Stochastic Convergence Analysis of Inverse Potential Problem
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908383706087424 |
|---|---|
| author | Jin, Bangti Quan, Qimeng Zhang, Wenlong |
| author_facet | Jin, Bangti Quan, Qimeng Zhang, Wenlong |
| contents | In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an $H^1(Ω)$ penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(Ω)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $γ$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14106 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Convergence Analysis of Inverse Potential Problem Jin, Bangti Quan, Qimeng Zhang, Wenlong Numerical Analysis In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an $H^1(Ω)$ penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(Ω)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $γ$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results. |
| title | Stochastic Convergence Analysis of Inverse Potential Problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2410.14106 |