# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021097
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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
##### References:

show all references

##### 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 $1.80$ $1.99$ $5.02 E-3$ $1.35 E-2$ $\bf{1.59 E-3}$
 [1] Qifeng Cheng, Xue Han, Tingting Zhao, V S Sarma Yadavalli. Improved particle swarm optimization and neighborhood field optimization by introducing the re-sampling step of particle filter. Journal of Industrial & Management Optimization, 2019, 15 (1) : 177-198. doi: 10.3934/jimo.2018038 [2] Jin-Won Kim, Amirhossein Taghvaei, Yongxin Chen, Prashant G. Mehta. Feedback particle filter for collective inference. Foundations of Data Science, 2021  doi: 10.3934/fods.2021018 [3] Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337 [4] Gerasimos G. Rigatos, Efthymia G. Rigatou, Jean Daniel Djida. Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1017-1035. doi: 10.3934/mbe.2015.12.1017 [5] David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117 [6] Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207 [7] Andrea Arnold, Daniela Calvetti, Erkki Somersalo. Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs. Conference Publications, 2015, 2015 (special) : 75-84. doi: 10.3934/proc.2015.0075 [8] Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021  doi: 10.3934/fods.2021023 [9] Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021  doi: 10.3934/fods.2021025 [10] Yao Lu, Rui Zhang, Dongdai Lin. Improved bounds for the implicit factorization problem. Advances in Mathematics of Communications, 2013, 7 (3) : 243-251. doi: 10.3934/amc.2013.7.243 [11] H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Hien T. Tran. A comparison of nonlinear filtering approaches in the context of an HIV model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 213-236. doi: 10.3934/mbe.2010.7.213 [12] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [13] Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227 [14] Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems & Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169 [15] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [16] Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure & Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861 [17] Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573 [18] Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial & Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105 [19] Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193 [20] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679

2020 Impact Factor: 2.425

## Tools

Article outline

Figures and Tables