Guannan Zhang

Stochastic finite element methods for partial differential equations with random input data

The quantification of probabilistic uncertainties in the outputs of physical, biological, and social systems governed by partial differential equations with random inputs require, in practice, the discretization of those equations. Stochastic finite element methods refer to an extensive class of algorithms for the approximate solution of partial differential equations having random input data, for which spatial discretization is effected by a finite element method. Fully discrete approximations require further discretization with respect to solution dependences on the random variables. For this purpose several approaches have been developed, including intrusive approaches such as stochastic Galerkin methods, for which the physical and probabilistic degrees of freedom are coupled, and non-intrusive approaches such as stochastic sampling and interpolatory-type stochastic collocation methods, for which the physical and probabilistic degrees of freedom are uncoupled. All these method classes are surveyed in this article, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates and complexity analyses of the algorithms are provided. Throughout, numerical examples are used to illustrate the theoretical results and to provide further insights into the methodologies.

Error Analysis of a Stochastic Collocation Method for Parabolic Partial Differential Equations with Random Input Data

A stochastic collocation method for solving linear parabolic partial differential equations with random coefficients, forcing terms, and initial conditions is analyzed. The input data are assumed to depend on a finite number of random variables. Unlike previous analyses, a wider range of situations are considered, including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element, Crank--Nicolson fully discrete space-time discretizations. Ingredients in the convergence analysis include the proof of the analyticity, with respect to the probabilistic parameters, of the semidiscrete and fully discrete approximate solutions. A numerical example is provided to illustrate the analyses.

Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients

In this work, we present a generalized methodology for analyzing the convergence of quasi-optimal Taylor and Legendre approximations, applicable to a wide class of parameterized elliptic PDEs with finite-dimensional deterministic and stochastic inputs. Such methods construct an optimal index set that corresponds to the sharp estimates of the polynomial coefficients. Our analysis, furthermore, represents a novel approach for estimating best M-term approximation errors by means of coefficient bounds, without the use of the standard Stechkin inequality. In particular, the framework we propose for analyzing asymptotic truncation errors is based on an extension of the underlying multi-index set into a continuous domain, and then an approximation of the cardinality (number of integer multi-indices) by its Lebesgue measure. Several types of isotropic and anisotropic (weighted) multi-index sets are explored, and rigorous proofs reveal sharp asymptotic error estimates in which we achieve sub-exponential convergence rates [of the form

An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Partial Differential Equations with Random Input Data

M \text {exp}({-(\kappa M)^{1/N}})

An adaptive sparse-grid iterative ensemble Kalman filter approach for parameter field estimation

Mexp(-(κM)1/N), with

Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients

\kappa