Sparse Quadrature Approach to Bayesian Inverse Problems

Abstract

We consider the parametric deterministic formulation of Bayesian inverse problems with distributed parameter uncertainty from infinite dimensional, separable Banach spaces X, with uniform prior probability measure on space X of all uncertainties. Under the assumption of given observation data δ subject to additive observation noise η ∼ N (0, Γ) with positive covariance Γ, an infinite-dimensional version of Bayes’ formula has been shown in [14]. For problems with uncertain, distributed parameters u ∈ X (which could be a diffusion coefficient, elastic moduli in solid mechanics, shape of the domain D of definition of the physical problem [1], kinetic parameters in stoichiometric models of reaction-systems in biological systems [4, 7], permeability in porous media or optimal control of uncertain systems [9]), we develop a practical, adaptive computational algorithm for the efficient approximation of the infinite-dimensional integrals with respect to the Bayesian posterior (conditional on given data δ) μδ which arise in Bayes’ formula in [14]