Parallel minimum norm solution of sparse block diagonal column overlapped underdetermined systems
Date
2017Source Title
ACM Transactions on Mathematical Software
Print ISSN
0098-3500
Publisher
Association for Computing Machinery
Volume
43
Issue
4
Language
English
Type
ArticleItem Usage Stats
186
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Abstract
Underdetermined systems of equations in which the minimum norm solution needs to be computed arise in many applications, such as geophysics, signal processing, and biomedical engineering. In this article, we introduce a new parallel algorithm for obtaining the minimum 2-norm solution of an underdetermined system of equations. The proposed algorithm is based on the Balance scheme, which was originally developed for the parallel solution of banded linear systems. The proposed scheme assumes a generalized banded form where the coefficient matrix has column overlapped block structure in which the blocks could be dense or sparse. In this article, we implement the more general sparse case. The blocks can be handled independently by any existing sequential or parallel QR factorization library. A smaller reduced system is formed and solved before obtaining the minimum norm solution of the original system in parallel. We experimentally compare and confirm the error bound of the proposed method against the QR factorization based techniques by using true single-precision arithmetic. We implement the proposed algorithm by using the message passing paradigm. We demonstrate numerical effectiveness as well as parallel scalability of the proposed algorithm on both shared and distributed memory architectures for solving various types of problems. © 2017 ACM.
Keywords
Minimum norm solutionParallel algorithms
Underdetermined least square problems
Biomedical engineering
Factorization
Linear systems
Memory architecture
Message passing
Signal processing
Balance methods
Least square problems
Message passing paradigms
Numerical effectiveness
Parallel scalability
Shared and distributed memory architectures
Underdetermined systems
Least squares approximations