Simultaneous spatial-parametric collocation approximation for parametric PDEs with log-normal random inputs
This provides improved theoretical guarantees for numerical methods in uncertainty quantification for parametric PDEs, though it appears incremental as it builds on existing collocation frameworks.
The paper tackles the problem of solving parametric elliptic PDEs with log-normal random inputs by establishing convergence rates for a fully discrete, multi-level linear collocation method. The result shows significantly improved rates that, up to logarithmic factors, match the rates of best n-term approximations.
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.