In this paper, we study efficient approximate sampling for probability
distributions known up to normalization constants. We specifically focus on a
problem class arising in Bayesian inference for large-scale inverse problems in
science and engineering applications. The computational challenges we address
with the proposed methodology are: (i) the need for repeated evaluations of
expensive forward models; (ii) the potential existence of multiple modes; and
(iii) the fact that gradient of, or adjoint solver for, the forward model might
not be feasible.
While existing Bayesian inference methods meet some of these challenges
individually, we propose a framework that tackles all three systematically. Our
approach builds upon the Fisher-Rao gradient flow in probability space,
yielding a dynamical system for probability densities that converges towards
the target distribution at a uniform exponential rate. This rapid convergence
is advantageous for the computational burden outlined in (i). We apply Gaussian
mixture approximations with operator splitting techniques to simulate the flow
numerically; the resulting approximation can capture multiple modes thus
addressing (ii). Furthermore, we employ the Kalman methodology to facilitate a
derivative-free update of these Gaussian components and their respective
weights, addressing the issue in (iii).
The proposed methodology results in an efficient derivative-free sampler
flexible enough to handle multi-modal distributions: Gaussian Mixture Kalman
Inversion (GMKI). The effectiveness of GMKI is demonstrated both theoretically
and numerically in several experiments with multimodal target distributions,
including proof-of-concept and two-dimensional examples, as well as a
large-scale application: recovering the Navier-Stokes initial condition from
solution data at positive times.

本文研究了精确抽样问题，特别关注涉及贝叶斯推断中的大规模逆问题的科学和工程应用。通过引入基于 Fisher-Rao 梯度流的动力学系统，提出了一种处理贝叶斯推断中的计算挑战的方法，并应用高斯混合逼近和 Kalman 方法以解决多模态分布的问题。所提出的方法可以高效地处理多模态分布，名为 Gaussian Mixture Kalman Inversion（GMKI）。在理论和数值实验中进行的多个实验证明了 GMKI 的有效性，包括概念验证、二维实例以及从正时间的解数据中恢复 Navier-Stokes 初始条件的大规模应用。