  The Bayesian approach to inverse problems provides a rigorous framework for
the incorporation and quantification of uncertainties in measurements,
parameters and models. We are interested in designing numerical methods which
are robust w.r.t. the size of the observational noise, i.e., methods which
behave well in case of concentrated posterior measures. The concentration of
the posterior is a highly desirable situation in practice, since it relates to
informative or large data. However, it can pose a computational challenge for
numerical methods based on the prior or reference measure. We propose to employ
the Laplace approximation of the posterior as the base measure for numerical
integration in this context. The Laplace approximation is a Gaussian measure
centered at the maximum a-posteriori estimate and with covariance matrix
depending on the logposterior density. We discuss convergence results of the
Laplace approximation in terms of the Hellinger distance and analyze the
efficiency of Monte Carlo methods based on it. In particular, we show that
Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods
are robust w.r.t. the concentration of the posterior for large classes of
posterior distributions and integrands whereas prior-based importance sampling
and plain quasi-Monte Carlo are not. Numerical experiments are presented to
illustrate the theoretical findings.
