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The Bayesian approach to inverse problems provides a rigorous framework forthe incorporation and quantification of uncertainties in measurements,parameters and models. We are interested in designing numerical methods whichare robust w.r.t. the size of the observational noise, i.e., methods whichbehave well in case of concentrated posterior measures. The concentration ofthe posterior is a highly desirable situation in practice, since it relates toinformative or large data. However, it can pose a computational challenge fornumerical methods based on the prior or reference measure. We propose to employthe Laplace approximation of the posterior as the base measure for numericalintegration in this context. The Laplace approximation is a Gaussian measurecentered at the maximum a-posteriori estimate and with covariance matrixdepending on the logposterior density. We discuss convergence results of theLaplace approximation in terms of the Hellinger distance and analyze theefficiency of Monte Carlo methods based on it. In particular, we show thatLaplace-based importance sampling and Laplace-based quasi-Monte-Carlo methodsare robust w.r.t. the concentration of the posterior for large classes ofposterior distributions and integrands whereas prior-based importance samplingand plain quasi-Monte Carlo are not. Numerical experiments are presented toillustrate the theoretical findings.

On the Convergence of the Laplace Approximation and Noise-Level-Robustness of Laplace-based Monte Carlo Methods for Bayesian Inverse Problems