A Multi-Resolution Approach to the Fokker-Planck-Kolmogorov Equation with Application to Stochastic Nonlinear Filtering and Optimal Design
Final rept. 15 Sep 2009-14 Sep 2012
TEXAS ENGINEERING EXPERIMENT STATION COLLEGE STATION
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In this paper, we consider the filtering of systems governed by partial differential equations PDE. We adopt a reduced order model ROM based strategy to solve the problem. We propose an iterative version of the snapshot proper orthogonal decomposition POD technique, termed I-POD, to sequentially construct a single ROM for PDEs that is capable of capturing their behavior over the entire state space of the system, and not just around the snapshot trajectory. Further, the technique is entirely data based, and is applicable to forced as well as unforced systems. The I-POD is compared to two other ROM techniques the Balanced POD BPOD and the dynamic mode decomposition DMD. We apply the ROM generated using the I-POD technique to construct reduced order Kalman filters to solve the filtering problem. The methodology is tested on several 1-dimensional PDEs of interest including the heat equation, the wave equation and 2-dimensional pollutant transport equation.
- Numerical Mathematics
- Statistics and Probability
- Operations Research