A note on sparse, adaptive Smolyak quadratures for Bayesian inverse problems

Abstract

We present a novel, deterministic approach to inverse problems for identification of parameters in differential equations from noisy measurements. Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite dimensional, separable Banach spaces, we develop a practical computational algorithm for the efficient approximation of the infinite-dimensional integrals with respect to the posterior measure. Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown, but are bounded independently of the number of random variables activated by the adaptive algorithm. Numerical experiments for a model problem of coefficient identification with point measurements in a diffusion problem based on uniform prior measure as well as lognormal Gaussian prior measure are presented.