A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations
This work addresses distributed computation challenges for systems of equations in tree-based networks, representing an incremental improvement over existing Kaczmarz methods.
The paper tackles the problem of solving systems of linear equations in a distributed network with a tree structure by introducing a modified Kaczmarz algorithm, proving convergence to the solution or minimal norm solution for consistent systems and to a weighted least squares solution for inconsistent systems as the relaxation parameter approaches zero.
The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e. the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least squares solution as the relaxation parameter approaches $0$.