Sparsegrid, reducedbasis Bayesian inversion: Nonaffineparametric nonlinear equations
Abstract
We extend the reduced basis (RB) accelerated Bayesian inversion methods for affineparametric, linear operator equations which are considered in [16,17] to nonaffine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin highfidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the socalled Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimensionindependent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinitedimensional parametric, deterministic forward map and operator admit Ntermmore »
 Authors:
 The Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, Stop C0200, Austin, TX 787121229 (United States)
 Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule, Römistrasse 101, CH8092 Zürich (Switzerland)
 Publication Date:
 OSTI Identifier:
 22572326
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 316; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CONVERGENCE; CURRENTS; EFFICIENCY; EQUATIONS; ERRORS; EXPERIMENT DESIGN; INTERPOLATION; NONLINEAR PROBLEMS; QUADRATURES; STOCHASTIC PROCESSES
Citation Formats
Chen, Peng, Email: peng@ices.utexas.edu, and Schwab, Christoph, Email: christoph.schwab@sam.math.ethz.ch. Sparsegrid, reducedbasis Bayesian inversion: Nonaffineparametric nonlinear equations. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.02.055.
Chen, Peng, Email: peng@ices.utexas.edu, & Schwab, Christoph, Email: christoph.schwab@sam.math.ethz.ch. Sparsegrid, reducedbasis Bayesian inversion: Nonaffineparametric nonlinear equations. United States. doi:10.1016/J.JCP.2016.02.055.
Chen, Peng, Email: peng@ices.utexas.edu, and Schwab, Christoph, Email: christoph.schwab@sam.math.ethz.ch. 2016.
"Sparsegrid, reducedbasis Bayesian inversion: Nonaffineparametric nonlinear equations". United States.
doi:10.1016/J.JCP.2016.02.055.
@article{osti_22572326,
title = {Sparsegrid, reducedbasis Bayesian inversion: Nonaffineparametric nonlinear equations},
author = {Chen, Peng, Email: peng@ices.utexas.edu and Schwab, Christoph, Email: christoph.schwab@sam.math.ethz.ch},
abstractNote = {We extend the reduced basis (RB) accelerated Bayesian inversion methods for affineparametric, linear operator equations which are considered in [16,17] to nonaffine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin highfidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the socalled Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimensionindependent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinitedimensional parametric, deterministic forward map and operator admit Nterm RB and EIM approximations which converge at rates which depend only on the sparsity of the parametric forward map. In several numerical experiments, the proposed algorithms exhibit dimensionindependent convergence rates which equal, at least, the currently known rate estimates for Nterm approximation. We propose to accelerate Bayesian estimation by first offline construction of reduced basis surrogates of the Bayesian posterior density. The parsimonious surrogates can then be employed for online data assimilation and for Bayesian estimation. They also open a perspective for optimal experimental design.},
doi = {10.1016/J.JCP.2016.02.055},
journal = {Journal of Computational Physics},
number = ,
volume = 316,
place = {United States},
year = 2016,
month = 7
}

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