Abstract: The Stochastic Landau-Lifshitz-Gilbert equation (SLLG) is a nonlinear stochastic PDE used to model a magnetic body immersed in a heat bath. Its strong nonlinearities and "roughness" of the stochastic component make its numerical approximation particularly challenging.
We present a novel methodology to reduce this stochastic PDE to a parametric coefficient PDE. The high dimensionality of the resulting parameter-to-solution map precludes the use of any scheme affected by the curse of dimensionality. To tackle this issue, we prove the existence of a holomorphic extension and estimate the size of its domain.
Based on this information, some schemes may circumvent the curse of dimensionality. We consider:

• Sparse grid interpolation: A high-dimensional interpolation method that allows fine-tuning the number of collocation nodes assigned to each parameter. When tuned well, it achieves the approximation error of tensor-product interpolation with dramatically fewer nodes.
• Proper Orthogonal Decomposition (POD) reduced basis: We compute a Singular Value Decomposition (SVD) of appropriate samples of the SLLG dynamics. The resulting basis of the solution manifold provides the approximation quality of a nodal finite elements' basis with fewer functions. This in turn allows faster Galerkin approximation.