Analytic Regularity and Polynomial Approximation of Parametric and Stochastic Elliptic PDEs
Abstract
Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space V = H1 0 (D) of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 31, 2010
- Accession Number
- ADA522076
Entities
People
- Albert Cohen
- Christoph Schwab
- Ronald DeVore
Organizations
- University of Paris