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

Abstract

In this paper, we propose a µ-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 µ-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional linear and nonlinear Schrödinger equations, and we show that the µ-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.

A $μ$-mode BLAS approach for multidimensional tensor-structured problems

“…Clearly, formula (14) can be employed to evaluate a spectral approximation (10) at a generic Cartesian grid of points, by properly defining the involved tensor C and matrices Φ µ . In the context of direct and inverse spectral transforms, for example for the effective numerical solution of differential equations (see [10]), one could be interested in the evaluation of (10) at the same grid of quadrature points (ξ k1 1 , . .…”

“…coupled again with suitable boundary conditions. If the partial differential equation ( 17) admits a Kronecker structure, such as for some linear Advection-Diffusion-Absorption (ADA) equations on tensor product domains or linear Schrödinger equations with a potential in Kronecker form (see [10] for more details and examples), then the method of lines yields the system of ordinary differential equations…”