SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 412-441, January 2020.
Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of overdamped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker--Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker--Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

中文翻译：

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相互作用的兰格文扩散：梯度结构和集合卡尔曼采样器

SIAM应用动力系统杂志，第19卷，第1期，第412-441页，2020年1月。
在科学和工程学中出现的许多应用中，非常需要在不使用正向模型的导数或伴随数的情况下解决反问题。在本文中，我们提出了这种方法的新版本，其分析框架以及所提出方法的实用性的数值证据。我们的出发点是过度阻尼的Langevin扩散的合奏，它通过单个预条件进行交互，该前置条件作为经验性合奏协方差进行计算。我们证明了源自相关随机微分方程（SDE）的平均场极限的非线性Fokker-Planck方程具有新颖的梯度流结构，该结构建立在Wasserstein度量和噪声流的协方差矩阵上。使用这种结构，我们研究了Fokker-Planck方程的较大时间特性，表明其不变度量与单个Langevin扩散的一致，并证明了在许多情况下指数收敛于不变度量。我们将原始SDE中发现的集合卡尔曼反演（EKI）算法引入了一个新的噪声变体，方法是将精确的梯度替换为集合差异。这定义了集成卡尔曼采样器（EKS）。数值结果表明，该方法可有效解决反问题引起的贝叶斯后验的无导数近似采样。我们将原始SDE中发现的集合卡尔曼反演（EKI）算法引入了一个新的噪声变体，方法是将精确的梯度替换为集合差异。这定义了集成卡尔曼采样器（EKS）。数值结果表明，该方法可有效解决反问题引起的贝叶斯后验的无导数近似采样。我们将原始SDE中发现的集合卡尔曼反演（EKI）算法引入了一个新的噪声变体，方法是将精确的梯度替换为集合差异。这定义了集成卡尔曼采样器（EKS）。数值结果表明，该方法可有效解决反问题引起的贝叶斯后验的无导数近似采样。