Sparse grid approximation of nonlinear SPDEs: The Landau--Lifshitz--Gilbert equation

TL;DR

This paper establishes convergence rates for a sparse grid method approximating the distribution of solutions to the complex, nonlinear stochastic Landau-Lifshitz-Gilbert equation, highlighting a novel approach for challenging uncertainty quantification problems.

Contribution

It introduces a new technique for proving regularity of the solution map, applicable to complex nonlinear SPDEs with high-dimensional noise.

Findings

Sparse grid approximation converges effectively for the equation.

Multilevel sparse grid scheme shows clear computational advantages.

The method is broadly applicable beyond the specific equation studied.

Abstract

We show convergence rates for a sparse grid approximation of the distribution of solutions of the stochastic Landau-Lifshitz-Gilbert equation. Beyond being a frequently studied equation in engineering and physics, the stochastic Landau-Lifshitz-Gilbert equation poses many interesting challenges that do not appear simultaneously in previous works on uncertainty quantification: The equation is strongly non-linear, time-dependent, and has a non-convex side constraint. Moreover, the parametrization of the stochastic noise features countably many unbounded parameters and low regularity compared to other elliptic and parabolic problems studied in uncertainty quantification. We use a novel technique to establish uniform holomorphic regularity of the parameter-to-solution map based on a Gronwall-type estimate and the implicit function theorem. This method is very general and based on a set of abstract assumptions. Thus, it can be applied beyond the Landau-Lifshitz-Gilbert equation as well. We demonstrate numerically the feasibility of approximating with sparse grid and show a clear advantage of a multilevel sparse grid scheme.