Multilevel Methods for Eigenspace Computations in Structural Dynamics

Size: px
Start display at page:

Download "Multilevel Methods for Eigenspace Computations in Structural Dynamics"

Transcription

1 Multilevel Methods for Eigenspace Computations in Structural Dynamics Ulrich Hetmaniuk & Rich Lehoucq Sandia National Laboratories, Computational Math and Algorithms, Albuquerque, NM Joint work with Peter Arbenz (ETH), Jeff Bennighof (UT Austin), Mark Muller (UT Austin), Bill Cochran (UIUC), Heidi Thornquist & Ray Tuminaro (Sandia)

2 Eigenvalue problem in structural dynamics Who/What is the problem? Multilevel approach 1 (use multilevel preconditioners) Multilevel approach 2 (component mode synthesis) Conclusions and future directions Thanks to the organizers

3 Eigen-gang Jeff Bennighof, University of Texas Ray Tuminaro, Sandia Mark Muller, University of Texas Heidi Thornquist Sandia (Jeff s Ph.D student) Peter Arbenz, ETH Bill Cochran, Sandia summer intern; Ph.d. student UIUC Ulrich Hetmaniuk, Sandia

4 Salinas: Implicit structural dynamics Salinas is a massively parallel implementation of structural dynamics finite element analysis This capability is required for high fidelity, validated models used in modal, vibration, static and shock analysis of weapons systems Gordon Bell Prize winner (C++ code).

5 Salinas and linear algebra Robust scalable iterative linear and eigensolvers are required Fundamental mode of an aircraft carrier

6 PDE of interest Consider the hyperbolic PDE on a domain u E( u) = f ( t) tt where E( u) is a linear elliptic coercive operator and is the mass density. Assume that appropriate initial and boundary conditions are specified and that is a two or three dimensional domain that represents a structure.

7 Problems of interest Frequency (or harmonic) response how does the structure respond to a prescribed load? Transient analysis how does the structure evolve in time? My assumption is that 100 plus eigenpairs are needed for large-scale 3D FEM discretizations the eigenpairs effect a reduced order model (ROM).

8 Helmholtz equation The Fourier transform of the hyperbolic PDE gives 2 u E( u ) = f The direct approach (Helmholtz) is to solve numerous linear PDEs as varies over the frequency range of interest. This approach is typically not undertaken when there are a large number of load terms or the frequency range of interest is large.

9 Modal approach (ROM) Compute the free vibrations or modal subspace of E ( z ) = z i i i over the domain with the same boundary conditions as the ROM. This standard approach computes numerous modes and then projects the PDE onto the modal subspace. This approach is useful when there are many load terms or the frequency range of interest is large.

10 Some details The i are the squares of the natural frequencies i Assume the ordering The eigenvectors z i are orthogonal

11 Discrete Modal Analysis A finite element discretization leads to the generalized eigenvalue problem K Kz M n h h h = Mz i i i where and are the stiffness and mass matrices of order respectively. Both are symmetric and semi-positive definite matrices and sparse. As in the continuous problem, the eigenvectors are used to project the PDE onto the span of the modal subspace.

12 Project with the Eigenvectors 2 u E( u ) = f h T h ˆ 2 h T h ˆ = h T m m m m m ( Z ) KZ u ( Z ) MZ u ( Z ) f Diagonal matrix containing the m eigenvalues Identity matrix of order m The solution is approximated with u h m Z uˆ

13 Why only m modes? Eigenvectors associated with the smallest eigenvalues have a mechanical significance they dominate the low frequency response FEM discretization only approximates a small number of fundamental modes of the PDE why compute more? (Error increases as the frequency increases) Keep in mind that thousands of eigenvectors may be needed when the modal density is large many eigenvalues within the desired frequency range.

14 Eigenvalue problem in structural dynamics Who/What is the problem? Multilevel approach 1 (use multilevel preconditioners) Multilevel approach 2 (component mode synthesis) Conclusions and future directions

15 Requirements of an algorithm and ensuing implementation Robust (vary data, parameters and still compute the same answers) Reliable (don t miss eigenvalues, compute a basis of eigenvectors) Return orthogonal eigenvectors Ability to efficiently compute ,000 modes (remember we to generate a ROM).

16 Standard computational approach Large-scale eigenvalue problem needs to be solved for 100 s to 1000 s of modes; the state of the art industrial solution is to run a block Lanczos algorithm (Boeing code Grimes, Lewis & Simon Simax 1994) on 1 h h h 1 i i i ( K M) Mz = z ( ) using an inner product. This is the approach traditionally used by MSC.NASTRAN and almost every engineering analysis package.

17 Key computational issue Solving the linear system Kx = Mv at every Lanczos iteration The Boeing code uses a sparse direct solver (and shifts into the spectrum) Salinas uses parallel ARPACK along with a FETI domain decomposition solver

18 Overview of multilevel approach 1 Kz h h h = Mz i i i 1. Shift-invert Lanczos method with a scalable preconditioned inner iteration 2. Preconditioned eigensolver (no inner iteration necessary)

19 Shift-invert Lanczos with a preconditioned inner iteration K m { m1 } ( T, x) = Span x, Tx,, T x T -1 K M Apply via a preconditioned conjugate iteration use a scalable preconditioner (multilevel/multigrid)

20 Preconditioned Eigensolver Only require an application of a preconditioner per (outer) iteration N D x m+ 1 { m m m } ( x ) = Span x,, x, N ( M K) x (0) (0) ( ) 1 ( ) ( ) x Kx ( x ) Kx = x Mx ( x ) Mx T ( m) min ( m) and T xd m ( m) T ( m) ( m) T ( m)

21 Preconditioned Eigensolver Replace shift-invert Lanczos attempt to minimize the Rayleigh-Quotient Newton s method (Davidson methods) (Davidson, Morgan, Scott, van der Vorst, Sleijpen, Stathopoulos, Saad, Notay) Nonlinear conjugate gradient iteration (Longsine, McCormick, Knyazev, Bergamaschi, Pini)

22 Separation of linear solver and the eigensolver Boeing Lanczos uses the sparse direct solver as a kernel. Preconditioned eigensolver replaces the eigenvalue algorithm and a preconditioner for the sparse direct solve. Quite often a good preconditioner is already available because a PDE required it. (See Knyazev Oxymoron paper (ETNA 1998))

23 Potential limitations of these methods Repeated use of stiffness, mass matrices and repeated preconditioning of the large-scale problem and cost of maintaining orthogonality. Clearly not an issue for a small number of modes or a limited frequency response (small frequency range). What about a large number of eigenvectors?

24 A Comparison of Eigensolvers for Largescale 3D Modal Analysis using AMG Preconditioned Iterative Methods Finished the revision of a paper for publication in IJNME (Arbenz, Hetmaniuk, Lehoucq, Tuminaro) Massively parallel modal analysis combine ML (multigrid), Anasazi (suite of codes implementing several of the previous algorithms including block extensions) and parallel ARPACK Extremely meticulous implementation, timings and verification on 3D FEM (2 elasticity) problems.

25 One conclusion from our comparison The cost of all the algorithms is asymptotic with the cost of maintaining orthogonality of the eigenvectors as the number of eigenpairs requested is increased. Are there alternate approaches?

26 Eigenvalue problem in structural dynamics Who/What is the problem? Multilevel approach 1 (use multilevel preconditioners) Multilevel approach 2 (component mode synthesis ideas) Conclusions and future directions

27 Component Mode Synthesis Component Mode Synthesis (CMS) techniques originated in the aerospace engineering community (Hurty 1965, Craig-Bampton 1968). The idea is simple. 1. Decompose the structure into component parts 2. Determine the modes of the parts 3. Synthesize these component modes to say something about the structure

28 Matrix view of CMS One way dissection on the union of the graphs of the mass and stiffness matrices produces K 0 K 0 K K K K K 1 2 T, 1 1 T, 2 2,, and M 0 M 0 M M M M M 1 2 T, 1 1 T, 2 2,,

29 Component Mode Synthesis (CMS) K Z = M Z Fixed-interface modes K Z = M Z K Z = Coupling modes M Z K = K K K K 1 T,, i i i i M = M K K M + M K K K K M K K 1 T 1 T 1 1 T ( ), i i, i, i i, i, i i i i, i i

30 Theorem The interface eigenvalue problem is equivalent to: 1/ 2 Find ( u, ) H ( ) such that 00 Su, v = Mu, v v H ( ) 1/2 00 where the Steklov-Poincaré and mass operators 1/ 2 act on H ( ) : 00 S & M 2 S = ( ( Ei ) i ) i= 1 2 M = ( ( Gi ( Ei )) i ) i= 1 Gi ( ) is the Green s function for the Dirichlet problem on and extends a trace function to. i E i i

31 CMS as a Rayleigh-Ritz process Project the global eigenvalue problem onto the span of Rectangular not square I 0 Z I Z 0 2 K K K K 2 2 Z I 1 T 1, 1 1 T, Modal truncation only retain the low order modes These three spaces are K orthogonal and so the projection of the stiffness matrix is diagonal but the mass matrix is NOT diagonal.

32 Previous/related work Bourquin (1992, 1993) casts CMS in a variational setting Asymptotic error analysis for 1,2,3 dimensional elliptic equations and their discretization Error is sum of modal truncation and discretization errors Can also consider free-interface methods (introduce a Lagrange multiplier at the interface)

33 0,1 1,1 1,2 AMLS Don t stop at one level! (Bennighof and graduate students, AIAA proceedings papers early 1990 s) 2,1 1,1 0,1 2,3 1,2 2,5 2,4 Recursively apply above procedures to obtain may small fixed interface eigenvalue problems. 2,2 Nested dissection automates the recursive reordering of the stiffness and mass matrices and so partitions the structure

34 Reference An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics SISC, 2004, by Bennighof & L. AMLS cast in a multilevel variational formulation (extension of the work by Bourquin) Showed that discrete AMLS represents a matrix decomposition of the stiffness matrix. The resulting decomposition performs a change of coordinates AMLS is a Rayleigh-Ritz method that efficiently computes an approximation to the modal subspace reduced order modeling

35 Example AMLS Calculation Recent paper Efficient Broadband Vibro- Acoustic Analysis of Passenger Car bodies Using an FE-based Component Mode Synthesis approach (Kropp & Heiserer, proceedings of the World Congress in Computational Mechanics, Vienna, 2002). Compared the industry standard Block Lanczos as (embedded in MSC.Nastran) approach against AMLS on modal analysis on the BMW 3 series.

36 BMW comparison result 1600 Frequency range (Hz) Direct (Helmholtz) AMLS Boeing Lanczos DOFS (Matrix order) mesh refinement

37 Cost of the computations These computations were performed on an HP 9000 (800 MHz) with 2 gigabytes of memory. The largest calculation took just under 3 days (2,500 eigenvectors for a problem with 13.5 million DOFs!). A 2.3 million DOF computation up through 400 Hz took 4 days of cputime and over a week turnaround time on a CRAY SV1 using the block Lanczos code. This calculation is not even feasible on the HP 9000.

38 Some stats on the BMW problem Order 13.5 million problem substructured into 38 levels 46,767 substructures Order 37,848 coarse problem During the past three years, AMLS has replaced Boeing Lanczos within the automotive industry. Cray supercomputers have been replaced by PC/workstations

39 Limitations of AMLS Not a good technique for 3D problems because of the size of the interface eigenvalue problem. AMLS assumes that the interface matrix operators are formed and factored. Important to point out that AMLS has the same complexity of one sparse direct factorization of the stiffness matrix.

40 Mass interface operator M = M K K M + M K K K K M K K 1 T 1 T 1 1 T ( ), i i, i, i i, i, i i i i, i i How can we approximate the above matrix operator? It s expensive to apply because of the amount of data.

41 Multilevel approach 2 alternatives CMS technique where the interface operators are not formed and instead preconditioned eigensolvers are employed. For example, use the BDDC (Dohrmann) preconditioner at the interface. Problem: Multilevel approach 1 issues come to bear. Algebraic multilevel?

42 Influence of the number of coupling modes Pencil (S, Mcplt) - 160,380 nodes - 16 subdomains Influence of the number of coupling modes Pencil (S, Mint) - 160,380 nodes - 16 subdomains 1.E E E-01 1.E-01 1.E-02 1.E-02 1.E-03 1.E-03 Relative error 1.E-04 EV 1 EV 5 EV 10 EV 50 EV 100 Relative error 1.E-04 EV 1 EV 5 EV 10 EV 50 EV E-05 1.E-05 1.E-06 1.E-06 1.E-07 1.E-07 1.E-08 Number of coupling eigenmodes 1.E-08 Number of coupling modes M = M K K M + M K K K K M K K 1 T 1 T 1 1 T ( ), i i, i, i i, i, i i i i, i i How can we approximate the above matrix operator? It s expensive to apply because of the amount of data.

43 Algebraic Multilevel Start with RQMG (Mandel and McCormick 1989) that approximately minimizes the Rayleigh Quotient over a sequence of grids. We re replacing the geometric information and smoothers of RQMG with algebraic info and better smoothers resulting in RQAMG. Goal of RQAMG is to overcome the cost of maintaining numerical orthogonality of the Ritz vectors associated with multilevel approach 1. Work in progress, Hetmaniuk & L.

44 Summary of multilevel approaches Approach 1 separates the preconditioner from the eigensolver Approach 2 interleaves the preconditioner and the eigensolver Our view is that the decision to use an approach depends upon how dominant the cost of orthogonality is for the modal analysis at hand c n 2 (nev)

45 Future/ongoing work Error estimation and stopping criterion (joint work with Hetmaniuk, Knyazev and Ovtchinnikov) for multilevel aproach 1 Modal truncation criterion (recent reports by others) Regularity issues or the effect of partitions on the approximation of global modes Approximation of the mass interface operator RQAMG, preconditioned CMS Packaging the preconditioned eigensolvers for public release into a the Anasazi subpackage of Trilinos (joint with Hetmaniuk and Thornquist) DD16 proceedings paper

Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices

Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices Peter Gerds and Lars Grasedyck Bericht Nr. 30 März 2014 Key words: automated multi-level substructuring,

More information

Computation of Smallest Eigenvalues using Spectral Schur Complements

Computation of Smallest Eigenvalues using Spectral Schur Complements Computation of Smallest Eigenvalues using Spectral Schur Complements Constantine Bekas Yousef Saad January 23, 2004 Abstract The Automated Multilevel Substructing method (AMLS ) was recently presented

More information

Automated Multi-Level Substructuring CHAPTER 4 : AMLS METHOD. Condensation. Exact condensation

Automated Multi-Level Substructuring CHAPTER 4 : AMLS METHOD. Condensation. Exact condensation Automated Multi-Level Substructuring CHAPTER 4 : AMLS METHOD Heinrich Voss voss@tu-harburg.de Hamburg University of Technology AMLS was introduced by Bennighof (1998) and was applied to huge problems of

More information

SOLVING MESH EIGENPROBLEMS WITH MULTIGRID EFFICIENCY

SOLVING 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 information

have invested in supercomputer systems, which have cost up to tens of millions of dollars each. Over the past year or so, however, the future of vecto

have invested in supercomputer systems, which have cost up to tens of millions of dollars each. Over the past year or so, however, the future of vecto MEETING THE NVH COMPUTATIONAL CHALLENGE: AUTOMATED MULTI-LEVEL SUBSTRUCTURING J. K. Bennighof, M. F. Kaplan, y M. B. Muller, y and M. Kim y Department of Aerospace Engineering & Engineering Mechanics The

More information

Multilevel and Adaptive Iterative Substructuring Methods. Jan Mandel University of Colorado Denver

Multilevel and Adaptive Iterative Substructuring Methods. Jan Mandel University of Colorado Denver Multilevel and Adaptive Iterative Substructuring Methods Jan Mandel University of Colorado Denver The multilevel BDDC method is joint work with Bedřich Sousedík, Czech Technical University, and Clark Dohrmann,

More information

A METHOD FOR PROFILING THE DISTRIBUTION OF EIGENVALUES USING THE AS METHOD. Kenta Senzaki, Hiroto Tadano, Tetsuya Sakurai and Zhaojun Bai

A METHOD FOR PROFILING THE DISTRIBUTION OF EIGENVALUES USING THE AS METHOD. Kenta Senzaki, Hiroto Tadano, Tetsuya Sakurai and Zhaojun Bai TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 3A, pp. 839-853, June 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ A METHOD FOR PROFILING THE DISTRIBUTION OF EIGENVALUES USING THE

More information

Adaptive Coarse Space Selection in BDDC and FETI-DP Iterative Substructuring Methods: Towards Fast and Robust Solvers

Adaptive Coarse Space Selection in BDDC and FETI-DP Iterative Substructuring Methods: Towards Fast and Robust Solvers Adaptive Coarse Space Selection in BDDC and FETI-DP Iterative Substructuring Methods: Towards Fast and Robust Solvers Jan Mandel University of Colorado at Denver Bedřich Sousedík Czech Technical University

More information

Preconditioned Eigensolver LOBPCG in hypre and PETSc

Preconditioned Eigensolver LOBPCG in hypre and PETSc Preconditioned Eigensolver LOBPCG in hypre and PETSc Ilya Lashuk, Merico Argentati, Evgueni Ovtchinnikov, and Andrew Knyazev Department of Mathematics, University of Colorado at Denver, P.O. Box 173364,

More information

Multispace and Multilevel BDDC. Jan Mandel University of Colorado at Denver and Health Sciences Center

Multispace and Multilevel BDDC. Jan Mandel University of Colorado at Denver and Health Sciences Center Multispace and Multilevel BDDC Jan Mandel University of Colorado at Denver and Health Sciences Center Based on joint work with Bedřich Sousedík, UCDHSC and Czech Technical University, and Clark R. Dohrmann,

More information

A Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation

A Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation A Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation Tao Zhao 1, Feng-Nan Hwang 2 and Xiao-Chuan Cai 3 Abstract In this paper, we develop an overlapping domain decomposition

More information

EIGIFP: A MATLAB Program for Solving Large Symmetric Generalized Eigenvalue Problems

EIGIFP: A MATLAB Program for Solving Large Symmetric Generalized Eigenvalue Problems EIGIFP: A MATLAB Program for Solving Large Symmetric Generalized Eigenvalue Problems JAMES H. MONEY and QIANG YE UNIVERSITY OF KENTUCKY eigifp is a MATLAB program for computing a few extreme eigenvalues

More information

SOME PRACTICAL ASPECTS OF PARALLEL ADAPTIVE BDDC METHOD

SOME PRACTICAL ASPECTS OF PARALLEL ADAPTIVE BDDC METHOD Conference Applications of Mathematics 2012 in honor of the 60th birthday of Michal Křížek. Institute of Mathematics AS CR, Prague 2012 SOME PRACTICAL ASPECTS OF PARALLEL ADAPTIVE BDDC METHOD Jakub Šístek1,2,

More information

An Algebraic Multigrid Method for Eigenvalue Problems

An Algebraic Multigrid Method for Eigenvalue Problems An Algebraic Multigrid Method for Eigenvalue Problems arxiv:1503.08462v1 [math.na] 29 Mar 2015 Xiaole Han, Yunhui He, Hehu Xie and Chunguang You Abstract An algebraic multigrid method is proposed to solve

More information

On correction equations and domain decomposition for computing invariant subspaces

On correction equations and domain decomposition for computing invariant subspaces On correction equations and domain decomposition for computing invariant subspaces Bernard Philippe Yousef Saad February 1, 26 Abstract By considering the eigenvalue problem as a system of nonlinear equations,

More information

Arnoldi Methods in SLEPc

Arnoldi Methods in SLEPc Scalable Library for Eigenvalue Problem Computations SLEPc Technical Report STR-4 Available at http://slepc.upv.es Arnoldi Methods in SLEPc V. Hernández J. E. Román A. Tomás V. Vidal Last update: October,

More information

Multigrid absolute value preconditioning

Multigrid absolute value preconditioning Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical

More information

Is there life after the Lanczos method? What is LOBPCG?

Is 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 information

ETNA Kent State University

ETNA Kent State University Electronic Transactions on Numerical Analysis. Volume 7, 1998, pp. 4-123. Copyright 1998,. ISSN 68-9613. ETNA PRECONDITIONED EIGENSOLVERS AN OXYMORON? ANDREW V. KNYAZEV y Abstract. A short survey of some

More information

Short title: Total FETI. Corresponding author: Zdenek Dostal, VŠB-Technical University of Ostrava, 17 listopadu 15, CZ Ostrava, Czech Republic

Short title: Total FETI. Corresponding author: Zdenek Dostal, VŠB-Technical University of Ostrava, 17 listopadu 15, CZ Ostrava, Czech Republic Short title: Total FETI Corresponding author: Zdenek Dostal, VŠB-Technical University of Ostrava, 17 listopadu 15, CZ-70833 Ostrava, Czech Republic mail: zdenek.dostal@vsb.cz fax +420 596 919 597 phone

More information

Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures

Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures Stanimire Tomov 1, Julien Langou 1, Andrew Canning 2, Lin-Wang Wang 2, and Jack Dongarra 1 1 Innovative

More information

A Jacobi Davidson Method with a Multigrid Solver for the Hermitian Wilson-Dirac Operator

A Jacobi Davidson Method with a Multigrid Solver for the Hermitian Wilson-Dirac Operator A Jacobi Davidson Method with a Multigrid Solver for the Hermitian Wilson-Dirac Operator Artur Strebel Bergische Universität Wuppertal August 3, 2016 Joint Work This project is joint work with: Gunnar

More information

Contents. Preface... xi. Introduction...

Contents. 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 information

Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures

Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures Conjugate-Gradient Eigenvalue Solvers in Computing Electronic Properties of Nanostructure Architectures Stanimire Tomov 1, Julien Langou 1, Andrew Canning 2, Lin-Wang Wang 2, and Jack Dongarra 1 1 Innovative

More information

Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions

Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New

More information

Recent implementations, applications, and extensions of the Locally Optimal Block Preconditioned Conjugate Gradient method LOBPCG

Recent implementations, applications, and extensions of the Locally Optimal Block Preconditioned Conjugate Gradient method LOBPCG MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Recent implementations, applications, and extensions of the Locally Optimal Block Preconditioned Conjugate Gradient method LOBPCG Knyazev,

More information

A Jacobi Davidson Algorithm for Large Eigenvalue Problems from Opto-Electronics

A Jacobi Davidson Algorithm for Large Eigenvalue Problems from Opto-Electronics RANMEP, Hsinchu, Taiwan, January 4 8, 2008 1/30 A Jacobi Davidson Algorithm for Large Eigenvalue Problems from Opto-Electronics Peter Arbenz 1 Urban Weber 1 Ratko Veprek 2 Bernd Witzigmann 2 1 Institute

More information

Algebraic Multigrid as Solvers and as Preconditioner

Algebraic Multigrid as Solvers and as Preconditioner Ò Algebraic Multigrid as Solvers and as Preconditioner Domenico Lahaye domenico.lahaye@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ domenico/ Department of Computer Science Katholieke Universiteit Leuven

More information

Multilevel spectral coarse space methods in FreeFem++ on parallel architectures

Multilevel spectral coarse space methods in FreeFem++ on parallel architectures Multilevel spectral coarse space methods in FreeFem++ on parallel architectures Pierre Jolivet Laboratoire Jacques-Louis Lions Laboratoire Jean Kuntzmann DD 21, Rennes. June 29, 2012 In collaboration with

More information

Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners

Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department

More information

M.A. Botchev. September 5, 2014

M.A. Botchev. September 5, 2014 Rome-Moscow school of Matrix Methods and Applied Linear Algebra 2014 A short introduction to Krylov subspaces for linear systems, matrix functions and inexact Newton methods. Plan and exercises. M.A. Botchev

More information

Inexact inverse iteration with preconditioning

Inexact 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 information

Preconditioned Locally Minimal Residual Method for Computing Interior Eigenpairs of Symmetric Operators

Preconditioned Locally Minimal Residual Method for Computing Interior Eigenpairs of Symmetric Operators Preconditioned Locally Minimal Residual Method for Computing Interior Eigenpairs of Symmetric Operators Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University

More information

Stabilization and Acceleration of Algebraic Multigrid Method

Stabilization and Acceleration of Algebraic Multigrid Method Stabilization and Acceleration of Algebraic Multigrid Method Recursive Projection Algorithm A. Jemcov J.P. Maruszewski Fluent Inc. October 24, 2006 Outline 1 Need for Algorithm Stabilization and Acceleration

More information

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method

Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method Andrew Knyazev Department of Mathematics University of Colorado at Denver P.O. Box 173364, Campus Box 170 Denver, CO 80217-3364 Time requested: 45 Andrew.Knyazev@cudenver.edu http://www-math.cudenver.edu/~aknyazev/

More information

Preface to the Second Edition. Preface to the First Edition

Preface to the Second Edition. Preface to the First Edition n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................

More information

The Conjugate Gradient Method

The 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 information

Domain decomposition on different levels of the Jacobi-Davidson method

Domain 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 information

Max Planck Institute Magdeburg Preprints

Max Planck Institute Magdeburg Preprints Thomas Mach Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format MAX PLANCK INSTITUT FÜR DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG Max Planck Institute Magdeburg Preprints MPIMD/11-09

More information

Preconditioning Subspace Iteration for Large Eigenvalue Problems with Automated Multi-Level Sub-structuring

Preconditioning Subspace Iteration for Large Eigenvalue Problems with Automated Multi-Level Sub-structuring Preconditioning Subspace Iteration for Large Eigenvalue Problems with Automated Multi-Level Sub-structuring Heinrich Voss 1, and Jiacong Yin 2 and Pu Chen 2 1 Institute of Mathematics, Hamburg University

More information

Multigrid and Iterative Strategies for Optimal Control Problems

Multigrid and Iterative Strategies for Optimal Control Problems Multigrid and Iterative Strategies for Optimal Control Problems John Pearson 1, Stefan Takacs 1 1 Mathematical Institute, 24 29 St. Giles, Oxford, OX1 3LB e-mail: john.pearson@worc.ox.ac.uk, takacs@maths.ox.ac.uk

More information

Solving PDEs with Multigrid Methods p.1

Solving PDEs with Multigrid Methods p.1 Solving PDEs with Multigrid Methods Scott MacLachlan maclachl@colorado.edu Department of Applied Mathematics, University of Colorado at Boulder Solving PDEs with Multigrid Methods p.1 Support and Collaboration

More information

Solving Large Nonlinear Sparse Systems

Solving Large Nonlinear Sparse Systems Solving Large Nonlinear Sparse Systems Fred W. Wubs and Jonas Thies Computational Mechanics & Numerical Mathematics University of Groningen, the Netherlands f.w.wubs@rug.nl Centre for Interdisciplinary

More information

An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems

An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems WEIGUO GAO Fudan University XIAOYE S. LI and CHAO YANG Lawrence Berkeley National Laboratory and ZHAOJUN BAI University

More information

ADDITIVE SCHWARZ FOR SCHUR COMPLEMENT 305 the parallel implementation of both preconditioners on distributed memory platforms, and compare their perfo

ADDITIVE SCHWARZ FOR SCHUR COMPLEMENT 305 the parallel implementation of both preconditioners on distributed memory platforms, and compare their perfo 35 Additive Schwarz for the Schur Complement Method Luiz M. Carvalho and Luc Giraud 1 Introduction Domain decomposition methods for solving elliptic boundary problems have been receiving increasing attention

More information

Multilevel low-rank approximation preconditioners Yousef Saad Department of Computer Science and Engineering University of Minnesota

Multilevel low-rank approximation preconditioners Yousef Saad Department of Computer Science and Engineering University of Minnesota Multilevel low-rank approximation preconditioners Yousef Saad Department of Computer Science and Engineering University of Minnesota SIAM CSE Boston - March 1, 2013 First: Joint work with Ruipeng Li Work

More information

problem Au = u by constructing an orthonormal basis V k = [v 1 ; : : : ; v k ], at each k th iteration step, and then nding an approximation for the e

problem Au = u by constructing an orthonormal basis V k = [v 1 ; : : : ; v k ], at each k th iteration step, and then nding an approximation for the e A Parallel Solver for Extreme Eigenpairs 1 Leonardo Borges and Suely Oliveira 2 Computer Science Department, Texas A&M University, College Station, TX 77843-3112, USA. Abstract. In this paper a parallel

More information

Conjugate-gradient eigenvalue solvers in computing electronic properties of nanostructure architectures. Stanimire Tomov*

Conjugate-gradient eigenvalue solvers in computing electronic properties of nanostructure architectures. Stanimire Tomov* Int. J. Computational Science and Engineering, Vol. 2, Nos. 3/4, 2006 205 Conjugate-gradient eigenvalue solvers in computing electronic properties of nanostructure architectures Stanimire Tomov* Innovative

More information

Application of Lanczos and Schur vectors in structural dynamics

Application of Lanczos and Schur vectors in structural dynamics Shock and Vibration 15 (2008) 459 466 459 IOS Press Application of Lanczos and Schur vectors in structural dynamics M. Radeş Universitatea Politehnica Bucureşti, Splaiul Independenţei 313, Bucureşti, Romania

More information

From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes

From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes www.oeaw.ac.at From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes D. Copeland, U. Langer, D. Pusch RICAM-Report 2008-10 www.ricam.oeaw.ac.at From the Boundary Element

More information

FETI domain decomposition method to solution of contact problems with large displacements

FETI domain decomposition method to solution of contact problems with large displacements FETI domain decomposition method to solution of contact problems with large displacements Vít Vondrák 1, Zdeněk Dostál 1, Jiří Dobiáš 2, and Svatopluk Pták 2 1 Dept. of Appl. Math., Technical University

More information

Auxiliary space multigrid method for elliptic problems with highly varying coefficients

Auxiliary space multigrid method for elliptic problems with highly varying coefficients Auxiliary space multigrid method for elliptic problems with highly varying coefficients Johannes Kraus 1 and Maria Lymbery 2 1 Introduction The robust preconditioning of linear systems of algebraic equations

More information

Algebraic Coarse Spaces for Overlapping Schwarz Preconditioners

Algebraic Coarse Spaces for Overlapping Schwarz Preconditioners Algebraic Coarse Spaces for Overlapping Schwarz Preconditioners 17 th International Conference on Domain Decomposition Methods St. Wolfgang/Strobl, Austria July 3-7, 2006 Clark R. Dohrmann Sandia National

More information

A decade of fast and robust Helmholtz solvers

A decade of fast and robust Helmholtz solvers A decade of fast and robust Helmholtz solvers Werkgemeenschap Scientific Computing Spring meeting Kees Vuik May 11th, 212 1 Delft University of Technology Contents Introduction Preconditioning (22-28)

More information

Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures

Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Efficient Reduced Order Modeling of Low- to Mid-Frequency Vibration and Power Flow in Complex Structures Yung-Chang Tan Graduate Student Research Assistant Matthew P. Castanier Assistant Research Scientist

More information

APPLIED NUMERICAL LINEAR ALGEBRA

APPLIED NUMERICAL LINEAR ALGEBRA APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface 1 Introduction 1 1.1 Basic Notation

More information

BETI for acoustic and electromagnetic scattering

BETI for acoustic and electromagnetic scattering BETI for acoustic and electromagnetic scattering O. Steinbach, M. Windisch Institut für Numerische Mathematik Technische Universität Graz Oberwolfach 18. Februar 2010 FWF-Project: Data-sparse Boundary

More information

Matrix Iteration. Giacomo Boffi.

Matrix Iteration. Giacomo Boffi. http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 12, 2016 Outline Second -Ritz Method Dynamic analysis of MDOF

More information

Adaptive algebraic multigrid methods in lattice computations

Adaptive algebraic multigrid methods in lattice computations Adaptive algebraic multigrid methods in lattice computations Karsten Kahl Bergische Universität Wuppertal January 8, 2009 Acknowledgements Matthias Bolten, University of Wuppertal Achi Brandt, Weizmann

More information

A Parallel Implementation of the Davidson Method for Generalized Eigenproblems

A Parallel Implementation of the Davidson Method for Generalized Eigenproblems A Parallel Implementation of the Davidson Method for Generalized Eigenproblems Eloy ROMERO 1 and Jose E. ROMAN Instituto ITACA, Universidad Politécnica de Valencia, Spain Abstract. We present a parallel

More information

From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes

From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes From the Boundary Element Domain Decomposition Methods to Local Trefftz Finite Element Methods on Polyhedral Meshes Dylan Copeland 1, Ulrich Langer 2, and David Pusch 3 1 Institute of Computational Mathematics,

More information

PERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD 1. INTRODUCTION

PERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD 1. INTRODUCTION J. KSIAM Vol., No., -0, 009 PERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD Wanil BYUN AND Seung Jo KIM, SCHOOL OF MECHANICAL AND AEROSPACE ENG, SEOUL NATIONAL UNIV, SOUTH

More information

A CHEBYSHEV-DAVIDSON ALGORITHM FOR LARGE SYMMETRIC EIGENPROBLEMS

A CHEBYSHEV-DAVIDSON ALGORITHM FOR LARGE SYMMETRIC EIGENPROBLEMS A CHEBYSHEV-DAVIDSON ALGORITHM FOR LARGE SYMMETRIC EIGENPROBLEMS YUNKAI ZHOU AND YOUSEF SAAD Abstract. A polynomial filtered Davidson-type algorithm is proposed for solving symmetric eigenproblems. The

More information

Incomplete Cholesky preconditioners that exploit the low-rank property

Incomplete 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 information

A Hierarchy of Preconditioned Eigensolvers for Elliptic Differential Operators

A Hierarchy of Preconditioned Eigensolvers for Elliptic Differential Operators A Hierarchy of Preconditioned Eigensolvers for Elliptic Differential Operators Habilitationsschrift von Dr. Klaus Neymeyr Mathematische Fakultät Universität Tübingen September 2001 V02/02WWW CONTENTS

More information

THE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES. Lou Menglin* SUMMARY

THE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES. Lou Menglin* SUMMARY 264 THE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES Lou Mengl* SUMMARY In this paper, the static substructure method based on the Ritz vector direct superposition method is suggested

More information

Vibration Transmission in Complex Vehicle Structures

Vibration Transmission in Complex Vehicle Structures Vibration Transmission in Complex Vehicle Structures Christophe Pierre Professor Matthew P. Castanier Assistant Research Scientist Yung-Chang Tan Dongying Jiang Graduate Student Research Assistants Vibrations

More information

Solving Ax = b, an overview. Program

Solving Ax = b, an overview. Program Numerical Linear Algebra Improving iterative solvers: preconditioning, deflation, numerical software and parallelisation Gerard Sleijpen and Martin van Gijzen November 29, 27 Solving Ax = b, an overview

More information

DELFT UNIVERSITY OF TECHNOLOGY

DELFT 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 information

A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation

A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation Zih-Hao Wei 1, Feng-Nan Hwang 1, Tsung-Ming Huang 2, and Weichung Wang 3 1 Department of

More information

2C9 Design for seismic and climate changes. Jiří Máca

2C9 Design for seismic and climate changes. Jiří Máca 2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom

More information

Using 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 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 information

ETNA Kent State University

ETNA Kent State University Electronic Transactions on Numerical Analysis. Volume 15, pp. 38-55, 2003. Copyright 2003,. ISSN 1068-9613. ETNA EFFICIENT SOLUTION OF SYMMETRIC EIGENVALUE PROBLEMS USING MULTIGRID PRECONDITIONERS IN THE

More information

A FETI-DP method for the parallel iterative solution of indefinite and complex-valued solid and shell vibration problems

A FETI-DP method for the parallel iterative solution of indefinite and complex-valued solid and shell vibration problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1282 A FETI-DP method for the parallel iterative solution

More information

Partitioned Formulation with Localized Lagrange Multipliers And its Applications **

Partitioned Formulation with Localized Lagrange Multipliers And its Applications ** Partitioned Formulation with Localized Lagrange Multipliers And its Applications ** K.C. Park Center for Aerospace Structures (CAS), University of Colorado at Boulder ** Carlos Felippa, Gert Rebel, Yong

More information

DELFT UNIVERSITY OF TECHNOLOGY

DELFT 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 information

Electronic Transactions on Numerical Analysis Volume 49, 2018

Electronic Transactions on Numerical Analysis Volume 49, 2018 Electronic Transactions on Numerical Analysis Volume 49, 2018 Contents 1 Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems. Axel Klawonn, Martin Kühn,

More information

DISPENSA FEM in MSC. Nastran

DISPENSA FEM in MSC. Nastran DISPENSA FEM in MSC. Nastran preprocessing: mesh generation material definitions definition of loads and boundary conditions solving: solving the (linear) set of equations components postprocessing: visualisation

More information

The Deflation Accelerated Schwarz Method for CFD

The Deflation Accelerated Schwarz Method for CFD The Deflation Accelerated Schwarz Method for CFD J. Verkaik 1, C. Vuik 2,, B.D. Paarhuis 1, and A. Twerda 1 1 TNO Science and Industry, Stieltjesweg 1, P.O. Box 155, 2600 AD Delft, The Netherlands 2 Delft

More information

APVC2009. Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations

APVC2009. Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations Forced Vibration Analysis of the Flexible Spinning Disk-spindle System Represented by Asymmetric Finite Element Equations Kiyong Park, Gunhee Jang* and Chanhee Seo Department of Mechanical Engineering,

More information

An Efficient FETI Implementation on Distributed Shared Memory Machines with Independent Numbers of Subdomains and Processors

An Efficient FETI Implementation on Distributed Shared Memory Machines with Independent Numbers of Subdomains and Processors Contemporary Mathematics Volume 218, 1998 B 0-8218-0988-1-03024-7 An Efficient FETI Implementation on Distributed Shared Memory Machines with Independent Numbers of Subdomains and Processors Michel Lesoinne

More information

Parallel Discontinuous Galerkin Method

Parallel Discontinuous Galerkin Method Parallel Discontinuous Galerkin Method Yin Ki, NG The Chinese University of Hong Kong Aug 5, 2015 Mentors: Dr. Ohannes Karakashian, Dr. Kwai Wong Overview Project Goal Implement parallelization on Discontinuous

More information

LARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems

LARGE SPARSE EIGENVALUE PROBLEMS. General Tools for Solving Large Eigen-Problems LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization General Tools for Solving Large Eigen-Problems

More information

Scalable Domain Decomposition Preconditioners For Heterogeneous Elliptic Problems

Scalable Domain Decomposition Preconditioners For Heterogeneous Elliptic Problems Scalable Domain Decomposition Preconditioners For Heterogeneous Elliptic Problems Pierre Jolivet, F. Hecht, F. Nataf, C. Prud homme Laboratoire Jacques-Louis Lions Laboratoire Jean Kuntzmann INRIA Rocquencourt

More information

of dimension n 1 n 2, one defines the matrix determinants

of 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 information

A Novel Aggregation Method based on Graph Matching for Algebraic MultiGrid Preconditioning of Sparse Linear Systems

A Novel Aggregation Method based on Graph Matching for Algebraic MultiGrid Preconditioning of Sparse Linear Systems A Novel Aggregation Method based on Graph Matching for Algebraic MultiGrid Preconditioning of Sparse Linear Systems Pasqua D Ambra, Alfredo Buttari, Daniela Di Serafino, Salvatore Filippone, Simone Gentile,

More information

Partial Differential Equations and the Finite Element Method

Partial Differential Equations and the Finite Element Method Partial Differential Equations and the Finite Element Method Pavel Solin The University of Texas at El Paso Academy of Sciences ofthe Czech Republic iwiley- INTERSCIENCE A JOHN WILEY & SONS, INC, PUBLICATION

More information

State-of-the-art numerical solution of large Hermitian eigenvalue problems. Andreas Stathopoulos

State-of-the-art numerical solution of large Hermitian eigenvalue problems. Andreas Stathopoulos State-of-the-art numerical solution of large Hermitian eigenvalue problems Andreas Stathopoulos Computer Science Department and Computational Sciences Cluster College of William and Mary Acknowledgment:

More information

Lecture 17: Iterative Methods and Sparse Linear Algebra

Lecture 17: Iterative Methods and Sparse Linear Algebra Lecture 17: Iterative Methods and Sparse Linear Algebra David Bindel 25 Mar 2014 Logistics HW 3 extended to Wednesday after break HW 4 should come out Monday after break Still need project description

More information

LARGE SPARSE EIGENVALUE PROBLEMS

LARGE SPARSE EIGENVALUE PROBLEMS LARGE SPARSE EIGENVALUE PROBLEMS Projection methods The subspace iteration Krylov subspace methods: Arnoldi and Lanczos Golub-Kahan-Lanczos bidiagonalization 14-1 General Tools for Solving Large Eigen-Problems

More information

Fast algorithms for the inverse medium problem. George Biros University of Pennsylvania

Fast algorithms for the inverse medium problem. George Biros University of Pennsylvania Fast algorithms for the inverse medium problem George Biros University of Pennsylvania Acknowledgments S. Adavani, H. Sundar, S. Rahul (grad students) C. Davatzikos, D. Shen, H. Litt (heart project) Akcelic,

More information

Implementation of a preconditioned eigensolver using Hypre

Implementation of a preconditioned eigensolver using Hypre Implementation of a preconditioned eigensolver using Hypre Andrew V. Knyazev 1, and Merico E. Argentati 1 1 Department of Mathematics, University of Colorado at Denver, USA SUMMARY This paper describes

More information

Scientific Computing with Case Studies SIAM Press, Lecture Notes for Unit VII Sparse Matrix

Scientific 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 information

Spectral Processing. Misha Kazhdan

Spectral Processing. Misha Kazhdan Spectral Processing Misha Kazhdan [Taubin, 1995] A Signal Processing Approach to Fair Surface Design [Desbrun, et al., 1999] Implicit Fairing of Arbitrary Meshes [Vallet and Levy, 2008] Spectral Geometry

More information

The All-floating BETI Method: Numerical Results

The All-floating BETI Method: Numerical Results The All-floating BETI Method: Numerical Results Günther Of Institute of Computational Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria, of@tugraz.at Summary. The all-floating

More information

c 2006 Society for Industrial and Applied Mathematics

c 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 information

Toward black-box adaptive domain decomposition methods

Toward black-box adaptive domain decomposition methods Toward black-box adaptive domain decomposition methods Frédéric Nataf Laboratory J.L. Lions (LJLL), CNRS, Alpines Inria and Univ. Paris VI joint work with Victorita Dolean (Univ. Nice Sophia-Antipolis)

More information

HARMONIC RAYLEIGH RITZ EXTRACTION FOR THE MULTIPARAMETER EIGENVALUE PROBLEM

HARMONIC 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 information

Robust solution of Poisson-like problems with aggregation-based AMG

Robust solution of Poisson-like problems with aggregation-based AMG Robust solution of Poisson-like problems with aggregation-based AMG Yvan Notay Université Libre de Bruxelles Service de Métrologie Nucléaire Paris, January 26, 215 Supported by the Belgian FNRS http://homepages.ulb.ac.be/

More information

Numerical Methods in Matrix Computations

Numerical Methods in Matrix Computations Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices

More information