## First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

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Author

Chernov, A.

Schwab, Christoph

Date

2011-08Type

- Report

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Abstract

We develop and analyze a class of efficient algorithms for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, for a class of abstract nonlinear, parametric operator equations J(!, u) = 0 for random parameters ! with realizations in a neighborhood of a nominal parameter !0. Under some structural assumptions on the parameter dependence, by the implicit function theorem, J(!, u) = 0 admits locally unique solutions u = S(!) for all values ! in some neighborhood of !0. Random parameters !(") = !0 + r("), are shown to imply a unique random solution u(") = S(!(")). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the solution fluctuations u(") − S(!0), provided that statistical moments of the random parameter perturbation r(") are known. We present a sparse tensor Galerkin discretization for the tensorized first order perturbation equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary parabolic diffusion problems in random domains. We verify Fr´echet differentiability by means of shape calculus, and establish the Hadamard principle that the first order, k-th moment equation is completely specified in terms of data on the boundary of the nominal space-time cylinder. We perform boundary reduction of this parabolic evolution problem and propose a novel sparse tensor space-time Galerkin discretization. In conjunction with the sparse tensor Galerkin approximation of the k-point correlation, it reduces the complexity of the Galerkin discretization to O(N(logN)k−1) where N denotes the number of degrees of freedom for a stationary problem on the boundary of the nominal domain (rather than on the space-time cylinder), thereby generalizing [25] to the boundary reduction of parabolic problems Show more

Publication status

unpublishedJournal / series

Research reportsPublisher

Seminar für Angewandte Mathematik, ETHSubject

Nonlinear operator equations; Random parameters; Deterministic methods; Fréchet derivative; Sparse tensor approximation; Random domainOrganisational unit

03435 - Schwab, Christoph
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