Sparse grid approximation of stochastic dynamic micromagnetics

Abstract

We consider the Stochastic Landau-Lifshitz-Gilbert (SLLG) problem as an example of parabolic stochastic PDE (SPDE) driven by Gaussian noise. Beyond being a popular model for magnetic materials immersed in heat baths, the forward uncertainty quantification (UQ) task poses several interesting challenges that did not appear simultaneously in previous works: The equation is strongly nonlinear, time-dependent, and has a non-convex side constraint. We first use the Doss-Sussman transform to convert the SPDE into a random coefficient PDE. We then employ the Lévy-Ciesielski parametrization of the Wiener process to obtain a parametric coefficient PDE. We study the regularity and sparsity properties of the parameter-to-solution map, which features countably many unbounded parameters and low regularity compared to other elliptic and parabolic model problems in UQ. We use a novel technique to establish uniform holomorphic regularity of the parameter-to-solution map based on a Gronwall-type estimate combined with previously known methods that employ the implicit function theorem. This regularity result is used to design a piecewise-polynomial sparse grid approximation through a profit maximization approach. We prove algebraic dimension-independent convergence and validate the result with numerical experiments. If time allows, we discuss the finite element discretization and multi-level approximation.