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Parametric holomorphy of elliptic eigenvalue problems
  • Byeong-Ho Bahn
Byeong-Ho Bahn
University of Massachusetts Amherst

Corresponding Author:bban@umass.edu

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Abstract

not-yet-known not-yet-known not-yet-known unknown 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, improved 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.
28 Aug 2024Submitted to Mathematical Methods in the Applied Sciences
29 Aug 2024Submission Checks Completed
29 Aug 2024Assigned to Editor
04 Sep 2024Review(s) Completed, Editorial Evaluation Pending
24 Sep 2024Reviewer(s) Assigned