Simultaneous spatial-parametric collocation approximation for parametric PDEs with log-normal random inputs

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.