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

We consider approximate solutions f₍, to linear operator equations Kf = g, of the form: f₍, is the minimizer in H of (1 / n) ₉ = ₁ⁿ (Kh) (tⱼ) - y (tⱼ) ² + \| h \|², where H is a Hilbert space, and the data \ {y (tⱼ) \} satisfy y (tⱼ) = g (tⱼ) + (tⱼ), the \ { (tⱼ) \} being measurement errors. f₍, is the so-called regularized solution, and > 0 is the regularization parameter, to be chosen. It is important to choose correctly. The purpose of this paper is to propose the method of weighted cross-validation for choosing 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 estimates the value of which minimizes (1 / n) E₉ = ₁ⁿ (Kf₍, ) (tⱼ) - (Kf) (tⱼ) ². Results related to the convergence of \| f - f₍, \|, including rates, are obtained.