Efficient signed backward substitution for piecewise affine functions via path problems in a directed acyclic graph.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2021 SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)Efficient signed backward substitution for piecewise affine functions via path problems in a directed acyclic graphTorsten Bosse, Ralf Seidler, and H. Martin BückerTorsten Bosse, Ralf Seidler, and H. Martin Bückerpp.171 - 181Chapter DOI:https://doi.org/10.1137/1.9781611976830.16PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We introduce an efficient signed backward substitution for a highly-structured system. More precisely, the problem is to find a vector u of dimension s that solves the system of piecewise affine equations u = c + L|u|, where L is a strictly lower left triangular s × s matrix, c denotes a given vector of dimension s, and the notation | · | indicates the component-wise absolute value of a vector. The novel approach is based on a Neumann series reformulation and attempts to exploit a high degree of parallelism. We provide an analysis of its parallel run-time and show that it is suited for large, sparse systems whose triangular matrix has a small switching depth. The general idea behind this approach which is also used in the convergence proof is based on modelling the switching depth by a graph theoretic model. The key observation is that the computation of the switching depth corresponds to a single-pair shortest path problem in a directed acyclic graph. The proposed method is implemented and numerically evaluated using several examples whose problem structures are representative of various applications in scientific computing. Previous chapter Next chapter RelatedDetails Published:2021eISBN:978-1-61197-683-0 https://doi.org/10.1137/1.9781611976830Book Series Name:ProceedingsBook Code:PRACDA21Book Pages:1-239