@inbook{8887d49894364a2c94b4545e49f91efb,

title = "A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations",

abstract = "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.",

author = "Chinmay Hegde and Fritz Keinert and Weber, {Eric S.}",

note = "Funding Information: Acknowledgments This research was supported by the National Science Foundation and the National Geospatial-Intelligence Agency under awards DMS-1830254 and CCF-1750920. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

doi = "10.1007/978-3-030-69637-5_20",

language = "English (US)",

series = "Applied and Numerical Harmonic Analysis",

publisher = "Birkhauser",

pages = "385--411",

booktitle = "Applied and Numerical Harmonic Analysis",

}