problem. Special
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationThe Eigenvalue Problem: Perturbation Theory
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just
More informationFinding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems
Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems AMSC 663-664 Final Report Minghao Wu AMSC Program mwu@math.umd.edu Dr. Howard Elman Department of Computer
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 10-12 Large-Scale Eigenvalue Problems in Trust-Region Calculations Marielba Rojas, Bjørn H. Fotland, and Trond Steihaug ISSN 1389-6520 Reports of the Department of
More informationLinear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4
Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary
More informationConjugate gradient method. Descent method. Conjugate search direction. Conjugate Gradient Algorithm (294)
Conjugate gradient method Descent method Hestenes, Stiefel 1952 For A N N SPD In exact arithmetic, solves in N steps In real arithmetic No guaranteed stopping Often converges in many fewer than N steps
More informationMatrix Algebra: Summary
May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 31, No. 2, pp. 460 477 c 2009 Society for Industrial and Applied Mathematics CONTROLLING INNER ITERATIONS IN THE JACOBI DAVIDSON METHOD MICHIEL E. HOCHSTENBACH AND YVAN
More informationSOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY
SOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY KLAUS NEYMEYR ABSTRACT. Multigrid techniques can successfully be applied to mesh eigenvalue problems for elliptic differential operators. They allow
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 435 (20) 60 622 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Fast inexact subspace iteration
More informationNumerical behavior of inexact linear solvers
Numerical behavior of inexact linear solvers Miro Rozložník joint results with Zhong-zhi Bai and Pavel Jiránek Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic The fourth
More informationQR-decomposition. The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR
QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which In Matlab A = QR [Q,R]=qr(A); Note. The QR-decomposition is unique
More informationU AUc = θc. n u = γ i x i,
HARMONIC AND REFINED RAYLEIGH RITZ FOR THE POLYNOMIAL EIGENVALUE PROBLEM MICHIEL E. HOCHSTENBACH AND GERARD L. G. SLEIJPEN Abstract. After reviewing the harmonic Rayleigh Ritz approach for the standard
More informationLecture Note 13: Eigenvalue Problem for Symmetric Matrices
MATH 5330: Computational Methods of Linear Algebra Lecture Note 13: Eigenvalue Problem for Symmetric Matrices 1 The Jacobi Algorithm Xianyi Zeng Department of Mathematical Sciences, UTEP Let A be real
More informationUniversiteit-Utrecht. Department. of Mathematics. The convergence of Jacobi-Davidson for. Hermitian eigenproblems. Jasper van den Eshof.
Universiteit-Utrecht * Department of Mathematics The convergence of Jacobi-Davidson for Hermitian eigenproblems by Jasper van den Eshof Preprint nr. 1165 November, 2000 THE CONVERGENCE OF JACOBI-DAVIDSON
More informationOn the accuracy of saddle point solvers
On the accuracy of saddle point solvers Miro Rozložník joint results with Valeria Simoncini and Pavel Jiránek Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic Seminar at
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationLecture 9: Krylov Subspace Methods. 2 Derivation of the Conjugate Gradient Algorithm
CS 622 Data-Sparse Matrix Computations September 19, 217 Lecture 9: Krylov Subspace Methods Lecturer: Anil Damle Scribes: David Eriksson, Marc Aurele Gilles, Ariah Klages-Mundt, Sophia Novitzky 1 Introduction
More informationA Jacobi Davidson type method for the generalized singular value problem
A Jacobi Davidson type method for the generalized singular value problem M. E. Hochstenbach a, a Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600
More informationUsing the Karush-Kuhn-Tucker Conditions to Analyze the Convergence Rate of Preconditioned Eigenvalue Solvers
Using the Karush-Kuhn-Tucker Conditions to Analyze the Convergence Rate of Preconditioned Eigenvalue Solvers Merico Argentati University of Colorado Denver Joint work with Andrew V. Knyazev, Klaus Neymeyr
More informationDomain decomposition on different levels of the Jacobi-Davidson method
hapter 5 Domain decomposition on different levels of the Jacobi-Davidson method Abstract Most computational work of Jacobi-Davidson [46], an iterative method suitable for computing solutions of large dimensional
More informationComputation. For QDA we need to calculate: Lets first consider the case that
Computation For QDA we need to calculate: δ (x) = 1 2 log( Σ ) 1 2 (x µ ) Σ 1 (x µ ) + log(π ) Lets first consider the case that Σ = I,. This is the case where each distribution is spherical, around the
More informationTopics. The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems
Topics The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems What about non-spd systems? Methods requiring small history Methods requiring large history Summary of solvers 1 / 52 Conjugate
More informationIterative methods for symmetric eigenvalue problems
s Iterative s for symmetric eigenvalue problems, PhD McMaster University School of Computational Engineering and Science February 11, 2008 s 1 The power and its variants Inverse power Rayleigh quotient
More informationA Newton-Galerkin-ADI Method for Large-Scale Algebraic Riccati Equations
A Newton-Galerkin-ADI Method for Large-Scale Algebraic Riccati Equations Peter Benner Max-Planck-Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory
More informationA Jacobi Davidson Method for Nonlinear Eigenproblems
A Jacobi Davidson Method for Nonlinear Eigenproblems Heinrich Voss Section of Mathematics, Hamburg University of Technology, D 21071 Hamburg voss @ tu-harburg.de http://www.tu-harburg.de/mat/hp/voss Abstract.
More informationSymmetric matrices and dot products
Symmetric matrices and dot products Proposition An n n matrix A is symmetric iff, for all x, y in R n, (Ax) y = x (Ay). Proof. If A is symmetric, then (Ax) y = x T A T y = x T Ay = x (Ay). If equality
More informationInvariant Subspace Computation - A Geometric Approach
Invariant Subspace Computation - A Geometric Approach Pierre-Antoine Absil PhD Defense Université de Liège Thursday, 13 February 2003 1 Acknowledgements Advisor: Prof. R. Sepulchre (Université de Liège)
More informationPoint Distribution Models
Point Distribution Models Jan Kybic winter semester 2007 Point distribution models (Cootes et al., 1992) Shape description techniques A family of shapes = mean + eigenvectors (eigenshapes) Shapes described
More informationAn Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems
An Arnoldi Method for Nonlinear Symmetric Eigenvalue Problems H. Voss 1 Introduction In this paper we consider the nonlinear eigenvalue problem T (λ)x = 0 (1) where T (λ) R n n is a family of symmetric
More informationNumerical Methods I: Eigenvalues and eigenvectors
1/25 Numerical Methods I: Eigenvalues and eigenvectors Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 2, 2017 Overview 2/25 Conditioning Eigenvalues and eigenvectors How hard are they
More informationHARMONIC RAYLEIGH RITZ EXTRACTION FOR THE MULTIPARAMETER EIGENVALUE PROBLEM
HARMONIC RAYLEIGH RITZ EXTRACTION FOR THE MULTIPARAMETER EIGENVALUE PROBLEM MICHIEL E. HOCHSTENBACH AND BOR PLESTENJAK Abstract. We study harmonic and refined extraction methods for the multiparameter
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
More informationIncomplete Cholesky preconditioners that exploit the low-rank property
anapov@ulb.ac.be ; http://homepages.ulb.ac.be/ anapov/ 1 / 35 Incomplete Cholesky preconditioners that exploit the low-rank property (theory and practice) Artem Napov Service de Métrologie Nucléaire, Université
More informationECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems. Part I: Review of basic theory of eigenvalue problems
ECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems Part I: Review of basic theory of eigenvalue problems 1. Let A C n n. (a) A scalar λ is an eigenvalue of an n n A
More informationStructured Preconditioners for Saddle Point Problems
Structured Preconditioners for Saddle Point Problems V. Simoncini Dipartimento di Matematica Università di Bologna valeria@dm.unibo.it p. 1 Collaborators on this project Mario Arioli, RAL, UK Michele Benzi,
More informationSolution of eigenvalue problems. Subspace iteration, The symmetric Lanczos algorithm. Harmonic Ritz values, Jacobi-Davidson s method
Solution of eigenvalue problems Introduction motivation Projection methods for eigenvalue problems Subspace iteration, The symmetric Lanczos algorithm Nonsymmetric Lanczos procedure; Implicit restarts
More informationContents. Preface... xi. Introduction...
Contents Preface... xi Introduction... xv Chapter 1. Computer Architectures... 1 1.1. Different types of parallelism... 1 1.1.1. Overlap, concurrency and parallelism... 1 1.1.2. Temporal and spatial parallelism
More informationSTEEPEST DESCENT AND CONJUGATE GRADIENT METHODS WITH VARIABLE PRECONDITIONING
SIAM J. MATRIX ANAL. APPL. Vol.?, No.?, pp.?? c 2007 Society for Industrial and Applied Mathematics STEEPEST DESCENT AND CONJUGATE GRADIENT METHODS WITH VARIABLE PRECONDITIONING ANDREW V. KNYAZEV AND ILYA
More informationLinear Solvers. Andrew Hazel
Linear Solvers Andrew Hazel Introduction Thus far we have talked about the formulation and discretisation of physical problems...... and stopped when we got to a discrete linear system of equations. Introduction
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT -09 Computational and Sensitivity Aspects of Eigenvalue-Based Methods for the Large-Scale Trust-Region Subproblem Marielba Rojas, Bjørn H. Fotland, and Trond Steihaug
More informationIndefinite Preconditioners for PDE-constrained optimization problems. V. Simoncini
Indefinite Preconditioners for PDE-constrained optimization problems V. Simoncini Dipartimento di Matematica, Università di Bologna, Italy valeria.simoncini@unibo.it Partly joint work with Debora Sesana,
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More informationPreconditioning Techniques Analysis for CG Method
Preconditioning Techniques Analysis for CG Method Huaguang Song Department of Computer Science University of California, Davis hso@ucdavis.edu Abstract Matrix computation issue for solve linear system
More informationof dimension n 1 n 2, one defines the matrix determinants
HARMONIC RAYLEIGH RITZ FOR THE MULTIPARAMETER EIGENVALUE PROBLEM MICHIEL E. HOCHSTENBACH AND BOR PLESTENJAK Abstract. We study harmonic and refined extraction methods for the multiparameter eigenvalue
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit VII Sparse Matrix
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit VII Sparse Matrix Computations Part 1: Direct Methods Dianne P. O Leary c 2008
More informationNumerical Methods for Large Scale Eigenvalue Problems
MAX PLANCK INSTITUTE Summer School in Trogir, Croatia Oktober 12, 2011 Numerical Methods for Large Scale Eigenvalue Problems Patrick Kürschner Max Planck Institute for Dynamics of Complex Technical Systems
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationOn Lagrange multipliers of trust-region subproblems
On Lagrange multipliers of trust-region subproblems Ladislav Lukšan, Ctirad Matonoha, Jan Vlček Institute of Computer Science AS CR, Prague Programy a algoritmy numerické matematiky 14 1.- 6. června 2008
More informationNonlinear Eigenvalue Problems: An Introduction
Nonlinear Eigenvalue Problems: An Introduction Cedric Effenberger Seminar for Applied Mathematics ETH Zurich Pro*Doc Workshop Disentis, August 18 21, 2010 Cedric Effenberger (SAM, ETHZ) NLEVPs: An Introduction
More informationABSTRACT OF DISSERTATION. Ping Zhang
ABSTRACT OF DISSERTATION Ping Zhang The Graduate School University of Kentucky 2009 Iterative Methods for Computing Eigenvalues and Exponentials of Large Matrices ABSTRACT OF DISSERTATION A dissertation
More information