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

TL;DR

This paper compares structure-preserving Krylov subspace methods for efficiently approximating matrix exponentials of large Hamiltonian matrices, crucial for exponential integrators in Hamiltonian systems.

Contribution

It provides a comparative analysis of various structure-preserving Krylov methods, highlighting their accuracy, efficiency, and structure preservation capabilities.

Findings

J-orthogonal Krylov bases maintain Hamiltonian structure

Structure-preserving methods outperform standard Krylov methods in accuracy

Adaptive Krylov dimension selection improves computational efficiency

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.