Solvers for Sparse Linear Systems
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Arxiv:1911.09220V2 [Cs.MS] 13 Jul 2020
MFEM: A MODULAR FINITE ELEMENT METHODS LIBRARY ROBERT ANDERSON, JULIAN ANDREJ, ANDREW BARKER, JAMIE BRAMWELL, JEAN- SYLVAIN CAMIER, JAKUB CERVENY, VESELIN DOBREV, YOHANN DUDOUIT, AARON FISHER, TZANIO KOLEV, WILL PAZNER, MARK STOWELL, VLADIMIR TOMOV Lawrence Livermore National Laboratory, Livermore, USA IDO AKKERMAN Delft University of Technology, Netherlands JOHANN DAHM IBM Research { Almaden, Almaden, USA DAVID MEDINA Occalytics, LLC, Houston, USA STEFANO ZAMPINI King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Abstract. MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of dis- cretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM's goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element mesh- ing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. arXiv:1911.09220v2 [cs.MS] 13 Jul 2020 1. Introduction The Finite Element Method (FEM) is a powerful discretization technique that uses general unstructured grids to approximate the solutions of many partial differential equations (PDEs). It has been exhaustively studied, both theoretically and in practice, in the past several decades [1, 2, 3, 4, 5, 6, 7, 8]. MFEM is an open-source, lightweight, modular and scalable software library for finite elements, featuring arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing (HPC) efficiency [9]. -
Overview of Iterative Linear System Solver Packages
Overview of Iterative Linear System Solver Packages Victor Eijkhout July, 1998 Abstract Description and comparison of several packages for the iterative solu- tion of linear systems of equations. 1 1 Intro duction There are several freely available packages for the iterative solution of linear systems of equations, typically derived from partial di erential equation prob- lems. In this rep ort I will give a brief description of a numberofpackages, and giveaninventory of their features and de ning characteristics. The most imp ortant features of the packages are which iterative metho ds and preconditioners supply; the most relevant de ning characteristics are the interface they present to the user's data structures, and their implementation language. 2 2 Discussion Iterative metho ds are sub ject to several design decisions that a ect ease of use of the software and the resulting p erformance. In this section I will give a global discussion of the issues involved, and how certain p oints are addressed in the packages under review. 2.1 Preconditioners A go o d preconditioner is necessary for the convergence of iterative metho ds as the problem to b e solved b ecomes more dicult. Go o d preconditioners are hard to design, and this esp ecially holds true in the case of parallel pro cessing. Here is a short inventory of the various kinds of preconditioners found in the packages reviewed. 2.1.1 Ab out incomplete factorisation preconditioners Incomplete factorisations are among the most successful preconditioners devel- op ed for single-pro cessor computers. Unfortunately, since they are implicit in nature, they cannot immediately b e used on parallel architectures. -
Accelerating the LOBPCG Method on Gpus Using a Blocked Sparse Matrix Vector Product
Accelerating the LOBPCG method on GPUs using a blocked Sparse Matrix Vector Product Hartwig Anzt and Stanimire Tomov and Jack Dongarra Innovative Computing Lab University of Tennessee Knoxville, USA Email: [email protected], [email protected], [email protected] Abstract— the computing power of today’s supercomputers, often accel- erated by coprocessors like graphics processing units (GPUs), This paper presents a heterogeneous CPU-GPU algorithm design and optimized implementation for an entire sparse iter- becomes challenging. ative eigensolver – the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) – starting from low-level GPU While there exist numerous efforts to adapt iterative lin- data structures and kernels to the higher-level algorithmic choices ear solvers like Krylov subspace methods to coprocessor and overall heterogeneous design. Most notably, the eigensolver technology, sparse eigensolvers have so far remained out- leverages the high-performance of a new GPU kernel developed side the main focus. A possible explanation is that many for the simultaneous multiplication of a sparse matrix and a of those combine sparse and dense linear algebra routines, set of vectors (SpMM). This is a building block that serves which makes porting them to accelerators more difficult. Aside as a backbone for not only block-Krylov, but also for other from the power method, algorithms based on the Krylov methods relying on blocking for acceleration in general. The subspace idea are among the most commonly used general heterogeneous LOBPCG developed here reveals the potential of eigensolvers [1]. When targeting symmetric positive definite this type of eigensolver by highly optimizing all of its components, eigenvalue problems, the recently developed Locally Optimal and can be viewed as a benchmark for other SpMM-dependent applications. -
MFEM: a Modular Finite Element Methods Library
MFEM: A Modular Finite Element Methods Library Robert Anderson1, Andrew Barker1, Jamie Bramwell1, Jakub Cerveny2, Johann Dahm3, Veselin Dobrev1,YohannDudouit1, Aaron Fisher1,TzanioKolev1,MarkStowell1,and Vladimir Tomov1 1Lawrence Livermore National Laboratory 2University of West Bohemia 3IBM Research July 2, 2018 Abstract MFEM is a free, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. Its mission is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers. MFEM also enables researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. Contents 1 Introduction 3 2 Overview of the Finite Element Method 4 3Meshes 9 3.1 Conforming Meshes . 10 3.2 Non-Conforming Meshes . 11 3.3 NURBS Meshes . 12 3.4 Parallel Meshes . 12 3.5 Supported Input and Output Formats . 13 1 4 Finite Element Spaces 13 4.1 FiniteElements....................................... 14 4.2 DiscretedeRhamComplex ................................ 16 4.3 High-OrderSpaces ..................................... 17 4.4 Visualization . 18 5 Finite Element Operators 18 5.1 DiscretizationMethods................................... 18 5.2 FiniteElementLinearSystems . 19 5.3 Operator Decomposition . 23 5.4 High-Order Partial Assembly . 25 6 High-Performance Computing 27 6.1 Parallel Meshes, Spaces, and Operators . 27 6.2 Scalable Linear Solvers . -
A Geometric Theory for Preconditioned Inverse Iteration. III: a Short and Sharp Convergence Estimate for Generalized Eigenvalue Problems
A geometric theory for preconditioned inverse iteration. III: A short and sharp convergence estimate for generalized eigenvalue problems. Andrew V. Knyazev Department of Mathematics, University of Colorado at Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364 1 Klaus Neymeyr Mathematisches Institut der Universit¨atT¨ubingen,Auf der Morgenstelle 10, 72076 T¨ubingen,Germany 2 Abstract In two previous papers by Neymeyr: A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient, LAA 322: (1-3), 61-85, 2001, and A geometric theory for preconditioned inverse iteration II: Convergence estimates, LAA 322: (1-3), 87-104, 2001, a sharp, but cumbersome, convergence rate estimate was proved for a simple preconditioned eigensolver, which computes the smallest eigenvalue together with the corresponding eigenvector of a symmetric positive def- inite matrix, using a preconditioned gradient minimization of the Rayleigh quotient. In the present paper, we discover and prove a much shorter and more elegant, but still sharp in decisive quantities, convergence rate estimate of the same method that also holds for a generalized symmetric definite eigenvalue problem. The new estimate is simple enough to stimulate a search for a more straightforward proof technique that could be helpful to investigate such practically important method as the locally optimal block preconditioned conjugate gradient eigensolver. We demon- strate practical effectiveness of the latter for a model problem, where it compares favorably with two -
16 Preconditioning
16 Preconditioning The general idea underlying any preconditioning procedure for iterative solvers is to modify the (ill-conditioned) system Ax = b in such a way that we obtain an equivalent system Aˆxˆ = bˆ for which the iterative method converges faster. A standard approach is to use a nonsingular matrix M, and rewrite the system as M −1Ax = M −1b. The preconditioner M needs to be chosen such that the matrix Aˆ = M −1A is better conditioned for the conjugate gradient method, or has better clustered eigenvalues for the GMRES method. 16.1 Preconditioned Conjugate Gradients We mentioned earlier that the number of iterations required for the conjugate gradient algorithm to converge is proportional to pκ(A). Thus, for poorly conditioned matrices, convergence will be very slow. Thus, clearly we will want to choose M such that κ(Aˆ) < κ(A). This should result in faster convergence. How do we find Aˆ, xˆ, and bˆ? In order to ensure symmetry and positive definiteness of Aˆ we let M −1 = LLT (44) with a nonsingular m × m matrix L. Then we can rewrite Ax = b ⇐⇒ M −1Ax = M −1b ⇐⇒ LT Ax = LT b ⇐⇒ LT AL L−1x = LT b . | {z } | {z } |{z} =Aˆ =xˆ =bˆ The symmetric positive definite matrix M is called splitting matrix or preconditioner, and it can easily be verified that Aˆ is symmetric positive definite, also. One could now formally write down the standard CG algorithm with the new “hat- ted” quantities. However, the algorithm is more efficient if the preconditioning is incorporated directly into the iteration. To see what this means we need to examine every single line in the CG algorithm. -
XAMG: a Library for Solving Linear Systems with Multiple Right-Hand
XAMG: A library for solving linear systems with multiple right-hand side vectors B. Krasnopolsky∗, A. Medvedev∗∗ Institute of Mechanics, Lomonosov Moscow State University, 119192 Moscow, Michurinsky ave. 1, Russia Abstract This paper presents the XAMG library for solving large sparse systems of linear algebraic equations with multiple right-hand side vectors. The library specializes but is not limited to the solution of linear systems obtained from the discretization of elliptic differential equations. A corresponding set of numerical methods includes Krylov subspace, algebraic multigrid, Jacobi, Gauss-Seidel, and Chebyshev iterative methods. The parallelization is im- plemented with MPI+POSIX shared memory hybrid programming model, which introduces a three-level hierarchical decomposition using the corre- sponding per-level synchronization and communication primitives. The code contains a number of optimizations, including the multilevel data segmen- tation, compression of indices, mixed-precision floating-point calculations, arXiv:2103.07329v1 [cs.MS] 12 Mar 2021 vector status flags, and others. The XAMG library uses the program code of the well-known hypre library to construct the multigrid matrix hierar- chy. The XAMG’s own implementation for the solve phase of the iterative methods provides up to a twofold speedup compared to hypre for the tests ∗E-mail address: [email protected] ∗∗E-mail address: [email protected] Preprint submitted to Elsevier March 15, 2021 performed. Additionally, XAMG provides extended functionality to solve systems with multiple right-hand side vectors. Keywords: systems of linear algebraic equations, Krylov subspace iterative methods, algebraic multigrid method, multiple right-hand sides, hybrid programming model, MPI+POSIX shared memory Nr. -
Numerical Linear Algebra Solving Ax = B , an Overview Introduction
Solving Ax = b, an overview Numerical Linear Algebra ∗ A = A no Good precond. yes flex. precond. yes GCR Improving iterative solvers: ⇒ ⇒ ⇒ yes no no ⇓ ⇓ ⇓ preconditioning, deflation, numerical yes GMRES A > 0 CG software and parallelisation ⇒ ⇓ no ⇓ ⇓ ill cond. yes SYMMLQ str indef no large im eig no Bi-CGSTAB ⇒ ⇒ ⇒ no yes yes ⇓ ⇓ ⇓ MINRES IDR no large im eig BiCGstab(ℓ) ⇐ yes Gerard Sleijpen and Martin van Gijzen a good precond itioner is available ⇓ the precond itioner is flex ible IDRstab November 29, 2017 A + A∗ is strongly indefinite A 1 has large imaginary eigenvalues November 29, 2017 2 National Master Course National Master Course Delft University of Technology Introduction Program We already saw that the performance of iterative methods can Preconditioning be improved by applying a preconditioner. Preconditioners (and • Diagonal scaling, Gauss-Seidel, SOR and SSOR deflation techniques) are a key to successful iterative methods. • Incomplete Choleski and Incomplete LU In general they are very problem dependent. • Deflation Today we will discuss some standard preconditioners and we will • Numerical software explain the idea behind deflation. • Parallelisation We will also discuss some efforts to standardise numerical • software. Shared memory versus distributed memory • Domain decomposition Finally we will discuss how to perform scientific computations on • a parallel computer. November 29, 2017 3 November 29, 2017 4 National Master Course National Master Course Preconditioning Why preconditioners? A preconditioned iterative solver solves the system − − M 1Ax = M 1b. 1 The matrix M is called the preconditioner. 0.8 The preconditioner should satisfy certain requirements: 0.6 Convergence should be (much) faster (in time) for the 0.4 • preconditioned system than for the original system. -
Preconditioned 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 Systems Magdeburg 3rd May 2011 joint work with Alastair Spence (Bath) 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions Outline 1 Introduction 2 Inexact inverse iteration 3 Inexact Shift-invert Arnoldi method 4 Inexact Shift-invert Arnoldi method with implicit restarts 5 Conclusions Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Iterative solves Power method Simultaneous iteration Arnoldi method Jacobi-Davidson method repeated application of the matrix A to a vector Generally convergence to largest/outlying eigenvector Problem and iterative methods Find a small number of eigenvalues and eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Iterative solves Power method Simultaneous iteration Arnoldi method Jacobi-Davidson method The first three of these involve repeated application of the matrix A to a vector Generally convergence to largest/outlying eigenvector Shift-invert strategy Wish to find a few eigenvalues close to a shift σ σ λλ λ λ λ 3 1 2 4 n Shift-invert strategy Wish to find a few eigenvalues close to a shift σ σ λλ λ λ λ 3 1 2 4 n Problem becomes − 1 (A − σI) 1x = x λ − σ each step of the iterative method involves repeated application of A =(A − σI)−1 to a vector Inner iterative solve: (A − σI)y = x using Krylov method for linear systems. -
Preconditioning the Coarse Problem of BDDC Methods—Three-Level, Algebraic Multigrid, and Vertex-Based Preconditioners
Electronic Transactions on Numerical Analysis. Volume 51, pp. 432–450, 2019. ETNA Kent State University and Copyright c 2019, Kent State University. Johann Radon Institute (RICAM) ISSN 1068–9613. DOI: 10.1553/etna_vol51s432 PRECONDITIONING THE COARSE PROBLEM OF BDDC METHODS— THREE-LEVEL, ALGEBRAIC MULTIGRID, AND VERTEX-BASED PRECONDITIONERS∗ AXEL KLAWONNyz, MARTIN LANSERyz, OLIVER RHEINBACHx, AND JANINE WEBERy Abstract. A comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver using the same software building blocks is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner. It is new that all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc to allow for a simple and meaningful comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for a vertex-based approximate BDDC method. Key words. approximate BDDC, three-level BDDC, multilevel BDDC, vertex-based BDDC AMS subject classifications. 68W10, 65N22, 65N55, 65F08, 65F10, 65Y05 1. Introduction. During the last decade, approximate variants of the BDDC (Balanc- ing Domain Decomposition by Constraints) and FETI-DP (Finite Element Tearing and Interconnecting-Dual-Primal) methods have become popular for the solution of various linear and nonlinear partial differential equations [1,8,9, 12, 14, 15, 17, 19, 21, 24, 25]. -
The Preconditioned Conjugate Gradient Method with Incomplete Factorization Preconditioners
Computers Math. Applic. Vol. 31, No. 4/5, pp. 1-5, 1996 Pergamon Copyright©1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/96 $15.00 4- 0.00 0898-1221 (95)00210-3 The Preconditioned Conjugate Gradient Method with Incomplete Factorization Preconditioners I. ARANY University of Miskolc, Institute of Mathematics 3515 Miskolc-Egyetemv~iros, Hungary mat arany©gold, uni-miskolc, hu Abstract--We present a modification of the ILUT algorithm due to Y. Sand for preparing in- complete factorization preconditioner for positive definite matrices being neither diagonally dominant nor M-matrices. In all tested cases, the rate of convergence of the relevant preconditioned conjugate gradient method was at a sufficient level, when linear systems arising from static problem for bar structures with 3D beam elements were solved. Keywords--Preconditioned conjugate gradient method, Incomplete factorization. 1. INTRODUCTION For solving large sparse positive definite systems of linear equations, the conjugate gradient method combined with a preconditoning technique seems to be one of the most efficient ways among the simple iterative methods. As known, for positive definite symmetric diagonally dom- inant M-matrices, the incomplete factorization technique may be efficiently used for preparing preconditioners [1]. However, for more general cases, when the matrix is neither diagonally domi- nant nor M-matrix, but it is a positive definite one, some incomplete factorization techniques [2,3] seem to be quite unusable, since usually, during the process, a diagonal element is turning to zero. We present a modification to the ILUT algorithm of Sand [2] for preparing preconditioners for the above mentioned matrices. -
Performance of H-Lu Preconditioning for Sparse Matrices
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.8(2008), No.4, pp.336–349 c 2008 Institute of Mathematics of the National Academy of Sciences of Belarus PERFORMANCE OF H-LU PRECONDITIONING FOR SPARSE MATRICES L. GRASEDYCK1 W. HACKBUSCH2, AND R. KRIEMANN2 Abstract — In this paper we review the technique of hierarchical matrices and put it into the context of black-box solvers for large linear systems. Numerical examples for several classes of problems from medium- to large-scale illustrate the applicability and efficiency of this technique. We compare the results with those of several direct solvers (which typically scale quadratically in the matrix size) as well as an iterative solver (algebraic multigrid) which scales linearly (if it converges in O(1) steps). 2000 Mathematics Subject Classification: 65F05, 65F30, 65F50, 65N55. Keywords: hierarchical matrices, black-box clustering, preconditioning, LU, direct solver, iterative solver, performance. 1. Introduction Many methods for the fast solution of large-scale linear systems exploit some hierarchical structure of the system. For example, when the system stems from the finite element (FE) discretization of a partial differential equation (PDE) posed in a domain Ω ⊂ Rd, then one can exploit the knowledge of the underlying geometrical problem. Whereas multigrid meth- ods use a discretization of the pde on different levels of resolution, hierarchical (H-) matrices use the geometry information associated to the degrees of freedom on a single level in order to find a distance measure for degrees of freedom. Once this distance measure is available, an almost black-box type arithmetic can be applied for the inversion or factorization of the system matrix.