A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence Analysis
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911343361130496 |
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| author | Ling, Duan-Peng Zhang, Wenlong |
| author_facet | Ling, Duan-Peng Zhang, Wenlong |
| contents | We investigate the statistical recovery of solutions to first-kind Fredholm integral equations with discrete, scattered, and noisy pointwise measurements. Assuming the forward operator's range belongs to the Sobolev space of order $m$, which implies algebraic singular-value decay $s_j\le Cj^{-m}$, we derive optimal upper bounds for the reconstruction error in the weak topology under an a priori choice of the regularization parameter. For bounded-variance noise, we establish mean-square error rates that explicitly quantify the dependence on sample size $n$, noise level $σ$, and smoothness index $m$; under sub-Gaussian noise, we strengthen these to exponential concentration bounds. The analysis yields an explicit a priori and a posteriori rule for the regularization parameter. Numerical experiments validate the theoretical results and demonstrate the efficiency of our practical parameter choice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence Analysis Ling, Duan-Peng Zhang, Wenlong Numerical Analysis Mathematical Physics We investigate the statistical recovery of solutions to first-kind Fredholm integral equations with discrete, scattered, and noisy pointwise measurements. Assuming the forward operator's range belongs to the Sobolev space of order $m$, which implies algebraic singular-value decay $s_j\le Cj^{-m}$, we derive optimal upper bounds for the reconstruction error in the weak topology under an a priori choice of the regularization parameter. For bounded-variance noise, we establish mean-square error rates that explicitly quantify the dependence on sample size $n$, noise level $σ$, and smoothness index $m$; under sub-Gaussian noise, we strengthen these to exponential concentration bounds. The analysis yields an explicit a priori and a posteriori rule for the regularization parameter. Numerical experiments validate the theoretical results and demonstrate the efficiency of our practical parameter choice. |
| title | A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence Analysis |
| topic | Numerical Analysis Mathematical Physics |
| url | https://arxiv.org/abs/2512.23362 |