Sparse, adaptive Smolyak algorithms for Bayesian inverse problems

Abstract

Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite dimensional, separable Banach spaces proposed in [28], we develop a practical computational algorithm whose convergence rates are provably higher than those of Monte-Carlo (MC) and Markov-Chain Monte-Carlo methods, in terms of the number of solutions of the forward problem. In the formulation of [28], the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. Sparsity of the generalized polynomial chaos (gpc) representation of the posterior density being implied by sparsity assumptions on the class of the prior [28], we design, analyze and implement a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient approximate numerical evaluation of expectations under the posterior, given data. The proposed algorithm is based on a greedy, iterative identification of finite sets of most significant, “active” chaos polynomials in the the posterior density analogous to recently proposed algorithms for adaptive interpolation [7, 8]. 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 results for a model problem of coefficient identification with point measurements in a diffusion problem confirm the theoretical results.