Fast algorithms for iterative Bayesian PDE inversion

Abstract

In this article we present algorithms to perform Bayesian inversion based on physical models, in particular based on partial differential equations. We are interested in identifying parameters of the PDEs that affect functionals of the solutions for which experimental data are available. Markov-chain Monte-Carlo methods like the Metropolis algorithm provide the algorithmic foundation. We present an adaptation and extension of this procedure to be able to perform multi-dimensional Bayesian inversion where not all measurements have to be present prior to the estimation, but become available in batches as time passes. Namely, based on the Delayed-Rejection Adaptive-Metropolis (DRAM) algorithm, we introduce an iterative approach, where we use the posterior of the last Metropolis run as the prior for the new run, where we use new measurements in each iteration. This allows to examine some information about the parameters already during the estimation process. Therefore a density estimator needs to be introduced. We make use of the Improved Fast Gauss Transform (IFGT) which allows us to perform a faster evaulation of the kernel density estimator, reducing the runtime from quadratic to nearly linear. Applications using a nano-capacitor sensor array are presented as well, where we estimate the radii of over 4000 nano-electrodes.