#### Date of Award

Spring 1-1-2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Xiao-Chuan Cai

#### Second Advisor

Karl Gustafson

#### Third Advisor

Robert Goodrich

#### Fourth Advisor

Congming Li

#### Fifth Advisor

Robert Leben

#### Abstract

We introduce and study parallel space-time domain decomposition methods for solving deterministic and stochastic parabolic equations. Traditional parallel algorithms solve parabolic problems time step by time step, i.e., the simulation of the later time step is based on the solution of the earlier time step. Therefore, the parallelism is restricted to each time step, and the algorithms are purely sequential in time. Recently, there are several attempts to develop time-parallel algorithms, such as parareal, waveform relaxation, and space-time multigrid. In this thesis, we develop some overlapping Schwarz methods whose subdomains cover both space and time variables, and we show numerically that the methods work well for stochastic parabolic equations discretized with an implicit stochastic Galerkin method. The main components of the stochastic Galerkin method are Karhunen-Loeve expansion and double orthogonal polynomials, which are used to decouple the stochastic parabolic problem into a sequence of deterministic parabolic equations. In order to solve the sequence of equations efficiently, one- and two-level Schwarz preconditioned recycling GMRES methods are carefully investigated such that some components of the methods are reused to maximize the benefit of the recycling Krylov subspace when a large number of linear systems are solved. The key elements of this approach include an ordering algorithm and two grouping algorithms. We present some experimental results obtained on a parallel computer with more than one thousand processors.

#### Recommended Citation

Cong, Cui, "Implicit Space-Time Domain Decomposition Methods for Stochastic Parabolic Partial Differential Equations" (2013). *Mathematics Graduate Theses & Dissertations*. 23.

https://scholar.colorado.edu/math_gradetds/23