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arXiv · 2606.30972

Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains

Dinh Dũng·Tùng Lê·Alexey Chernov
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Abstract

We establish local Gevrey regularity for the weak solution to parametric divergence-form diffusion elliptic PDEs, assuming the diffusion coefficient itself possesses local Gevrey parametric regularity over a non-compact domain. Here "local Gevrey regularity" means that the regularity is determined in a neighborhood of each parametric point and depends on that point. Explicit bounds on parametric derivatives of the weak solution are proved in the $H^1$-norm. Building on this local Gevrey framework, we develop a novel theoretical treatment of the dimension truncation error for infinite-dimensional integration. We prove convergence rates of quasi-Monte Carlo quadrature for the finite-dimensional Gaussian-weighted integration of integrands having local Gevrey regularity. Together with the finite element discretization error, the errors arising from dimension truncation and quasi-Monte Carlo quadrature yield a complete error analysis and convergence rates for the fully discrete approximation of a bounded linear functional of the weak solution. Numerical experiments confirming the theoretical convergence rates are presented.

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Dinh Dũng, Tùng Lê, Alexey Chernov. 2026-06-29. Local Gevrey regularity and Quasi-Monte Carlo quadrature for PDEs parameterized on non-compact domains. https://arxiv.org/abs/2606.30972

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