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Structure-preserving Krylov Subspace Approximations for the Matrix Exponential of Hamiltonian Matrices: A Comparative Study

Peter Benner, Heike Faßbender, Michel-Niklas Senn

Abstract

We study structure-preserving Krylov subspace methods for approximating the matrix-vector products f(H)b, where H is a large Hamiltonian matrix and f denotes either the matrix exponential or the related phi-function. Such computations are central to exponential integrators for Hamiltonian systems. Standard Krylov methods generally destroy the Hamiltonian structure under projection, motivating the use of Krylov bases with J-orthogonal columns that yield Hamiltonian projected matrices and symplectic reduced exponentials. We compare several such structure-preserving Krylov methods on representative Hamiltonian test problems, focusing on accuracy, efficiency, and structure preservation, and briefly discuss adaptive strategies for selecting the Krylov subspace dimension.

Structure-preserving Krylov Subspace Approximations for the Matrix Exponential of Hamiltonian Matrices: A Comparative Study

Abstract

We study structure-preserving Krylov subspace methods for approximating the matrix-vector products f(H)b, where H is a large Hamiltonian matrix and f denotes either the matrix exponential or the related phi-function. Such computations are central to exponential integrators for Hamiltonian systems. Standard Krylov methods generally destroy the Hamiltonian structure under projection, motivating the use of Krylov bases with J-orthogonal columns that yield Hamiltonian projected matrices and symplectic reduced exponentials. We compare several such structure-preserving Krylov methods on representative Hamiltonian test problems, focusing on accuracy, efficiency, and structure preservation, and briefly discuss adaptive strategies for selecting the Krylov subspace dimension.
Paper Structure (20 sections, 3 theorems, 53 equations, 4 figures, 3 tables, 6 algorithms)

This paper contains 20 sections, 3 theorems, 53 equations, 4 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Fundamentals on Hamiltonian and symplectic matrices
  3. Krylov subspace projection methods
  4. Arnoldi Method (A)
  5. Hamiltonian Lanczos Method (HL)
  6. Symplectic Arnoldi Method (SA)
  7. Isotropic Arnoldi Method (IA)
  8. Hamiltonian Extended Krylov Subspace Method (HEKS)
  9. Block $J$-orthogonal Basis Method (BJ)
  10. Arnoldi projection method using inner product x'Hy
  11. Re-$J$-orthogonalization
  12. Summary of Krylov subspace methods considered
  13. Numerical Examples
  14. Linear wave equation
  15. Nonlinear Sine-Gordon Equation
  16. ...and 5 more sections

Key Result

Lemma 1

\newlabellem11

Figures (4)

  • Figure 1: Relative solution error for different matrices in the approximation of $e^{H}b$.
  • Figure 2: Relative solution error for different matrices in the approximation of $\varphi_\text{expl}(H)b$ using \ref{['eq:varphi']}.
  • Figure 3: Relative solution error for different matrices in the approximation of $\varphi_\text{impl}(H)b$ using \ref{['eq:trickexp']}.
  • Figure 4: Timings for different matrices for generating the subspaces of dimension $2r$.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 3: S92