2011/01/01 by Albert Cohen, ALBERT COHEN, Ronald DeVore +3 · 349 citations
Decision Sciences · Mathematics · Physics and Astronomy · #Applied mathematics #Bounded function #Curse of dimensionality #Discretization #Hilbert space #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Parametric statistics #Polynomial #Probabilistic and Robust Engineering Design #Rate of convergence #Scientific Research and Discoveries
paper · doi:10.1142/s0219530511001728
published in Analysis and Applications 09(01), 11-47 (World Scientific)
crossref issued 2011/01/01 · crossref published 2011/01/01 · crossref published-print 2011/01/01 · openalex publication_date 2011/01/01 · crossref created 2011/01/19 · crossref published-online 2011/11/20 · crossref deposited 2019/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22 · crossref indexed 2026/08/05
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 PDE's 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 [Formula: see text] 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. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number N dof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.