Model reduction methods for a class of PDES in multiscale and random media

Abstract

In this thesis, for elliptic PDEs with multiscale and random coeffcients, we first propose an optimization approach to construct multiscale data-driven stochastic basis functions based on the structure of the finite element method (FEM) and generalized polynomial chaos (gPC). This optimization approach will enable us to construct a set of localized multiscale data-driven stochastic basis functions that give an optimal approximation property of the solution operator.

Except for the first proposed method, when focusing on physical space reduction, we also propose a proper orthogonal decomposition (POD) method which applied to multiscale finite element method (MsFEM) for constructing our reduced multiscale basis functions. For the random space, we still adopt gPC as the intrusive method for discretizing solution space. Both methods are used for multi-query setting and composited by two stages. In the offline stage, we construct reduced multiscale basis functions. In the online stage, we use the reduced multiscale basis functions to derive the weak formulation and discretize the solution by Galerkin method. Error analysis and numerical experiments are also presented.

Apart from elliptic PDEs, we propose reduction method for Schrodinger equation in the semiclassical regime with multiscale potentials. In this thesis, a multiscale finite element method based on sparse compression of the Hamiltonian operator is proposed. The constructed multiscale basis functions are localized and "blind" to the form of the potential. Then we solve the obtained ordinary differential equations (ODE) system explicitly for the time evolution by using a one-shot eigendecomposition. Numerical examples are elaborated to verify the performance and analysis.

When the electrostatic potential contains multiscale and random features, for such kind of semiclassical Schrodinger equation, the wavefunction has high-frequency oscillations in both physical and random spaces. Therefore, we propose another multiscale reduced basis method that combines optimization method with POD method to construct multiscale reduced basis functions in the physical space and make use of quasi-Monte Carlo (qMC) method for the random space. The famous Anderson localization phenomena for Schrodinger equation with correlated random potentials is also investigated for both 1D and 2D physical space.

Adopting the similar idea of combining POD method with qMC method, we also develop a model reduction method for solving the random Helmholtz equation which is used to describe wave propagation in random media. The generalizability and robustness of our method to a series of random media are verified by numerical examples.