The row-oriented form of the regularized Kaczmarz's method

• Main Authors: Alexander I. Zhdanov, Yury V. Sidorov
• Format: Article
• Language: English
• Published: Samara State Technical University 2017-11-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Subjects: iterative methods; projection algorithms; Tikhonov's regularization; Kaczmarz algorithm; row-oriented form of the regularized Kaczmarz's algorithm
• Online Access: http://mi.mathnet.ru/eng/vsgtu1548

This paper presents the new iterative method for solving the standard Tikhonov regularization problem. The basis of the method is the application the projection Kaczmarz algorithm to the augmented regularized normal system of equations. The use of the augmented regularized normal system of equations, instead the system of regularized normal equations, makes it possible to significantly reduce the spectral condition number of the original problem. The paper presents the row-oriented form of the regularized Kaczmarz algorithm. This form of the regularized Kaczmarz algorithm allows to solve problems in which the data are received sequentially (line by line). The proposed algorithm makes it possible to effectively calculate solutions of problems with sparse matrices of large and superlarge dimensions. The comparison's results of the proposed row-oriented form of the algorithm with the column-oriented form of this algorithm are presented. By considering a certain classes of problems, the paper demonstrates that the proposed form of the regularized algorithm allows to reduce the number of iterations in comparison with the column-oriented form of the algorithm.