  Minimization of a stochastic cost function is commonly used for approximate
sampling in high-dimensional Bayesian inverse problems with Gaussian prior
distributions and multimodal posterior distributions. The density of the
samples generated by minimization is not the desired target density, unless the
observation operator is linear, but the distribution of samples is useful as a
proposal density for importance sampling or for Markov chain Monte Carlo
methods. In this paper, we focus on applications to sampling from multimodal
posterior distributions in high dimensions. We first show that sampling from
multimodal distributions is improved by computing all critical points instead
of only minimizers of the objective function. For applications to
high-dimensional geoscience problems, we demonstrate an efficient approximate
weighting that uses a low-rank Gauss-Newton approximation of the determinant of
the Jacobian. The method is applied to two toy problems with known posterior
distributions and a Darcy flow problem with multiple modes in the posterior.
