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Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion

by P. Chen and Ch. Schwab

(Report number 2015-08)

Abstract
We present new sparse-grid based algorithms for fast Bayesian estimation and inversion of parametric operator equations. We propose Reduced Basis (RB) acceleration of numerical integration based on Smolyak sparse grid quadrature. To tackle the curse-of-dimensionality in high-dimensional Bayesian inversion, we exploit sparsity of the parametric forward solution map as well as of the Bayesian posterior density with respect to the random parameters. We employ an dimension adaptive Sparse Grid method (aSG) for both, offline-training the reduced basis as well as for deterministic quadrature of the conditional expectations which arise in Bayesian estimates. For the forward problem with nonaffine dependence on the random variables, we perform further affine approximation based on the Empirical Interpolation Method (EIM) proposed in [1]. A combined algorithm to adaptively refine the sparse grid quadrature, reduced basis approximation and empirical interpolation is proposed and its computational efficiency is demonstrated in a set of numerical experiments, in parameter space dimensions up to $1024$.

Keywords: Sparse Grid, Model Order Reduction, Reduced Basis, Empirical Interpolation, Bayesian inversion, Best N-term Approximation

@Techreport{CS15_598,
  author = {P. Chen and Ch. Schwab},
  title = {Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-08.pdf },
  year = {2015}
}

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