A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations
| Autor: | Fritz Keinert, Chinmay Hegde, Eric Weber |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
0209 industrial biotechnology
Iterative method Computer science Relaxation (iterative method) 010103 numerical & computational mathematics 02 engineering and technology System of linear equations 01 natural sciences Kaczmarz algorithm 020901 industrial engineering & automation Tree structure Norm (mathematics) Applied mathematics Tree based 0101 mathematics |
| Zdroj: | Applied and Numerical Harmonic Analysis ISBN: 9783030696368 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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