SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 3, Page 1090-1117, January 2020.
Feedback particle filter (FPF) is a numerical algorithm to approximate the solution of the nonlinear filtering problem in continuous-time settings. In any numerical implementation of the FPF algorithm, the main challenge is to numerically approximate the so-called gain function. A numerical algorithm for gain function approximation is the subject of this paper. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian $\Delta_\rho$. The numerical problem is to approximate this solution using only finitely many particles sampled from the probability distribution $\rho$. A diffusion map-based algorithm was proposed by the authors in prior works [A. Taghvaei and P. G. Mehta, Gain function approximation in the feedback particle filter, in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, 2016, pp. 5446--5452], [A. Taghvaei, P. G. Mehta, and S. P. Meyn, Error estimates for the kernel gain function approximation in the feedback particle filter, in American Control Conference (ACC), IEEE, 2017, pp. 4576--4582] to solve this problem. The algorithm is named as such because it involves, as an intermediate step, a diffusion map approximation of the exact semigroup $e^{\Delta_\rho}$. The original contribution of this paper is to carry out a rigorous error analysis of the diffusion map-based algorithm. The error is shown to include two components: bias and variance. The bias results from the diffusion map approximation of the exact semigroup. The variance arises because of finite sample size. Scalings and upper bounds are derived for bias and variance. These bounds are then illustrated with numerical experiments that serve to emphasize the effects of problem dimension and sample size. The proposed algorithm is applied to two filtering examples and comparisons provided with the sequential importance resampling (SIR) particle filter.

中文翻译：

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基于扩散图的反馈粒子滤波中增益函数逼近算法

SIAM / ASA不确定性量化期刊，第8卷，第3期，第1090-1117页，2020年1月。
反馈粒子滤波器（FPF）是一种数值算法，用于在连续时间设置中近似求解非线性滤波问题。在FPF算法的任何数值实现中，主要挑战是在数值上近似所谓的增益函数。增益函数逼近的数值算法是本文的主题。精确的增益函数是涉及概率加权拉普拉斯算子$ \ Delta_ \ rho $的Poisson方程的解。数值问题是仅使用从概率分布$ \ rho $中采样的有限多个粒子来近似此解决方案。作者在先前的工作中提出了一种基于扩散图的算法[A．Taghvaei和PG Mehta，反馈粒子滤波器中的增益函数逼近，在2016年IEEE第55届决策与控制会议（CDC），IEEE，2016，pp.5446--5452]，[A. Taghvaei，PG Mehta和SP Meyn，“反馈粒子滤波器中内核增益函数近似的误差估计”，在美国控制会议（ACC），IEEE，2017年，第4576--4582页]中解决了此问题。该算法之所以这样命名，是因为该算法作为中间步骤涉及精确半群$ e ^ {\ Delta_ \ rho} $的扩散图近似。本文的主要贡献是对基于扩散图的算法进行了严格的误差分析。误差显示为包括两个分量：偏差和方差。偏差是由精确半群的扩散图近似引起的。由于样本数量有限而产生差异。得出比例和上限的偏差和方差。然后用数值实验说明这些界限，以强调问题尺寸和样本量的影响。所提出的算法应用于两个滤波示例，并与顺序重要性重采样（SIR）粒子滤波器进行了比较。