A $\mu$-mode integrator for solving evolution equations in Kronecker form

TL;DR

This paper introduces a $0$-mode integrator for efficiently solving stiff evolution equations using a splitting approach and matrix exponentials, demonstrating significant performance gains especially on GPUs.

Contribution

The paper presents a novel $0$-mode integrator that leverages splitting and tensor products for efficient computation of evolution equations, outperforming existing methods.

Findings

Significant performance improvements on GPUs, up to 20x faster.

Efficient implementation of spectral transforms without fast transforms.

Successful application to 3D Schr46dinger equations.

Abstract

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of $10$ and $20$, depending on the problem.