Stationary splitting iterative methods for the matrix equation AX B = C

Stationary splitting iterative methods for solving AXB = Care considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M −N by a matrix H such that (I −H) invertible. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications, our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer-aided geometric design (CAGD).The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, and the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

A sketch-and-project method for solving the matrix equation AXB = C

In this paper, based on an optimization problem, a sketch-and-project method for solving the linear matrix equation AXB = C is proposed. We provide a thorough convergence analysis for the new method and derive a lower bound on the convergence rate and some convergence conditions including the case that the coefficient matrix is rank deficient. By varying three parameters in the new method and convergence theorems, the new method recovers an array of well-known algorithms and their convergence results. Meanwhile, with the use of Gaussian sampling, we can obtain the Gaussian global randomized Kaczmarz (GaussGRK) method which shows some advantages in solving the matrix equation AXB = C. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods.Comment: arXiv admin note: text overlap with arXiv:1506.03296, arXiv:1612.06013, arXiv:2204.13920 by other author

Matrix equation solving of PDEs in polygonal domains using conformal mappings

We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz-Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation