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.

翻译：在高山先前的分布和多式场外分布中,高位贝叶斯山反向问题中,通常使用最小化成本功能的大致取样。通过最小化生成的样品密度不是理想的目标密度,除非观测操作员是线性,但样本的分布有助于作为重要取样或Markov连锁Monte Carlo方法的建议密度。在本文中,我们侧重于多式后方分布物的取样应用。我们首先表明,通过计算所有临界点,而不是仅仅减少目标功能的最小化,多式联运分布物的采样会通过计算所有临界点而得到改善。对于高度地球科学问题,我们展示了高效的近似加权,使用的是雅各山决定因素的低位高斯-纽顿近似值。这种方法适用于已知后方分布物和多种模式的达斯流问题。