Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems

The Bayesian methodology has proven to be a convenient framework to solve inverse problems from available data with limited information. The main computational challenges in the Bayesian solution of inverse problems arise from the need for repeated evaluations of the forward model, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. In this paper, we present an efficient technique for constructing stochastic surrogate models to accelerate the Bayesian inference approach for statistical inverse problems. The stochastic collocation algorithms using $l_1$-minimization (L1-SCM), based on generalized polynomial chaos, are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that is inexpensive to evaluate. A rigorous error analysis shows that the convergence rate of the posterior density is at least twice as fast as the convergence rate of the gPC expa...

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