Sampling recovery in Bochner spaces and applications to parametric PDEs
Felix Bartel, Dinh Dũng
TL;DR
The paper tackles non-intrusive, linear sampling recovery for parametric PDEs with random inputs by formulating the problem in Bochner spaces $L_2(U,X;\mu)$ and reducing Hilbert-valued recovery to a scalar RKHS setting. It establishes near-optimal convergence rates, showing the sampling width $\varrho_n(B_{X,\boldsymbol{\sigma}}, L_2(U,X;\mu))$ decays as $n^{-1/q}$ for $0<q<2$ under $\|\boldsymbol{\sigma}^{-1}\|_{\ell_q}\le1$, unifying log-normal and affine input models and improving over prior results. The authors extend least-squares sampling to Bochner spaces, derive quantitative convergence rates, and apply the theory to parametric elliptic PDEs and to infinite-dimensional holomorphic function spaces, achieving substantial rate improvements (e.g., $n^{-1/q}$ in many cases and a $n^{-1/2}$-level improvement over previous holomorphic-function results). They also develop constructive sampling schemes with high-probability error guarantees, enabling scalable non-intrusive approximations and providing a unified framework across log-normal, affine, and holomorphic settings with practical implications for computational uncertainty quantification.
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 two core problems in Computational Uncertainty Quantification. First, we address non-intrusive approximations of solutions to parametric elliptic or parabolic PDEs with log-normal inputs, using a finite set of particular solvers. Second, we consider approximating infinite-dimensional holomorphic functions that arise as solutions to more general parametric PDEs with Gaussian random field inputs. This approach yields substantial improvements in the state of the art for these problems. Importantly, our framework unifies log-normal and affine input models. In the affine case, we obtain convergence rates that improve known results by a logarithmic factor.