Binned Multilevel Monte Carlo for Bayesian Inverse Problems with Large Data

Abstract

We consider Bayesian inversion of parametric operator equations for the case of a large number of measurements. Increased computational efficiency over standard averaging approaches, per measurement, is obtained by binning the data and applying a multilevel Monte Carlo method, specifying optimal forward solution tolerances per level. Based on recent bounds of convergence rates of adaptive Smolyak quadratures in Bayesian inversion for single observation data, the bin sizes in large sets of measured data are optimized and a rate of convergence of the error vs. work is derived analytically and confirmed by numerical experiments.