Sampling recovery in Bochner spaces and applications to parametric PDEs

Abstract: We prove convergence rates of linear sampling recovery of functions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algorithm is a function-valued extension of the least squares method widely used and thoroughly studied in scalar-valued function recovery. We apply our theory to collocation approximation of solutions to parametric elliptic or parabolic PDEs with log-normal random inputs and to relevant approximation of infinite dimensional holomorphic functions on $\mathbb R^\infty$. The application allows us to significantly improve known results in Computational Uncertainty Quantification for these problems. Our results are also applicable for parametric PDEs with affine inputs, where they match the known rates.