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

03/02/2021
by   Marco Caliari, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/31/2019

Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators

We introduce some general tools to design exact splitting methods to com...
research
05/29/2023

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

In this paper, we evaluate the performance of novel numerical methods fo...
research
10/13/2017

On Parallel Solution of Sparse Triangular Linear Systems in CUDA

The acceleration of sparse matrix computations on modern many-core proce...
research
10/27/2021

Spectral splitting method for nonlinear Schrödinger equations with quadratic potential

In this paper we propose a modified Lie-type spectral splitting approxim...
research
09/19/2017

Magnus integrators on multicore CPUs and GPUs

In the present paper we consider numerical methods to solve the Schrödin...
research
11/14/2021

Numerical methods to evaluate Koopman matrix from system equations

A method that is employed to evaluate a Koopman matrix from a data set o...
research
10/14/2022

A μ-mode approach for exponential integrators: actions of φ-functions of Kronecker sums

We present a novel method for computing actions of the so-called φ-funct...

Please sign up or login with your details

Forgot password? Click here to reset