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.

翻译：我们证明了随机Landau-Lifshitz-Gilbert方程解分布稀疏网格逼近的收敛率。作为工程与物理学中频繁研究的方程，随机Landau-Lifshitz-Gilbert方程提出了许多有趣挑战，这些挑战在以往不确定性量化工作中并未同时出现：该方程具有强非线性、时间依赖性，并带有一个非凸的侧约束条件。此外，与不确定性量化研究中其他椭圆和抛物问题相比，随机噪声参数化具有可数无限个无界参数及低正则性。我们采用一种基于Gronwall型估计和隐函数定理的新技术，建立了参数到解映射的一致全纯正则性。该方法具有高度普适性，建立在抽象假设集之上，因此亦可应用于Landau-Lifshitz-Gilbert方程之外的问题。我们通过数值实验验证了稀疏网格逼近的可行性，并展示了多层稀疏网格方案的显著优势。