Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data

Abstract

Given data $y_i = (Kg)(u_i) + \varepsilon_i$ where the $\varepsilon$'s are random errors, the $u$'s are known, $g$ is an unknown function in a reproducing kernel space with kernel $r$ and $K$ is a known integral operator, it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points $\{u_i\}$ become dense in the interval of interest. These rates are shown to depend on the eigenvalue asymptotics of $KRK^\ast$, where $R$ is the integral operator with kernel $r$. The theory is applied to Abel's equation and the estimation of particle size densities in stereology. Rates of convergence of regularized histogram estimates of the particle size density are given.