International audienceWe consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalized polynomial chaos discretizations to random and stochastic PDEs. 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 computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on standard principle of tensorization of a one dimensional interpolation scheme and sparsifi-cation. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE's, we have shown in [11] that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE's

The numerical approximation of parametric partial differential equations D(u,y)=0 is a computational challenge when the dimension d of of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown 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 u -> u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for some classes parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z->u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying radii of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case where d is infinite. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDE's that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDE's, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allows us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions