Claude Jeffrey Gittelson

A STOCHASTIC COLLOCATION ALGORITHM WITH MULTIFIDELITY MODELS

We present a numerical method for utilizing stochastic models with differing fideli- ties to approximate parameterized functions. A representative case is where a high-fidelity and a low-fidelity model are available. The low-fidelity model represents a coarse and rather crude ap- proximation to the underlying physical system. However, it is easy to compute and consumes little simulation time. On the other hand, the high-fidelity model is a much more accurate representation of the physics but can be highly time consuming to simulate. Our approach is nonintrusive and is therefore applicable to stochastic collocation settings where the parameters are random variables. We provide sufficient conditions for convergence of the method, and present several examples that are of practical interest, including multifidelity approximations and dimensionality reduction.

An adaptive stochastic Galerkin method for random elliptic operators

We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence of the random solution, which combines with any discretization on the spatial domain. In addition to selecting active polynomial modes, this solver can adaptively construct a separate spatial discretization for each of their coefficients. We show convergence of the solver in this general setting, along with a computable bound for the mean square error, and an optimality property in the case of a single spatial discretization. Numerical computations demonstrate convergence of the solver and compare it to a sparse tensor product construction. Introduction Stochastic Galerkin methods have emerged in the past decade as an efficient solution procedure for boundary value problems depending on random data; see [14, 32, 2, 30, 23, 18, 31, 28, 6, 5]. These methods approximate the random solution by a Galerkin projection onto a finite-dimensional space of random fields. This requires the solution of a single coupled system of deterministic equations for the coefficients of the Galerkin projection with respect to a predefined set of basis functions on the parameter domain. A major remaining obstacle is the construction of suitable spaces in which to compute approximate solutions. These should be adapted to the stochastic structure of the equation. Simple tensor product constructions are infeasible due to the high dimensionality of the parameter domain in the case of input random fields with low regularity. Parallel to but independently from the development of stochastic Galerkin methods, a new class of adaptive methods has emerged, which are set not in the continuous framework of a boundary value problem, but rather on the level of coefficients with respect to a hierarchic Riesz basis, such as a wavelet basis. Due to the norm equivalences constitutive of Riesz bases, errors and residuals in appropriate sequence spaces are equivalent to those in physically meaningful function spaces. This permits adaptive wavelet methods to be applied directly to a large class of equations, provided that a suitable Riesz basis is available. Received by the editor March 2, 2011 and, in revised form, September 24, 2011. 2010 Mathematics Subject Classification. Primary 35R60, 47B80, 60H25, 65C20, 65N12, 65N22, 65N30, 65J10, 65Y20. This research was supported in part by the Swiss National Science Foundation grant No. 200021-120290/1. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication

LOCAL POLYNOMIAL CHAOS EXPANSION FOR LINEAR DIFFERENTIAL EQUATIONS WITH HIGH DIMENSIONAL RANDOM INPUTS

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be h...

The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209–259,1990. doi:10.1007/BF00375065, Arch Rational Mech Anal 113(3): 261–298, 1990. doi:10.1007/BF00375066) with a lognormal random permeability. Specifically, the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size

Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE

d\times d

High-order methods as an alternative to using sparse tensor products for stochastic Galerkin FEM

, where

PLANE WAVE DISCONTINUOUS GALERKIN METHODS: ANALYSIS OF THE h-VERSION ∗, ∗∗

d