Abstract
The non-intrusive generalized Polynomial Chaos (gPC) method is a popular
computational approach for solving partial differential equations (PDEs) with
random inputs. The main hurdle preventing its efficient direct application for
high-dimensional input parameters is that the size of many parametric sampling
meshes grows exponentially in the number of inputs (the "curse of
dimensionality"). In this paper, we design a weighted version of the reduced
basis method (RBM) for use in the non-intrusive gPC framework. We construct an
RBM surrogate that can rigorously achieve a user-prescribed error tolerance,
and ultimately is used to more efficiently compute a gPC approximation
non-intrusively. The algorithm is capable of speeding up traditional
non-intrusive gPC methods by orders of magnitude without degrading accuracy,
assuming that the solution manifold has low Kolmogorov width. Numerical
experiments on our test problems show that the relative efficiency improves as
the parametric dimension increases, demonstrating the potential of the method
in delaying the curse of dimensionality. Theoretical results as well as
numerical evidence justify these findings.