In this paper, we investigate the limiting behavior of the RK algorithm for solving doubly noisy inconsistent linear systems without imposing any additional initial assumptions. The proposed bounds effectively characterize the convergence behavior of these algorithms when applied to doubly noisy linear systems. Furthermore, to the best of our knowledge, this work provides the first convergence analysis of the randomized extended Kaczmarz (REK), randomized block Kaczmarz (RBK), and randomized double block Kaczmarz (RDBK) algorithms for doubly noisy linear systems. We prove that these algorithms converge to a neighborhood of the least-squares solution of the underlying noiseless system. These results show that RK and RBK, designed for consistent systems, outperform REK and RDBK, designed for inconsistent systems, in both the convergence rate and the convergence horizon. What's more, the analytical skills to the sketch-and-project method completely overcome the limitation when analyzing RK and RBK. Finally, numerical experiments are conducted to validate the theoretical results.
The classical method of Kaczmarz, introduced in 1937, is a textbook iterative method for solving linear systems $A x = b.$ Despite its widespread use, particularly in the context of solving inverse problems where it is commonly included in software packages within Matlab, Python, and Julia, the precise characterization...
A globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems that replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian.
In order to solve large sparse linear systems, we introduce a relaxation factor into the iterative formula based on the randomized Kaczmarz method. According to this idea, a parameterized randomized Kaczmarz method is constructed. Then, the convergence of this method is proved when the linear systems are consistent. Nu...
Chen-Xiao Gao, Fang Chen· Journal of Mathematical Stud...· 0 citations
Stochastic gradient methods have gained increasing attention for solving large-scale inverse problems due to their computational efficiency. However, their theoretical justification in the context of ill-posed problems remains underdeveloped, particularly regarding convergence rate analysis, where existing results...
Qi-Nian Jin, Xi-Liang Lu, Lin Tian· Numerische Mathematik· 0 citations
We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Pol...
Taeho Yoon, Sayantan Choudhury, Ezra Greenberg et al.· 1 citation
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