High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We 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 generalised polynomial chaos discretisations 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 the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. 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 Chkifa et al. (Mod,l. Math. Anal. Num,r. 47(1):253-280, 2013) 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.