Quantitative Analysis in Stochastic Homogenization and Hypocoercive PDEs in Sampling

In this dissertation, we study two types of equations that involve randomness, and conduct quantitative analysis to study their ergodicity: (I) stochastic homogenization; (II) hypocoercive PDEs that arise from sampling.

In Chapter 2, we study the optimal numerical algorithms to compute the electrical field generated by a localized charge and an infinite random heterogeneous medium in dimension 3, only using the information of the medium in a large finite box. The algorithm is optimal if the medium is a sample from a stationary ensemble with a finite range of dependence. The algorithm is motivated by multipole expansion for stochastic homogenization, and the proof relies on estimates of second-order, next to first-order, correctors, which in turn relies on a parabolic semigroup estimate.

In Chapter 3, we provide an explicit estimate of exponential decay rate for underdamped Langevin dynamics in $L^2$ distance. To achieve this, we prove a Poincar\'{e}-type inequality in time-augmented state space, so that the exponential convergence is reduced to a standard energy dissipation estimate. Our new estimate provides a more explicit and simpler expression of decay rate; moreover, our estimate is the first that shows, for a wide variety of convex potentials, after optimizing the choice of friction coefficient, the underdamped Langevin dynamics converge much faster than the overdamped Langevin dynamics.

In Chapter 4, we establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Similar to Chapter 3, our analysis is based on the variational framework for hypocoercivity, and also provides explicit convergence rate estimates that are more quantitative than existing results.

Finally in Chapter 5, we identify possible open problems and future research directions for both topics that follow the work established in this dissertation.