Abstract
In a recently developed approach to the optical inverse scattering problem, the need arose to solve very large systems of linear equations with a nonsparse matrix. The entries in the matrix are determined by the specifications for the data collection pattern. The matrix is not a Discrete Fourier Transform matrix, and it not anenable to FFT methods. In this paper we show how a combination of the techniques of “redundancies” and “Kronecker decompositions” may be used with a specification for the data collection to produce a fast, accurate algorithm which solves the linear system. This algorithm has been implemented on a sequential computer, but parallel computation is clearly feasible.
Zusammenfassung
In einer neueren Darstellung des optischen inversen Streuungsproblems tauchte das Problem auf, große lineare Gleichungssysteme mit vollbesetzten Matrizen zu lösen. Die Matriexelemente sind durch die Beschreibungen der Muster für die Datenerfassung festgelegt. Die Matrix beschreibt keine diskrete Fourier-Transformation und ist nicht auf FFT-Methoden anwendbar. In diesem Artikel zeigen wir, wie man eine Kombination der “redundancies”-Techniken und der Kronecker-Zerlegungen dazu benutzen kann, mit einer Beschreibung der Datenerfassung einen schnellen und genauen Algorithmus zur Lösung linearer Gleichungssysteme zu gewinnen. Dieser Algorithmus wurde auf einem sequentiellen Computer implementiert; parallele Berechnung ist natürlich auch durchführbar.
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Lam, D.K., Wouk, A. Redundancy techniques and fast algorithms for a special large linear system. Computing 18, 317–327 (1977). https://doi.org/10.1007/BF02244018
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DOI: https://doi.org/10.1007/BF02244018