A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations

Feng Bao; Yanzhao Cao; Clayton Webster; Guannan Zhang

doi:10.1137/140952910

A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations

Feng Bao, Yanzhao Cao, Clayton Webster, Guannan Zhang

Data Analysis and Machine Learning

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

A hybrid finite difference algorithm for the Zakai equation is constructed to solve nonlinear filtering problems. The algorithm combines the splitting-up finite difference scheme and hierarchical sparse grid method to solve moderately high-dimensional nonlinear filtering problems. When applying hierarchical sparse-grid methods to approximate bell-shaped solutions in most applications of nonlinear filtering problems, we introduce a logarithmic approximation to reduce the approximation errors. Some space adaptive methods are also introduced to make the algorithm more efficient. Numerical experiments are carried out to demonstrate the performance and efficiency of our algorithm.

Original language	English
Pages (from-to)	784-804
Number of pages	21
Journal	SIAM-ASA Journal on Uncertainty Quantification
Volume	2
Issue number	1
DOIs	https://doi.org/10.1137/140952910
State	Published - 2014

Funding

∗Received by the editors January 16, 2014; accepted for publication (in revised form) October 3, 2014; published electronically December 23, 2014. This material is based upon work supported in part by the U.S. Air Force Office of Scientific Research under grants FA9550-12-1-0281 and 1854-V521-12; by the National Science Foundation under grant DMS0914554; by the Guangdong Provincial Government of China through the Computational Science Innovative Research Team program; by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts ERKJ259 and ERKJE45; and by the Laboratory of Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle for the U.S. Department of Energy under contract DE-AC05-00OR22725. http://www.siam.org/journals/juq/2/95291.html †Department of Mathematics and Statistics, Auburn University, and Department of Computational and Applied Mathematics, Oak Ridge National Laboratory, MS-6164, Oak Ridge, TN 37831 ([email protected]). ‡Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 ([email protected]).

Funders	Funder number
Guangdong Provincial Government of China
National Science Foundation	DMS0914554
U.S. Department of Energy
Air Force Office of Scientific Research	1854-V521-12, FA9550-12-1-0281
Office of Science
Advanced Scientific Computing Research	ERKJ259, ERKJE45
Laboratory Directed Research and Development
UT-Battelle	DE-AC05-00OR22725

Keywords

Logarithmic approximation
Nonlinear filtering problems
Space adaptive methods
Sparse-grid approximation
Splitting-up method
Zakai equation

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abstract = "A hybrid finite difference algorithm for the Zakai equation is constructed to solve nonlinear filtering problems. The algorithm combines the splitting-up finite difference scheme and hierarchical sparse grid method to solve moderately high-dimensional nonlinear filtering problems. When applying hierarchical sparse-grid methods to approximate bell-shaped solutions in most applications of nonlinear filtering problems, we introduce a logarithmic approximation to reduce the approximation errors. Some space adaptive methods are also introduced to make the algorithm more efficient. Numerical experiments are carried out to demonstrate the performance and efficiency of our algorithm.",

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A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations

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