Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy

Abstract

We consider approximate solutions $f_{n,\lambda } $ to linear operator equations $\mathcal{K}f = g$, of the form: $f_{n,\lambda } $ is the minimizer in $\mathcal{H}$ of $({1 / n})\sum _{j = 1}^n {[(\mathcal{K}h)(t_j ) - y(t_j )]} ^2 + \lambda \| h \|^2 $, where $\mathcal{H}$ is a Hilbert space, and the data $\{ {y(t_j )} \}$ satisfy $y(t_j ) = g(t_j ) + \varepsilon (t_j )$, the $\{ {\varepsilon (t_j )} \}$ being measurement errors. $f_{n,\lambda } $ is the so-called regularized solution, and $\lambda > 0$ is the regularization parameter, to be chosen. It is important to choose $\lambda $ correctly. The purpose of this paper is to propose the method of weighted cross-validation for choosing $\lambda $from the data. We suppose that g is very smooth and the errors are white noise. It is shown that the weighted cross-validation estimate $\hat \lambda $ estimates the value of $\lambda $ which minimizes $({1 / n})E\sum\nolimits_{j = 1}^n {[(\mathcal{K}f_{n,\lambda } )(t_j ) - (\mathcal{K}f)(t_j )]} ^2 $ . Results related to the convergence of $\| {f - f_{n,\hat \lambda } } \|$, including rates, are obtained.