One of the challenges in uncertainty quantification (UQ) is to solve stochastic partial differential equations (SPDEs) with both multiscale features in the physical space and high dimensional random input variables. To solve these problems, one not only needs to use a very fine mesh to resolve the small scales of the solution in the physical space, but also needs to approximate the solution in the stochastic space with high input dimension. In this talk, I will introduce two effective model reduction methods to solve these problems.

 

The first one is the multiscale date-driven stochastic method (MsDSM). I will first derive an effective (upscaled) stochastic equation that can be well-resolved on a coarse grid. Then, I will construct data-driven stochastic basis functions under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. By applying the model reduction in both the physical and stochastic spaces, the MsDSM offers considerable savings over traditional methods.

 

The second one is the multiscale multilevel Monte Carlo method (MsMLMC). The multilevel Monte Carlo (MLMC) method is an effective method in solving SPDEs. For the multiscale problem, however, it is still very expensive since the variance decay property holds only if the coarsest grid resolves the smallest scale feature. To overcome this difficulty, I will construct a small number of reduced basis functions within each coarse grid, which can be used to approximate the multiscale finite element basis functions efficiently. Since these multiscale basis functions contain the multiscale information of the solution, I can apply the MLMC to multiscale SPDEs starting from a relatively coarse grid, without requiring the coarsest grid to resolve the smallest scale of the solution.

 

Numerical results are presented to demonstrate the accuracy and efficiency of the proposed methods for several multiscale stochastic problems without scale separation.