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 nonlinear, 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 combined with previously known methods that use the implicit function theorem. We demonstrate numerically the feasibility of the stochastic collocation method and show a clear advantage of a multi-level stochastic collocation scheme for the stochastic Landau-Lifshitz-Gilbert equation.

翻译：我们展示了随机Landau-Lifshitz-Gilbert方程解的分布的稀疏网格逼近的收敛速率。该方程作为工程和物理学中频繁研究的对象，提出了许多有趣且前所未有的挑战——这些挑战未在先前的不确定性量化研究中同时出现：方程具有强非线性、时间依赖性以及非凸的边值约束。此外，随机噪声的参数化特征包含可数无穷多个无界参数，且与不确定性量化中研究的其他椭圆和抛物问题相比，其正则性较低。我们采用一种基于Gronwall型估计与先前隐函数定理方法相结合的新技术，建立了参数到解映射的均匀全纯正则性。通过数值实验验证了随机配置法的可行性，并展示了多层级随机配置方案在随机Landau-Lifshitz-Gilbert方程中的显著优势。