Preconditioning techniques for elliptic partial differential equations with random data

Youngnoi, Rawin (2021). Preconditioning techniques for elliptic partial differential equations with random data. University of Birmingham. Ph.D.

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Abstract

The stochastic Galerkin finite element method (SGFEM) is a well-established numerical method for approximating solutions of partial differential equations with parametric or random data. The advantage of SGFEM over traditional sampling methods is its convergence rate. However, this approach yields large-scale, often intractable systems of linear equations. Therefore, a powerful iterative solver equipped with a suitable preconditioner is required to approximate the solution for such linear systems.

In this thesis, we focus on designing preconditioners for stochastic Galerkin matrices that arise when solving the steady-state diffusion equation with random data. We consider two parametric representations of the diffusion coefficient: affine and non-affine.

For the case of affine-parametric diffusion coefficient, we present two preconditioners. Truncation preconditioners for affine-parametric diffusion problems form a new class of preconditioners that generalise the mean-based preconditioner by including additional information from the diffusion coefficient. Next, the domain decomposition technique for the parametric domain is introduced. This technique provides a framework for designing preconditioners which are capable of parallelism. We present a new concept of parametric mesh to represent the structure of the parametric space. Moreover, a so-called even-odd partitioning strategy for the parametric mesh is introduced. This strategy results in three versions of the even-odd preconditioners.

We provide spectral analyses of the preconditioned systems both for the truncation preconditioners and domain decomposition preconditioners, which confirm the optimality of the preconditioners with respect to discretisation parameters.

For the case of non-affine parametric diffusion coefficient, the truncation preconditioners and domain decomposition preconditioners are presented. They generalise the idea of truncation preconditioners and domain decomposition preconditioners for affine-parametric coefficients by capturing the important terms and finding a structure which can utilise parallelism. We also design a preconditioner for log-transformed coefficients.

Finally, the performance of each preconditioner is illustrated by numerical experiments. We compare the efficiency (in terms of iteration counts and total complexity) of our purposed preconditioners with some existing preconditioners such as the mean-based preconditioner and the Kronecker product preconditioner.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):