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A Parallel Bisection and Inverse Iteration Solver for a Subset of Eigenpairs of Symmetric Band Matrices 1

引用

2nd International Workshop on Eigenvalue Problems - Algorithms, Software and Applications in Petascale Computing (EPASA)

作者： Ishigami, Hiroyuki Hasegawa, Hidehiko Kimura, Kinji Nakamura, Yoshimasa Kyoto Univ Grad Sch Informat Sakyo Ku Kyoto Kyoto Japan Yahoo Japan Corp Chiyoda Ku Kioi Tower1-3 Kioicho Tokyo Japan Univ Tsukuba Fac Lib Informat & Media Sci Kasuga 1-2 Tsukuba Ibaraki Japan

ISBN: (数字)9783319624266

ISBN: (纸本)9783319624266;9783319624242

The tridiagonalization and its back-transformation for computing eigen-pairs of real symmetric dense matrices are known to be the bottleneck of the execution time in parallel processing owing to the communication cost and the number of floating-point operations. To overcome this problem, we focus on real symmetric band eigensolvers proposed by Gupta and Murata since their eigensolvers are composed of the bisection and inverse iteration algorithms and do not include neither the tridiagonalization of real symmetric band matrices nor its back-transformation. In this paper, the following parallel solver for computing a subset of eigenpairs of real symmetric band matrices is proposed on the basis of Murata's eigensolver: the desired eigenvalues of the target band matrices are computed directly by using parallel Murata's bisection algorithm. The corresponding eigenvectors are computed by using block inverse iteration algorithm with reorthogonalization, which can be parallelized with lower communication cost than the inverse iteration algorithm. Numerical experiments on shared-memory multi-core processors show that the proposed eigensolver is faster than the conventional solvers.

关键词： Eigenvalues and eigenfunctions

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Space-Time CFOSLS Methods with AMGe Upscaling 23rd

Space-Time CFOSLS Methods with AMGe Upscaling

引用

23rd International Conference on Domain Decomposition Methods

作者： Neumuller, Martin Vassilevski, Panayot S. Villa, Umberto E. Johannes Kepler Univ Linz Inst Computat Math Altenberger Str 69 A-4040 Linz Austria Lawrence Livermore Natl Lab Ctr Appl Sci Comp POB 808L-561 Livermore CA 94551 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97201 USA Univ Texas Austin ICES 201 E 24th StStop C0200 Austin TX 78712 USA

ISBN: (纸本)9783319523897;9783319523880

This work considers the combined space-time discretization of time-dependent partial differential equations by using first order least square methods. We also impose an explicit constraint representing space-time mass conservation. To alleviate the restrictive memory demand of the method, we use dimension reduction via accurate element agglomeration AMG coarsening, referred to as AMGe upscaling. Numerical experiments demonstrating the accuracy of the studied AMGe upscaling method are provided. © Springer International Publishing AG 2017.

关键词： Numerical methods

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Generalizations of Simple Kriging Methods in Spatial Data Analysis 1

引用

8th International Workshop on Meshfree Methods for Partial Differential Equations

作者： Ye, Qi South China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China

ISBN: (数字)9783319519548

ISBN: (纸本)9783319519548;9783319519531

In this article, we use the theory of meshfree approximation to generalize the simple kriging methods by kernel-based probabilities. The main idea is that the new kriging estimations are modeled by the Gaussian fields indexed by bounded linear functionals defined on Sobolev spaces. Moreover, the covariances of the Gaussian fields at the observed functionals can be computed by the given co-variance kernels with respect to the related functionals, for example, Gaussian kernels evaluated at points and gradients. This guarantees that the generalized kriging estimations can be obtained by the same techniques of the simple kriging methods and the generalized kriging estimations can cover many kinds of the complex observed information. By the generalized kriging methods, we can model the geostatistics with the additional observations of gradients at the uncertain locations.

关键词： Sobolev spaces

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Embedding Enriched Partition of Unity Approximations in Finite Element Simulations 1

引用

8th International Workshop on Meshfree Methods for Partial Differential Equations

作者： Schweitzer, Marc Alexander Ziegenhagel, Albert Rhein Friedrich Wilhelms Univ Bonn Inst Numer Simulat Wegelerstr. 6 D-53115 Bonn Germany Fraunhofer Inst Algorithms & Sci Comp SCAI Sankt St Augustin Germany Fraunhofer Inst Algorithmen & Wissensch Rechnen S Schloss Birlinghoven D-53754 St Augustin Germany

ISBN: (数字)9783319519548

ISBN: (纸本)9783319519548;9783319519531

In this paper we present a general approach to embed arbitrary approximation spaces into classical finite element simulations in a non-intrusive fashion. To this end, we employ a global partition of unity method to splice the two independent approximation spaces together. The main goal of this research is to enable the timely evaluation of novel discretization approaches and meshfree techniques in an industrial context by embedding them into large scale finite element simulations. We present some numerical results showing the generality and effectiveness of our approach.

关键词： Finite element method

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Parallel Solver for H(div) Problems Using Hybridization and AMG 23rd

Parallel Solver for <i>H</i>(div) Problems Using Hybridizati...

引用

23rd International Conference on Domain Decomposition Methods

作者： Lee, Chak S. Vassilevski, Panayot S. Texas A&M Univ Dept Math Mailstop 3368 College Stn TX 77843 USA Lawrence Livermore Natl Lab Ctr Appl Sci Comp POB 808L-561 Livermore CA 94551 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97201 USA

ISBN: (纸本)9783319523897;9783319523880

A scalable parallel solver for (div) problems discretized by arbitrary order finite elements on general unstructured meshes is proposed. The solver is based on hybridization and algebraic multigrid (AMG). The hybridization part of the solver requires the fine-grid element matrix information. Weak and strong scaling are examined through several numerical tests which demonstrate that the proposed solver provides a competitive alternative to ADS (Kolev and Vassilevski, SIAM J Sci Comput 34(6):A3079-A3098, 2012), a state-of-the-art solver for problems (div) problems. In fact, it outperforms ADS for higher order elements. © Springer International Publishing AG 2017.

关键词： Computers

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Algebraic Adaptive Multipreconditioning Applied to Restricted Additive Schwarz 23rd

Algebraic Adaptive Multipreconditioning Applied to Restricte...

引用

23rd International Conference on Domain Decomposition Methods

作者： Spillane, Nicole Univ Paris Saclay CNRS Ecole Polytech CMAP F-91128 Palaiseau France

ISBN: (纸本)9783319523897;9783319523880

In 2006 the Multipreconditioned Conjugate Gradient (MPCG) algorithm was introduced by Bridson and Greif. It is an iterative linear solver, adapted from the Preconditioned Conjugate Gradient (PCG) algorithm which can be used in cases where several preconditioners are available or the usual preconditioner is a sum of contributions. There, it was already pointed out that Domain Decomposition algorithms are ideal candidates to benefit from MPCG. The question posed by this work is whether an adaptive MPCG algorithm can be developed for Restricted Additive Schwarz. The goal is to design an adaptive algorithm that is robust at a minimal cost. One great feature of Additive Schwarz is that it is algebraic (all the components in the preconditioner can be computed from the knowledge of the matrix) and we will aim to preserve this property. © Springer International Publishing AG 2017.

关键词： Adaptive algorithms

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An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration 1

引用

2nd International Workshop on Eigenvalue Problems - Algorithms, Software and Applications in Petascale Computing (EPASA)

作者： Yamamoto, Yusaku Univ Electrocommun 1-5-1 Chofugaoka Chofu Tokyo 1828585 Japan

ISBN: (数字)9783319624266

ISBN: (纸本)9783319624266;9783319624242

The Sakurai-Sugiura (SS) projection method for the generalized eigenvalue problem has been extended to the nonlinear eigenvalue problem A(z)w = 0, where A(z) is an analytic matrix valued function, by several authors. To the best of the authors' knowledge, existing derivations of these methods rely on canonical forms of an analytic matrix function such as the Smith form or the theorem of Keldysh. While these theorems are powerful tools, they require advanced knowledge of both analysis and linear algebra and are rarely mentioned even in advanced textbooks of linear algebra. In this paper, we present an elementary derivation of the SS-type algorithm for the nonlinear eigenvalue problem, assuming that the wanted eigenvalues are all simple. Our derivation uses only the analyticity of the eigenvalues and eigenvectors of a parametrized matrix A(z), which is a standard result in matrix perturbation theory. Thus we expect that our approach will provide an easily accessible path to the theory of nonlinear SS-type methods.

关键词： Matrix algebra

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Newton-Krylov-FETI-DP with Adaptive Coarse Spaces 23rd

Newton-Krylov-FETI-DP with Adaptive Coarse Spaces

引用

23rd International Conference on Domain Decomposition Methods

作者： Klawonn, Axel Lanser, Martin Niehoff, Balthasar Radtke, Patrick Rheinbach, Oliver Univ Cologne Math Inst Weyertal 86-90 D-50931 Cologne Germany Tech Univ Bergakad Freiberg Fak Math & Informat Inst Numer Math & Optimierung Akad Str 6 D-09596 Freiberg Germany

ISBN: (纸本)9783319523897;9783319523880

A Newton-Krylov-FETI-DP method for solving nonlinear partial differential equations is presented. The FETI-DP method, which is applied in each Newton step, has an adaptively enriched coarse space to deal with ill-conditioned linearized operators. The adaptive coarse spaces are obtained by solving local generalized eigenvalue problems. Heuristic strategies to reduce the overhead caused by the adaptive computation of constraints are discussed and numerical examples for the p-Laplace equation are presented. © Springer International Publishing AG 2017.

关键词： Discontinuous coefficients Helmholtz equation Maxwell’s equations Optimized Schwarz methods

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Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations 1

引用

8th International Workshop on Meshfree Methods for Partial Differential Equations

作者： Brown, Donald L. Gallistl, Dietmar Peterseim, Daniel Univ Nottingham Sch Math Sci Univ Pk Nottingham England Karlsruher Inst Technol Dietmar Gallistl Inst Angew & Numer Math Englerstr 2 D-76131 Karlsruhe Germany Univ Bonn Inst Numer Simulat Wegelerstr 6 D-53115 Bonn Germany

ISBN: (数字)9783319519548

ISBN: (纸本)9783319519548;9783319519531

This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution-free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of Melenk (Ph.D. Thesis, 1995) and Hetmaniuk (Commun. Math. Sci. 5, 2007) for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients.

关键词： Galerkin methods

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Optimized Schwarz Methods for Domain Decompositions with Parabolic Interfaces 23rd

Optimized Schwarz Methods for Domain Decompositions with Par...

引用

23rd International Conference on Domain Decomposition Methods

作者： Gander, Martin J. Xu, Yingxiang Univ Geneva Sect Math 2-4 Rue LievreCP 64 CH-1211 Geneva Switzerland Northeast Normal Univ Sch Math & Stat Changchun 130024 Peoples R China

ISBN: (纸本)9783319523897;9783319523880

Optimized Schwarz methods have attracted tremendous attention because of their excellent performance. For circular domain decompositions, Gander and Xu (SIAM J. Numer. Anal. 52(4): 1981-2004, 2014;Math. Comput., 2016b) showed that the interface curvature enters the optimized transmission condition and the corresponding convergence rate estimate. Since where the curvature is the only interface characteristic, how the curved interface affects the performance of the optimized Schwarz method is still not completely clear. In this short paper, we show, for an example of domain decomposition with parabolic interfaces, that not only the interface curvature, but also other characteristics of the interfaces will enter as well the optimized transmission parameter. We use numerical experiments to illustrate our theoretical findings. © Springer International Publishing AG 2017.

关键词： Domain decomposition methods

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