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Sparse-Grid, Reduced-Basis Bayesian Inversion: Nonaffine-Parametric Nonlinear Equations

by P. Chen and Ch. Schwab

(Report number 2015-21)

Abstract
We extend the reduced basis accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [15, 16] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [18] and of Bayesian inversion in [41, 42] to the fully discrete setting, including Petrov-Galerkin high-fidelity (“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 nonlinearity with respect to the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the so-called Empiricial Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations with respect to the posterior are computed by adaptive quadratures with dimension-independent convergence rates proposed in [42]; the present work generalizes [42] to account for the impact of the PG discretization in the forward maps on the expectation of the Quantities of Interest (QoI). Second, we propose to perform the Bayesian estimation only with respect to a parsimonious, reduced basis approximation of the posterior density. In [42], under general conditions on the forward map, the infinite-dimensional parametric, deterministic Bayesian posterior was shown to admit N-term approximations which converge at rates which depend only on the sparsity of the parametric forward map. We present dimension-adaptive collocation algorithms to build finite-dimensional parametric surrogates. In several numerical experiments, the proposed algorithms exhibit dimension-independent convergence rates which equal, at least, the currently known rate estimates for N-term approximation. We propose to accelerate Bayesian estimation by offline computation of reduced basis surrogates of the Bayesian posterior density. The parsimonious surrogates can be employed for online data assimilation and for Bayesian estimation. They also open a perspective for optimal experimental design.

Keywords: Bayesian inversion, sparse grid, reduced basis, generalized empirical interpolation, greedy algorithm, high-fidelity, Petrov-Galerkin Finite Elements, goal-oriented a-posteriori error estimate, a-priori error estimate, uncertainty quantification

@Techreport{CS15_611,
  author = {P. Chen and Ch. Schwab},
  title = {Sparse-Grid, Reduced-Basis Bayesian Inversion: Nonaffine-Parametric Nonlinear Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-21.pdf },
  year = {2015}
}

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