An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations

Abstract

We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic diffierential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized Polynomial Chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan-Orszag three-mode problem, as well as advection-diffusion problems. The new method is similar to spectral element method for deterministic problems but with h-p discretization of the random space.

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Document Details

Document Type
Technical Report
Publication Date
Mar 09, 2005
Accession Number
ADA458984

Entities

People

  • George Karniadakis
  • Xiaoliang Wan

Organizations

  • Brown University

Tags

Communities of Interest

  • Air Platforms
  • Energy and Power Technologies

DTIC Thesaurus Topics

  • Algorithms
  • Applied Mathematics
  • Differential Equations
  • Distribution Functions
  • Equations
  • Fluid Flow
  • Mathematics
  • Monte Carlo Method
  • Partial Differential Equations
  • Polynomials
  • Probability
  • Probability Distributions
  • Random Variables
  • Stochastic Processes
  • Three Dimensional
  • Time Intervals
  • Two Dimensional

Fields of Study

  • Mathematics

Readers

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

Technology Areas

  • Space