A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence Analysis

TL;DR

This paper develops a statistical method for solving first-kind Fredholm integral equations from noisy data, providing optimal error bounds, regularization rules, and validating results with numerical experiments.

Contribution

It introduces a data-driven regularization approach with explicit error bounds and parameter choice rules for solving Fredholm integral equations from noisy measurements.

Findings

Optimal error bounds in weak topology

Explicit regularization parameter selection rules

Numerical validation of theoretical results

Abstract

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 $\sigma$, 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.