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High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs
(2012)SAM Research ReportWe consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solution to parametric/stochastic PDE’s. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the ...Report -
Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs
(2009)SAM Research ReportDeterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D_Rd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(!)=(yi(!)). This yields an equivalent parametric deterministic PDE whose ...Report -
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs
(2010)SAM Research ReportParametric 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 ...Report -
Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs
(2013)SAM Research ReportThe numerical approximation of parametric partial differential equations $D(u,y)$ =0 is a computational challenge when the dimension $d$ of the parameter vector $y$ is large, due to the so-called $curse$ $of$ $dimensionality$. It was recently shown in [5, 6] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map $y$ ...Report -
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
(2011)Research ReportsThe numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [11, 12] that under very weak assumptions on the diffusion ...Report