Estimates of the accuracy of numerical solutions using regularization

Abstract

This article explores the accuracy of numerical solutions, and suggests methods for analyzing accuracy depending on the properties of the problem. Numerical studies of complex expensive objects of technology and physics require that the computational results be obtained with guaranteed accuracy. It also depends on the fact that in the work of technical objects there are large intervals of operation time not observed experimentally. Therefore, there is a need to describe the location of the observed and calculated values, as well as the accuracy with which they are calculated. The effect of strong growth in estimates of error bounds is manifested for a large number of methods used to estimate the error of a numerical solution. This means the lack of correctness of algorithms for evaluating the accuracy of numerical solutions due to the failure of the stability conditions with respect to perturbations of the right-hand side. For many problems, among all the algorithms, the backward analysis of errors turned out to be the most effective method for assessing the accuracy of numerical solutions. The backward analysis of errors consists in the fact that when assessing the accuracy (error) of a numerical solution, the numerical solution is considered as an exact solution to a problem close to the original problem. A backward error analysis was proposed and developed in the algorithms of J Wilkinson in the context of the numerical solution of problems of linear algebra, and in the algorithms of V V Voevodin, who widely distributed it to many areas of numerical analysis. In the framework of the backward error analysis, the regularization of the algorithm for estimating the error of a numerical solution is reasonably applied. This article explores methods for the backward analysis of errors of numerical solutions.