Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

Guettel, Stefan

doi:10.1002/gamm.201310002

Cited by 155 publications

(185 citation statements)

References 101 publications

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“…Similarly, Langevin dynamics [32] could be used to replace solvent forces by homogenized and stochastic forces while preserving symplecticity [56]. Finally, we also plan to test out an approach to obtain coarse-grained parallelism using rational Krylov subspaces instead: the associated rational Arnoldi method, in which a linear system is solved independently for each pole of the polynomial denominator [20], could offer a coarse-grained way to parallelize the matrix function computations involved in the integrator.…”

Section: Discussionmentioning

confidence: 99%

A semi-analytical approach to molecular dynamics

Michels

Desbrun

2015

Journal of Computational Physics

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Despite numerous computational advances over the last few decades, molecular dynamics still favors explicit (and thus easily-parallelizable) time integrators for large scale numerical simulation. As a consequence, computational efficiency in solving its typically stiff oscillatory equations of motion is hampered by stringent stability requirements on the time step size. In this paper, we present a semianalytical integration scheme that offers a total speedup of a factor 30 compared to the Verlet method on typical MD simulation by allowing over three orders of magnitude larger step sizes. By efficiently approximating the exact integration of the strong (harmonic) forces of covalent bonds through matrix functions, far improved stability with respect to time step size is achieved without sacrificing the explicit, symplectic, time-reversible, or fine-grained parallelizable nature of the integration scheme. We demonstrate the efficiency and scalability of our integrator on simulations ranging from DNA strand unbinding and protein folding to nanotube resonators.

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Section: Discussionmentioning

confidence: 99%

A semi-analytical approach to molecular dynamics

Michels

Desbrun

2015

Journal of Computational Physics

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“…Among the large literature on Krylov methods for matrix functions we cite just the recent survey by Güttel [25].…”

Section: Methods For F (A)bmentioning

confidence: 99%

Matrix Functions: A Short Course

Higham

Lin

2015

Series in Contemporary Applied Mathematics

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“…by Lu & Wachspress [14], and their computation by a few lines of MATLAB code is described by Sabino [19, p.43]. Other selections associated with quasi-optimal rational approximation of matrix functions could be used instead; see for example, Beckermann & Reichel [3] or Güttel [11]. In our setting (with m > 1), given an estimate for the interval S that contains the eigenvalues of each of the matrices A 1 , .…”

Section: Multiple Parameter Strategymentioning

confidence: 99%

An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations

Powell

Silvester

Simoncini³

2017

SIAM J. Sci. Comput.

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Abstract. Stochastic Galerkin finite element approximation of PDEs with random inputs leads to linear systems of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multi-term linear matrix equations, we develop an efficient solution algorithm which generalizes ideas from rational Krylov subspace approximation. The new approach determines a low-rank approximation to the solution matrix by performing a projection onto a low-dimensional space and provides an efficient solution strategy whose convergence rate is independent of the spatial approximation. Moreover, it requires far less memory than the standard preconditioned conjugate gradient method applied to the Kronecker formulation of the linear systems.Key words. generalized matrix equations, PDEs with random data, stochastic finite elements, iterative solvers, rational Krylov subspace methods.AMS subject classifications. 35R60 , 60H35, 65N30, 65F101. Introduction. This paper is concerned with the design and implementation of efficient iterative solution algorithms for high-dimensional linear algebra systems that arise from Galerkin approximation of elliptic PDE problems with correlated random inputs. The tensor product structure of the Galerkin approximation space and the low-rank structure that is inherent in the discrete systems is exploited in our innovative solution strategy. This strategy builds on recent progress in rational Krylov subspace approximation (see, for example, Simoncini [21]) and generates an accurate reduced basis approximation of the spatial component of the Galerkin solution.We focus on stochastic steady-state diffusion equations with homogeneous Dirichlet boundary conditions. To this end, let D ⊂ R 2 be a sufficiently regular spatial domain and let Ω be a sample space associated with a probability space (Ω, F , P). Our goal is to approximate u : D × Ω → R such that P-a.s.,

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Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

Cited by 155 publications

References 101 publications

A semi-analytical approach to molecular dynamics

A semi-analytical approach to molecular dynamics

Matrix Functions: A Short Course

An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations

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