Reflecting uncertainty in inverse problems: a Bayesian solution using Lévy processes

Abstract

We formulate the inverse problem of solving Fredholm integral equations of the first kind as a nonparametric Bayesian inference problem, using Lévy random fields (and their mixtures) as prior distributions. Posterior distributions for all features of interest are computed employing novel Markov chain Monte Carlo numerical methods in infinite-dimensional spaces, based on generalizations and extensions of the authors' inverse Lévy measure (ILM) algorithm. The method is also well suited for deconvolution problems, for inverting Laplace and Fourier transforms, and for other linear and nonlinear problems in which the unknown feature is high (or even infinite) dimensional and where the corresponding forward problem may be solved rapidly. The methods are illustrated for an application to an important problem in rheology: that of inferring the molecular weight distribution of polymers from conventional rheological measurements, in which we achieve not just a point estimate but a posterior probability density plot representing all uncertainty about the weight.