Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs

In this work we provide a convergence analysis for the quasi-optimal version of the Stochastic Sparse Grid Collocation method we had presented in our previous work “On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods” [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This argument is very general and can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasioptimal sparse grid to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the “inclusions problem”: we detail the convergence estimate obtained in this case, using polynomial interpolation on either nested (Clenshaw–Curtis) or non-nested (Gauss–Legendre) abscissas, verify its sharpness numerically, and compare the performance of the resulting quasioptimal grids with a few alternative sparse grids construction schemes recently proposed in literature.