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Sparse Quadrature Approach to Bayesian Inverse Problems

by Ch. Schwab and C. Schillings

(Report number 2013-27)

Abstract
We survey recent results and references on the parametric deterministic formulation of Bayesian inverse problems with distributed parameter uncertainty from infinite dimensional, separable spaces, with uniform prior probability measure. The underlying forward problems are parametric, deterministic operator equations, and computational Bayesian inversion is to evaluate expectations of quantities of interest (QoIs) under the Bayesian posterior, conditional on given data. In this extended abstract, we review sparsity results of the Bayesian posterior from SAM Report 2013-17. These results imply dimension independent convergence rates for adaptive Smolyak integration algorithms.

Keywords: Bayesian Inverse Problems, Parametric Operator Equations, Smolyak Quadrature, Sparsity, Uniform Prior Measures

@Techreport{SS13_524,
  author = {Ch. Schwab and C. Schillings},
  title = {Sparse Quadrature Approach to Bayesian Inverse Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-27},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-27.pdf },
  year = {2013}
}

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