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

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 novel combined algorithm to adaptively refine the sparse grid used for quadrature approximation of the Bayesian estimates, of the reduced basis approximation and to compress the parametric forward solutions by empirical interpolation is proposed. The theoretically predicted computational efficiency which is independent of the number of active parameters is demonstrated in numerical experiments for a model, nonaffine-parametric, stationary, elliptic diffusion problem, in two spacial and in parameter space dimensions up to 1024.