  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 multi-level sparse grid
scheme.
