A Multi-Resolution Approach to the Fokker-Planck-Kolmogorov Equation with Application to Stochastic Nonlinear Filtering and Optimal Design

Abstract

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.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
Dec 14, 2012
Accession Number
ADA582272

Entities

People

  • Dong Yu
  • Suman Chakravorty

Organizations

  • Texas Engineering Experiment Station

Tags

Communities of Interest

  • Materials and Manufacturing Processes
  • Space

DTIC Thesaurus Topics

  • Algorithms
  • Boltzmann Equation
  • Computational Complexity
  • Computational Fluid Dynamics
  • Computational Science
  • Computer Programs
  • Control Systems
  • Difference Equations
  • Differential Equations
  • Kalman Filters
  • Mathematical Filters
  • Model Predictive Control
  • Partial Differential Equations
  • Random Variables
  • Space Situational Awareness
  • Two Dimensional
  • Wave Equations

Fields of Study

  • Mathematics

Readers

  • Adaptive Control and Estimation with Uncertainty in Dynamic Systems.
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Immunology and Pathology

Technology Areas

  • Space