Semiconvergence analysis of the randomized row iterative method and its extended variants

Abstract

The row iterative method is popular in solving the large-scale ill-posed problems due to its simplicity and efficiency. In this work we consider the randomized row iterative (RRI) method to tackle this issue. First, we present the semiconvergence analysis of RRI method for the overdetermined and inconsistent system, and derive upper bounds for the noise error propagation in the iteration vectors. To achieve a least squares solution, we then propose an extended version of the RRI (ERRI) method, which in fact can converge in expectation to the solution of the overdetermined or underdetermined, consistent or inconsistent systems. Finally, some numerical examples are given to demonstrate the convergence behaviors of the RRI and ERRI methods for these types of linear system.

On two-subspace randomized extended Kaczmarz method for solving large linear least-squares problems

“…This system is underdetermined and consistent with infinitely many solutions. It has been shown that the orthogonal projection of b onto N (A T ), equals b N , is one of the least-squares solutions with minimum Euclidean norm; see, e.g., [28]. In the following, we give a convergence result of the algorithm.…”

“…This fact tells us that RMR may not converge to the least-squares solution due to the existence of b N . To resolve this problem, inspired by the extended row method in [10,19,20,28], we provide the following extended variant of RMR (ERMR) and analyze its convergence and computational complexity.…”

“…The selection of A i,: equals to use of an independent random variable µ T i to multiply the left of A, where µ i denotes the ith standard unit coordinate vector with size m. Wu and Xiang generalized it into matrix form and developed a framework for the analysis and design of REK. Some new iteration schemes were also given, such as the Gaussian extended Kaczmarz (GEK) method [28], defined by…”

“…The convergence analysis reveals that the resulting algorithm has a linear (sometimes referred to as exponential) convergence rate, which is bounded by explicit expression. As with many iterative methods in the row family [3,9,10,22,28,29,31], our method relies on the information on the righthand side. The underlying idea is that first using an additional row iterate to output an approximation of b R and then utilizing another row iterate to an asymptotical consistent linear system.…”

On the relaxed greedy deterministic row and column iterative methods

“…Projection algorithm [16] is a kind of classic while effective iterative solver for computing an approximate solution for (1.1). As we know, many existing practical projection iterative algorithms are under the framework of Petrov-Galerkin conditions, such as the Kaczmarz and coordinate descent (CD) methods, see [8,16,18,19,20] and the references therein. Let the constrained subspace and the search subspace correspond to L = span{Y } and K = span{Z}, respectively, where Y and Z are two parameter matrices.…”