Achievement

We propose a meshfree approximation method for the implicit filter, which is a novel numerical algorithm for nonlinear filtering problems. The main challenge of the implicit filter method is that the conditional PDF of the nonlinear filtering solution is estimated at grid points. As such the method suffers the so called “the curse of dimensionality” when the dimension of the state variable is high. In addition, the efficiency of the method may be significantly reduced when the domain of the PDF is unbounded. In this paper, we propose to construct a meshfree implicit filter algorithm to alleviate the aforementioned challenges. Motivated by the particle filter method, we first generate a set of random particles and propagate these particles through the system model and use these particles to replace the grid points in the state space. After that we generate other necessary points through the Shepard’s method which constructs the interpolant by the weighted average of the values on state points [21]. In order to prevent particle degeneracy in the generation of random state points, we introduce a resample step in the particle propagation. In addition we choose state points according to the system state, which make them adaptively located in the high probability region of the PDF of state. In this way, we solve the nonlinear filtering problem in a relatively small region in the state space at each time step and approximate the solution on a set of meshfree state points distributed adaptively to the desired PDF of the state. Furthermore, since we approximate the PDF as a function on each state point, instead of using state points themselves to describe the empirical distribution, the implicit filter algorithm requires much fewer points than the particle filter method to depict the PDF of the state. The construction of the algorithm includes generation of random state space points and a meshfree interpolation method. Numerical experiments show the effectiveness and efficiency of our algorithm.

Publications:

• F. Bao, Y. Cao, C. Webster and G. Zhang, An efficient meshfree implicit filter for nonlinear filtering problems, International Journal for Uncertainty Quantification, 6 (2016), pp. 19-33.

• C. Webster, G. Zhang and M. Gunzburger, An adaptive sparse-grid-based iterative ensemble Kalman filter approach for parameter field estimation, International Journal of Computer Mathematics, 91 (2014), pp. 798-817.