Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler
Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler
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DOI：
10.1137/19m1251655
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发表时间：
2019-03
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通讯作者：
A. Garbuno-Iñigo;F. Hoffmann;Wuchen Li;A. Stuart
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作者：
A. Garbuno-Iñigo;F. Hoffmann;Wuchen Li;A. Stuart
在科学和工程领域的许多应用中，在不使用正演模型的导数或伴随的情况下解决反问题是非常理想的。在本文中，我们提出了这种方法的一个新版本、一个用于分析它的框架以及所提方法实用性的数值证据。我们的出发点是一组过阻尼朗之万扩散，它们通过一个作为经验集合协方差计算的单一预处理器相互作用。我们证明，由相关随机微分方程（SDE）的平均场极限产生的非线性福克 - 普朗克方程具有一种新颖的梯度流结构，该结构建立在瓦瑟斯坦度量和噪声流的协方差矩阵之上。利用这种结构，我们研究了福克 - 普朗克方程的长时间特性，表明其不变测度与单个朗之万扩散的不变测度一致，并在多种情况下证明了向不变测度的指数收敛。我们通过用集合差代替精确梯度，在集合卡尔曼反演（EKI）算法的基础上引入了一种新的基于原始SDE的含噪变体；这定义了集合卡尔曼采样器（EKS）。给出的数值结果证明了它作为反问题产生的贝叶斯后验的无导数近似采样器的有效性。
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 over-damped 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.