Sparsity in Bayesian Inversion of Parametric Operator Equations

Abstract

We establish posterior sparsity in Bayesian inversion for systems with distributed parameter uncertainty subject to noisy data. We generalize the particular case of scalar diffusion problems with random coefficients in [29] to broad classes of operator equations. For countably parametric, deterministic representations of uncertainty in the forward problem which belongs to a certain sparsity class, we quantify analytic regularity of the (countably parametric) Bayesian posterior density and prove that the parametric, deterministic density of the Bayesian posterior belongs to the same sparsity class. Generalizing [32, 29], the considered 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. The sparsity results imply, on the one hand, sparsity of Legendre (generalized) polynomial chaos expansions of the Bayesian posterior and, on the other hand, convergence rates for data-adaptive Smolyak integration algorithms for computational Bayesian estimation which are independent of dimension of the parameter space. The convergence rates are, in particular, superior to Markov Chain Monte-Carlo sampling of the posterior, in terms of the number N of instances of the parametric forward problem to be solved.