Parametric holomorphy of elliptic eigenvalue problems
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917915868004352 |
|---|---|
| author | Bahn, Byeong-Ho |
| author_facet | Bahn, Byeong-Ho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parametric holomorphy of elliptic eigenvalue problems Bahn, Byeong-Ho Numerical Analysis Analysis of PDEs 35A17 (Primary) 35J61, 35P30, 35A20, 35P05 (Secondary) 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. |
| title | Parametric holomorphy of elliptic eigenvalue problems |
| topic | Numerical Analysis Analysis of PDEs 35A17 (Primary) 35J61, 35P30, 35A20, 35P05 (Secondary) |
| url | https://arxiv.org/abs/2408.01227 |