Incremental kernel based approximations for Bayesian inverse problems

Abstract

We provide an interpretation for the covariance of the predictive process of Bayesian Gaussian process regression as reproducing kernel of a subset of the Cameron Martin space of the prior. We demonstrate that this deterministic viewpoint enables us to relate particular greedy methods using that subset kernel to instances of powerful low-rank matrix approximation techniques such as <em>adaptive cross approximation</em> or <em>pivoted Cholesky decomposition</em>. In particular, we can show convergence results for such algorithms which appear to be novel in the case of finitely smooth kernels. <br> Moreover, we consider the inverse problem to reconstruct a parametrized diffusion coefficient from point evaluations of the solution to a diffusion equation with that parametrized coefficient. To this end, we present a Gaussian process regression based approach to approximate the observati- on operator. The error estimates for this approximation methods are capable to take deterministic model errors explicitly into account. Finally, we show how the findings about incremental low rank approximations can be applied to these reconstruction problems.