Simultaneous spatial-parametric collocation approximation for parametric PDEs with log-normal random inputs
Dinh Dũng
TL;DR
The paper develops a comprehensive Bochner-space framework for fully discrete, multi-level spatial-parametric collocation of parametric PDEs with log-normal inputs. It combines finite element spatial discretization, Hermite-Gaussian collocation or extended least-squares sampling for the parametric variables, and sparse-grid interpolation to achieve convergence rates that closely match best $n$-term approximations, up to logarithmic factors. The key contributions include new multilevel sampling guarantees in Bochner spaces, sharp rate regimes depending on spatial regularity, and practical algorithms with near-optimal computational costs. This work advances efficient uncertainty quantification for high-dimensional parametric PDEs and offers a broadly applicable methodology for holomorphic function classes arising in stochastic settings.
Abstract
We establish convergence rates for a fully discrete, multi-level, linear collocation method solving parametric elliptic PDEs on bounded polygonal domains with log-normal inputs. The method uses a finite set of function evaluations in the spatial-parametric domain. Compared with the best-known fully discrete collocation rates, these rates are significantly improved and, up to logarithmic factors, match the rates of best n-term approximations. The results follow from applying general multi-level linear sampling recovery theory in abstract Bochner spaces -- via extended least-squares -- to infinite-dimensional holomorphic functions. The abstract multi-level recovery in Bochner spaces guarantees yield the improved rates when specialized to the parametric PDE setting.