First order k-th moment finite element analysis of nonlinear operator equations with stochastic data
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Date
2011-08Type
- Report
ETH Bibliography
yes
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Abstract
We develop and analyze a class of efficient algorithms for uncertainty quantification of nonlinear operator equations. The algorithm are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, for a class of abstract nonlinear, parametric operator equations J(α,u) = 0 for random parameters α ith realizations in a neighborhood of a nominal parameter α0. Under some structural assumptions on the parame- ter dependence, by the implicit function theorem, J(α,u) = 0 admits locally unique solutions u=S(α) for all values α in some neighborhood of α0. Random parameters α(ω) = α$_0$ + r(ω), are shown to imply a unique random solution u(ω) = S(α(ω)). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the solution fluctuations u(ω) - S(α$_0$ provided that statistical moments of the random parameter perturbation r(ω) 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 Fréchet 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(log (N)^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
Permanent link
https://doi.org/10.3929/ethz-a-010398893Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichSubject
Nonlinear operator equations; Random parameters; Deterministic methods; Fréchet derivative; Sparse tensor approximation; Random domainOrganisational unit
03435 - Schwab, Christoph / Schwab, Christoph
Funding
247277 - Automated Urban Parking and Driving (EC)
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Is previous version of: http://hdl.handle.net/20.500.11850/59246
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