Catherine E. Powell

An Introduction to Computational Stochastic PDEs

This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of the art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modeling and materials science.

Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data

We introduce a stochastic Galerkin mixed formulation of the steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use with the minimum residual method. We derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters.

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

Image reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head.

Optimal Preconditioning for Raviart--Thomas Mixed Formulation of Second-Order Elliptic Problems

We evaluate two preconditioning strategies for the indefinite linear system obtained from Raviart--Thomas mixed finite element formulation of a second-order elliptic problem with variable diffusion coefficients. It is known that the underlying saddle-point problem is well-posed in two function spaces, H(div) × L2 and L2 × H1, leading to the possibility of two distinct types of preconditioner. For homogeneous Dirichlet boundary conditions, the discrete problems are identical. This motivates our use of Raviart--Thomas approximation in both frameworks, yielding a nonconforming method in the second case. The focus is on linear algebra; we establish the optimality of two parameter-free block-diagonal preconditioners using basic properties of the finite element matrices. Uniform eigenvalue bounds are established and the impact of the PDE coefficients is explored in numerical experiments. A practical scheme is discussed, the key building block for which is a fast solver for a scalar diffusion operator based on algebraic multigrid. Trials of preconditioned minres illustrate that both preconditioning schemes are optimal with respect to the discretization parameter and robust with respect to the PDE coefficients.

Preconditioning Stochastic Galerkin Saddle Point Systems

Mixed finite element discretizations of deterministic second-order elliptic PDEs lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations, and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense, and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (MINRES). By introducing so-called Kronecker product preconditioners, we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

Preconditioning Steady-State Navier--Stokes Equations with Random Data

We consider the numerical solution of the steady-state Navier--Stokes equations with uncertain data. Specifically, we treat the case of uncertain viscosity, which results in a flow with an uncertain Reynolds number. After linearization, we apply a stochastic Galerkin finite element method, combining standard inf-sup stable Taylor--Hood approximation on the spatial domain (on highly stretched grids) with orthogonal polynomials in the stochastic parameter. This yields a sequence of nonsymmetric saddle-point problems with Kronecker product structure. The novel contribution of this study lies in the construction of efficient block triangular preconditioners for these discrete systems, for use with GMRES. Crucially, the preconditioners are robust with respect to the discretization and statistical parameters, and we exploit existing deterministic solvers based on the so-called pressure convection-diffusion and least-squares commutator approximations.

Energy Norm A Posteriori Error Estimation for Parametric Operator Equations

Stochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional (

A priori error analysis of stochastic galerkin mixed approximations of elliptic pdes with random data

h

Black-Box Preconditioning for Mixed Formulation of Self-Adjoint Elliptic PDEs

-)finite element approximation on the spatial domain with spectral (

H(div) Preconditioning for a Mixed Finite Element Formulation of the diffusion problem with random data

p