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.

翻译：本文研究针对归一化常数未知的概率分布的高效近似采样方法，特别关注科学与工程应用中大规模反问题贝叶斯推断所产生的问题类型。本文所提方法旨在解决以下计算挑战：(i) 需要重复评估昂贵的前向模型；(ii) 可能存在多个模态；(iii) 前向模型的梯度或伴随求解器可能不可行。虽然现有贝叶斯推断方法能单独应对部分挑战，我们提出了一个系统解决所有三个问题的框架。该方法建立在概率空间中Fisher-Rao梯度流的基础上，产生了一个概率密度的动力系统，该系统以均匀指数速率收敛于目标分布。这种快速收敛有利于缓解挑战(i)所述的计算负担。我们采用高斯混合近似与算子分裂技术对流动进行数值模拟；所得近似能够捕获多个模态，从而应对挑战(ii)。此外，我们运用卡尔曼方法实现高斯分量及其对应权重的无导数更新，以解决挑战(iii)。所提出的方法最终形成了一种高效的无导数采样器，其灵活性足以处理多模态分布：高斯混合卡尔曼反演(GMKI)。通过多模态目标分布的若干实验（包括概念验证、二维示例以及大规模应用案例：从正时间解数据中恢复Navier-Stokes方程初始条件），GMKI的有效性在理论与数值层面均得到了验证。