Research On Randomized Iterative Algorithm With Mismatched Adjoint

Numerical linear algebra(NLA)is a major achievement in the field of scientific computing,and it is widely used in various scientific fields.Randomized iterative al-gorithms plays an important role in NLA,and it has made outstanding contributions in the fields of solving large-scale linear equations,numerical optimization,machine learning,and artificial intelligence.The well-known algorithms are:Stochastic Gradi-ent Descent(SGD),Random Coordinate Descent(RCD),Randomized Kaczmarz(RK)algorithm and so on.Kaczmarz algorithm is an iterative algorithm for solving linear equations.Because of its simplicity,it is widely used in image reconstruction,distributed computing,and signal processing.Compared with the classic Kaczmarz algorithm,the RK algorithm has a complete theoretical proof and excellent numerical performance.At the same time,the randomized version algorithm has more room for design and improvement.This paper is based on the previous research on RK algorithm,and further research and generalize the algorithm.The main work of this article is as follows:Fisrstly,based on the mismatch adjoint projection algorithm,this paper generalizes the randomized iteration algorithm framework to obtain a more generalized algorithm framework.The framework introduces the idea of a mismatched adjoint projection algorithm(RKMA),using VT as a"backprojector".At the same time,the convergence theorem of the framework is given and proved.Finally,some classic random iterations of the algorithm are restored by designing different parameter matrices.Secondly,in this paper,a homogenized variant of the block-RK algorithm(RBKv)is applied to the block-algorithm(RBKMA)in the mismatched adjoint framework.And proved the convergence of the algorithm in the compatibility system and the noise sys-tem,respectively,from a theoretical point of view.Finally,the validity of the algorithm was verified using numerical experiments...