Dimension reduction methods for PDEs with multiscale and random natures

Abstract

In this thesis, I focus on elliptic PDEs, Bayesian elliptic inverse problems and Schrödinger equations. Specifically, I reviewed some key results that are typi- cally researched for dimension reduction for PDEs with multiscale and random natures. I also developed several new efficient numerical methods to achieve dimension reduction for solving these problems. For the elliptic PDEs with multiscale and random coefficients, the proposed data-driven approach is based on the intrinsic approximate low dimensional structure of the underlying elliptic differential operators. The method consists of offline and online stages. At the offline stage, a low dimensional space and its basis are extracted from solution samples to achieve significant dimension reduc- tion in the solution space. At the online stage, the extracted data-driven basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of approximate low dimensional structure is established in two scenarios based on: (1) high separability of the underlying Green’s functions; and (2) smooth depen- dence of the parameters in the random coefficients. Various online construction methods are proposed for different problem setups. Error analysis and extensive numerical experiments are provided. Apart from the forward elliptic problems, I also study a Bayesian inverse problem modeled by elliptic PDEs. The data-driven and model-based approach for forward problems is applied and generalized to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is similar to that in the forward problems. Two main parts are training component that computes a set of data-driven basis to achieve signifi- cant dimension reduction in the solution space, and fast solving component that computes the solution and its derivatives for a newly sampled elliptic PDE with the constructed data-driven basis. Hence we can overcome the typical compu- tational bottleneck of HMC – repeated evaluation of the Hamiltonian involving the solution (and its derivatives) modeled by a complex system, a multiscale elliptic PDE in this case. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method. Another problem studied in this thesis is semiclassical Schrödinger equa- tion with time-dependent potentials. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrödinger equa- tion is approximated by these localized multiscale basis functions in space and is solved by the Crank-Nicolson method in time. These multiscale basis functions have compact supports in space, leading to the sparsity of the stiffness matrix, and thus the computational complexity of these two methods is comparable to that of the standard finite element method. Through a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation errors of two methods is significantly reduced.