Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos

Abstract

We present a generalized polynomial chaos algorithms for the solution of stochastic elliptic partial differential equations subject to uncertain inputs. In particular, were focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener (1938). A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations.

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Document Details

Document Type
Technical Report
Publication Date
Jul 25, 2002
Accession Number
ADA460658

Entities

People

  • Dongbin Xiu
  • George Karniadakis

Organizations

  • Brown University

Tags

Communities of Interest

  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Algorithms
  • Applied Mathematics
  • Computational Science
  • Differential Equations
  • Diffusion
  • Eigenvalues
  • Equations
  • Mathematics
  • Monte Carlo Method
  • Partial Differential Equations
  • Polynomials
  • Probability
  • Probability Density Functions
  • Random Variables
  • Steady State
  • Stochastic Processes
  • Weighting Functions

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