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

Duan-Peng Ling, Wenlong Zhang

TL;DR

This work tackles the statistical recovery of solutions to first-kind Fredholm integral equations from discrete noisy measurements by harnessing a data-driven Tikhonov regularization with fidelity $|Kx-\boldsymbol w|_n^2$. Under the assumption $R(K)\subset H^m(a,b)$, it leverages spectral-decay properties $s_j(K)\le C j^{-m}$ to derive sharp nonasymptotic mean-square bounds and high-probability tails (sub-Gaussian case), along with explicit a priori and a posteriori parameter-choice rules. A variational characterization reduces the problem to a finite-dimensional, well-posed subspace, and the analysis extends to convergence in the dual space $W^*=W^{*}$ with $W=R((K^*K)^{1/4})$, bypassing classical source conditions. Numerical experiments corroborate the theory, demonstrating near-optimal parameter tuning, confirmed rate predictions, and efficient adaptive schemes for practical data-driven inverse problems in ill-posed settings.

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 $σ$, 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.

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

TL;DR

This work tackles the statistical recovery of solutions to first-kind Fredholm integral equations from discrete noisy measurements by harnessing a data-driven Tikhonov regularization with fidelity . Under the assumption , it leverages spectral-decay properties to derive sharp nonasymptotic mean-square bounds and high-probability tails (sub-Gaussian case), along with explicit a priori and a posteriori parameter-choice rules. A variational characterization reduces the problem to a finite-dimensional, well-posed subspace, and the analysis extends to convergence in the dual space with , bypassing classical source conditions. Numerical experiments corroborate the theory, demonstrating near-optimal parameter tuning, confirmed rate predictions, and efficient adaptive schemes for practical data-driven inverse problems in ill-posed settings.

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 , which implies algebraic singular-value decay , 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 , noise level , and smoothness index ; 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.
Paper Structure (13 sections, 26 theorems, 146 equations, 1 figure, 1 algorithm)

This paper contains 13 sections, 26 theorems, 146 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Variational characterization of the data driven regularized solution
  3. Mean-square and weak convergence analysis with bounded-variance noise
  4. Preliminaries
  5. Convergence analysis in probability
  6. Convergence analysis with respect to expectation
  7. Weak convergence in the dual space
  8. Convergence for data with sub-Gaussian noise
  9. Preliminaries
  10. Main results with sub-Gaussian noise
  11. Numerical experiments and discussions
  12. Discretization and adaptive parameter choice
  13. Numerical results and discussions

Key Result

Lemma 2.1

Let $K:L^2(a,b)\to L^2(a,b)$ be the integral operator $(Kx)(s)=\int_a^b k(s,t)x(t)\,dt$ with the kernel $k(s,t)\in L^2((a,b)^2)$ and let observations be $w_i=y(s_i)+e(s_i)$ at sampling points $\{s_i\}_{i=1}^n$. Then for $\alpha>0$, $x_{n,\alpha}$ is the unique minimizer of the functional It is characterized in variational form

Figures (1)

  • Figure 5: Fitted slopes of $\log s_j$ vs. $\log j$ for different discretization sizes.

Theorems & Definitions (47)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1: Edmunds2008, p.11
  • Lemma 3.1: Edmunds2008, Lemma 2, p.11
  • Theorem 3.1: Edmunds1987, Theorem 5.10, p.88
  • Theorem 3.2: Edmunds2008, p.119
  • Theorem 3.3: Kirsch2021, p.324
  • Theorem 3.4: Kirsch2021, p.331
  • ...and 37 more