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Ensemble Kalman Sampler: Mean-field Limit and Convergence Analysis
SIAM Journal on Mathematical Analysis ( IF 2.2 ) Pub Date : 2021-03-23 , DOI: 10.1137/20m1339507 Zhiyan Ding , Qin Li
SIAM Journal on Mathematical Analysis ( IF 2.2 ) Pub Date : 2021-03-23 , DOI: 10.1137/20m1339507 Zhiyan Ding , Qin Li
SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1546-1578, January 2021.
The ensemble Kalman sampler (EKS) is a method introduced in [Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412--441] to find approximately independent and identically distributed samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the well posedness of the SDE system and justify its mean-field limit is a Fokker--Planck equation, whose long time equilibrium is the target distribution. We further demonstrate that the convergence rate is near optimal ($J^{-1/2}$ with $J$ being the number of particles). These results, combined with the in-time convergence of the Fokker--Planck equation to its equilibrium [J. A. Carrillo and U. Vaes, preprint, arXiv:1910.07555, 2019] justify the validity of the EKS, and provide the convergence rate as a sampling method.
中文翻译：
合奏Kalman采样器：均值场限制和收敛性分析
SIAM数学分析杂志，第53卷，第2期，第1546-1578页，2021年1月。
集成卡尔曼采样器（EKS）是[Garbuno-Inigo等，SIAM J. Appl。达因 Syst。，19（2020），pp。412--441]从目标分布中找到近似独立且分布均匀的样本。到目前为止，该算法为何起作用以及如何收敛仍然是未知之数。该算法的连续版本是一组耦合的随机微分方程（SDE）。在本文中，我们证明了SDE系统的适定性，并证明其平均场极限是Fokker-Planck方程，该方程的长时间平衡是目标分布。我们进一步证明了收敛速度接近最佳（$ J ^ {-1/2} $，其中$ J $是粒子数）。这些结果与福克-普朗克方程的及时收敛及其平衡相结合[JA Carrillo和U.Vaes，预印本，arXiv：
更新日期：2021-03-23
The ensemble Kalman sampler (EKS) is a method introduced in [Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412--441] to find approximately independent and identically distributed samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the well posedness of the SDE system and justify its mean-field limit is a Fokker--Planck equation, whose long time equilibrium is the target distribution. We further demonstrate that the convergence rate is near optimal ($J^{-1/2}$ with $J$ being the number of particles). These results, combined with the in-time convergence of the Fokker--Planck equation to its equilibrium [J. A. Carrillo and U. Vaes, preprint, arXiv:1910.07555, 2019] justify the validity of the EKS, and provide the convergence rate as a sampling method.
中文翻译：
合奏Kalman采样器：均值场限制和收敛性分析
SIAM数学分析杂志，第53卷，第2期，第1546-1578页，2021年1月。
集成卡尔曼采样器（EKS）是[Garbuno-Inigo等，SIAM J. Appl。达因 Syst。，19（2020），pp。412--441]从目标分布中找到近似独立且分布均匀的样本。到目前为止，该算法为何起作用以及如何收敛仍然是未知之数。该算法的连续版本是一组耦合的随机微分方程（SDE）。在本文中，我们证明了SDE系统的适定性，并证明其平均场极限是Fokker-Planck方程，该方程的长时间平衡是目标分布。我们进一步证明了收敛速度接近最佳（$ J ^ {-1/2} $，其中$ J $是粒子数）。这些结果与福克-普朗克方程的及时收敛及其平衡相结合[JA Carrillo和U.Vaes，预印本，arXiv：
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