Parametric holomorphy of elliptic eigenvalue problems

TL;DR

This paper develops a framework to verify parametric holomorphy in elliptic eigenvalue problems, enabling efficient high-dimensional approximation methods like neural networks and quasi-Monte Carlo techniques.

Contribution

It introduces a general approach using derivative bounds to establish parametric holomorphy for elliptic eigenvalue problems, including linear and semilinear cases.

Findings

Ground eigenpairs exhibit the desired holomorphy.

Bounds for mixed derivatives of eigenpairs are derived.

Results support error analysis in quasi-Monte Carlo methods.

Abstract

The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, the bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.