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

We consider approximate solutions fn,λ to linear operator equations Kf = g, of the form: fn,λ is the minimizer in H of (1 / n)∑ j = 1ⁿ [(Kh)(tⱼ ) - y(tⱼ )] ² + λ \| h \|², where H is a Hilbert space, and the data \ y(tⱼ ) \ satisfy y(tⱼ ) = g(tⱼ ) + ε (tⱼ ), the \ ε (tⱼ ) \ being measurement errors. fn,λ is the so-called regularized solution, and λ > 0 is the regularization parameter, to be chosen. It is important to choose λ correctly. The purpose of this paper is to propose the method of weighted cross-validation for choosing λfrom the data. We suppose that g is very smooth and the errors are white noise. It is shown that the weighted cross-validation estimate λ estimates the value of λ which minimizes (1 / n)E∑j = 1ⁿ [(Kfn,λ )(tⱼ ) - (Kf)(tⱼ )] ² . Results related to the convergence of \| f - fn, λ \|, including rates, are obtained.