Least Squares Method for Solving a System of Linear Equations Based on Multilevel Wavelet Decomposition of the Residual

Abstract

The solution of systems of linear equations has a large number of practical applications related to solving ill-conditioned problems. In paper, the technique of solving systems of linear equations based on the least squares method is considered. It is shown that the problem of the least squares method can have an alternative formulation. It consists in formulating the problem for elements of a multilevel wavelet decomposition of the residual vector. The proposed approach is demonstrated by the example of a linear quadratic problem. Experiments have shown that the wavelet decomposition of the residual can significantly improve the accuracy of solving the system of equations, making it comparable to the accuracy obtained by applying projection methods. The number of levels of the wavelet decomposition is determined by the structural parameters of the matrix. The quality of the solution can also depend on the type of wavelet used in the transformation of the residual.