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doi: 10.3934/dcdss.2021097
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## A drift homotopy implicit particle filter method for nonlinear filtering problems

 Department of Mathematics, Florida State University, Tallahassee, Florida

* Corresponding author

Received  February 2021 Revised  June 2021 Early access August 2021

In this paper, we develop a drift homotopy implicit particle filter method. The methodology of our approach is to adopt the concept of drift homotopy in the resampling procedure of the particle filter method for solving the nonlinear filtering problem, and we introduce an implicit particle filter method to improve the efficiency of the drift homotopy resampling procedure. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of our drift homotopy implicit particle filter.

Citation: Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021097
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##### References:
Double well potential case 1: $\alpha = 1$, $\sigma = 1.5$, $R = 1.5$
Double well potential case 2: $\alpha = 1$, $\sigma = 1$, $R = 1$ with state switch
Double well potential case 3: $\alpha = 10$, $\sigma = 1$, $R = 2$ with state switch
Tracking performance for $4000$ steps
Tracking errors for $4000$ steps
Tracking performance with rapid change in the state
Tracking errors with rapid change in the state
Mean square errors with respect to observation gaps
 Algorithm: Drift homotopy implicit particle filter (DHIPF) Initialize the particle cloud $\{x_0^{(i)}\}_{i=1}^{N_p}$, the number of drift homotopy levels $L$ with the intermediate drift function $b$ and the constant sequence $\{\beta_l\}_{l=0}^{L}$, and the reference random variable $\xi$ for the implicit particle filter procedure. while $n =0, 1, 2, \cdots$, do for: particles $i = 1, 2, \cdots, N_p$, for: drift homotopy levels $l = 0, 1, 2, \cdots, L-1$, -: Construct the drift homotopy dynamics (10);             -: Solve for $\psi_{l}^{n+1, i}$ in the equation (17) with the initial guess $\hat{x}_{n+1, l}^{(i)}$;             -: Generate the sample $\hat{x}_{n+1, l+1}^{(i)}$ through $\exp\big(- \frac{\xi^{T} \xi}{2}\big) J_{\psi_l^{n+1, i}}^{-1}$; end for end for The particles $\{x_{n+1}^{(i)}\}_{i = 1}^{N_p} : = \{\hat{x}_{n+1, L}^{(i)}\}_{i = 1}^{N_p}$ provide an empirical distribution for the filtering density $p(X_{n+1} | Y_{1:n+1})$ end while
 Algorithm: Drift homotopy implicit particle filter (DHIPF) Initialize the particle cloud $\{x_0^{(i)}\}_{i=1}^{N_p}$, the number of drift homotopy levels $L$ with the intermediate drift function $b$ and the constant sequence $\{\beta_l\}_{l=0}^{L}$, and the reference random variable $\xi$ for the implicit particle filter procedure. while $n =0, 1, 2, \cdots$, do for: particles $i = 1, 2, \cdots, N_p$, for: drift homotopy levels $l = 0, 1, 2, \cdots, L-1$, -: Construct the drift homotopy dynamics (10);             -: Solve for $\psi_{l}^{n+1, i}$ in the equation (17) with the initial guess $\hat{x}_{n+1, l}^{(i)}$;             -: Generate the sample $\hat{x}_{n+1, l+1}^{(i)}$ through $\exp\big(- \frac{\xi^{T} \xi}{2}\big) J_{\psi_l^{n+1, i}}^{-1}$; end for end for The particles $\{x_{n+1}^{(i)}\}_{i = 1}^{N_p} : = \{\hat{x}_{n+1, L}^{(i)}\}_{i = 1}^{N_p}$ provide an empirical distribution for the filtering density $p(X_{n+1} | Y_{1:n+1})$ end while
Example 1. Performance comparison for Case 1
 APF EnKF IPF DHPF DHIPF CPU Time $9.703$ $0.365625$ $0.0938$ $48.563$ $\bf{0.312}$ MSE $2.03 E-3$ $5.22 E-3$ $1.10 E-3$ $5.92 E-4$ $\bf{4.60 E-4}$
 APF EnKF IPF DHPF DHIPF CPU Time $9.703$ $0.365625$ $0.0938$ $48.563$ $\bf{0.312}$ MSE $2.03 E-3$ $5.22 E-3$ $1.10 E-3$ $5.92 E-4$ $\bf{4.60 E-4}$
Example 1. Performance comparison for Case 2
 APF EnKF IPF DHPF DHIPF CPU Time $9.391$ $0.578$ $0.109$ $43.344$ $\bf{0.297}$ MSE $6.29 E-1$ $1.36$ $5.28 E-3$ $1.78 E-3$ $\bf{1.08 E-3}$
 APF EnKF IPF DHPF DHIPF CPU Time $9.391$ $0.578$ $0.109$ $43.344$ $\bf{0.297}$ MSE $6.29 E-1$ $1.36$ $5.28 E-3$ $1.78 E-3$ $\bf{1.08 E-3}$
Example 1. Performance comparison for Case 3
 APF EnKF IPF DHPF DHIPF CPU Time $9.563$ $0.453$ $0.156$ $50.188$ $\bf{0.422}$ MSE $1.80$ $1.99$ $5.02 E-3$ $1.35 E-2$ $\bf{1.59 E-3}$
 APF EnKF IPF DHPF DHIPF CPU Time $9.563$ $0.453$ $0.156$ $50.188$ $\bf{0.422}$ MSE $1.80$ $1.99$ $5.02 E-3$ $1.35 E-2$ $\bf{1.59 E-3}$
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