The Technology of the Stable Solution for Discrete Ill-posed Problems by Modified Random Projection Method

Abstract

Introduction.Ill-posed problems solution is actual for many areas of science and technology. For example, discrete ill-posed problems (DIP) appears after discretization of the integral equations in the spectrometry, gravimetry, magnitometry, electrical prospecting and others.In the case of linear DIP the matrix, which model some measuring system, makes a linear transformation of input vector to the output vector. Usually DIP output vector contains noise and singular values series of the matrix smoothly decrease to zero. In this case, the solution (input vector estimation) using the inversion of the transformation matrix is unstable and inaccurate. To overcome instability and increase accuracy we use regularization methods.We develop an approach which uses regularizing properties of random projection to obtain a stable solution of DIP. However, the development of effective sustainable methods for solving DIP continues to be a problem of current interest.The purpose of the paper is to increase the accuracy of DIP solution by the random projection method.Results. In this paper we developed the method of stable solution of DIP by the modified method of random projection. For this modification the regularization by random projection is complemented by the regularization in the ridge regression style.For the our method we obtained expressions which connect in the direct way the solution error components with the matrix specter and the regularization parameter. For the developed method the experimental research of the accuracy is conducted on the test problems.