About A. N. Tikhonov’s regularized least squares method

Abstract

The Tikhonov’s regularized least squares method (RLS) and its applications to problems of noisy data processing are considered. The starting point of this research is method for solving approximate systems of linear algebraic equations (SLAE), proposed by A. N. Tikhonov in 1980 that he later named ’RLS’. The traditional method of finding sustainable solutions of the problems with approximate data is Tikhonov’s regularization: unconditional minimization of the smoothing functional. RLS differs from this approach. This method makes use of the solution approach to the mathematical programming problem of a special kind. Nowadays RLS has not become a common tool for solving approximate linear systems, and the theory of RLS is not well researched. The purpose of the article is to remedy this deficiency. A number of important theoretical and practical aspects of RLS are considered. New results are presented. One of the results is the method of constructing a model SLAE with the exact right-hand side and approximate matrix for which a priori lower bounds for the maximum relative error of RLS are achieved. It is shown that, under certain conditions, RLS reduces to the problem of minimizing the smoothing functional or to the least squares method or — and this is an unexpected result — to the problem of finding a stationary point of smoothing functional with a negative regularization parameter. The algebraic transformation which reduces the conditionality of the RLS problem to a numerical solution is constructed and substantiated. We propose illustrative example of RLS through an applied image restoration problem for an image, registered by a device with an inexact point-spread function. The results of computational experiments are given. Refs 18. Figs 3.