Convergence acceleration of Kaczmarz's method
For researchers and practitioners using Kaczmarz's method, this work offers acceleration techniques that improve convergence speed, though the improvements are incremental.
This paper addresses the slow convergence of Kaczmarz's method for solving linear systems and proposes acceleration techniques based on sequence transformations. Numerical examples demonstrate the effectiveness of both direct and restarting acceleration procedures.
The method of alternation projections (MAP) is an iterative procedure for finding the projection of a point on the intersection of closed subspaces of an Hilbert space. The convergence of this method is usually slow, and several methods for its acceleration have already been proposed. In this work, we consider a special MAP, namely Kaczmarz' method for solving systems of linear equations. The convergence of this method is discussed. After giving its matrix formulation and its projection properties, we consider several procedures for accelerating its convergence. They are based on sequence transformations whose kernels contain sequences of the same form as the sequence of vectors generated by Kaczmarz' method. Acceleration can be achieved either directly, that is without modifying the sequence obtained by the method, or by restarting it from the vector obtained by acceleration. Numerical examples show the effectiveness of both procedures.