Higher-Order Quasi-Monte Carlo for Bayesian Shape Inversion

In this article, we consider a Bayesian approach towards data assimilation and uncertainty quantification in diffusion problems on random domains. We provide a rigorous analysis of parametric regularity of the posterior distribution given that the data exhibit only limited smoothness. Moreover, we present a dimension truncation analysis for the forward problem, which is formulated in terms of the domain mapping method. Having these novel results at hand, we shall consider as a practical example electrical impedance tomography in the regime of constant conductivities. We are interested in computing moments, in particular, expectation and variance, of the contour of an unknown inclusion, given perturbed surface measurements. By casting the forward problem into the framework of elliptic diffusion problems on random domains, we can directly apply the presented analysis. This straightforwardly yields parametric regularity results for the system response and for the posterior measure, facilitating the application of higher-order quadrature methods for the approximation of moments of quantities of interest. As an example of such a quadrature method, we consider here recently developed higher-order quasi-Monte Carlo methods. To solve the forward problem numerically, we employ a fast boundary integral solver. Numerical examples are provided to illustrate the presented approach and validate the theoretical findings.