Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916665294323712 |
|---|---|
| author | Liu, Shu Zuo, Xinzhe Osher, Stanley Li, Wuchen |
| author_facet | Liu, Shu Zuo, Xinzhe Osher, Stanley Li, Wuchen |
| contents | In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method Liu, Shu Zuo, Xinzhe Osher, Stanley Li, Wuchen Numerical Analysis Optimization and Control In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section. |
| title | Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2401.14602 |