Abstract
The ART algorithm, an iterative technique for solving large systems of linear equations, is shown to converge even for inconsistent systems, provided the relaxation parameters are chosen appropriately. The limit is a weighted least squares solution.
Zusammenfassung
Die Konvergenz des ART-Algorithmus, ein iteratives Verfahren zur Lösung linearer Gleichungssysteme, wird bewiesen. Bei geeigneter Wahl der Relaxationsparameter konvergiert der Algorithmus selbst im Falle inkonsistenter Systeme, und zwar gegen eine Kleinste-Quadrate-Lösung.
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References
Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Review23, 444–446 (1981).
Censor, Y., Eggermont, P. P. B., Gordon, D.: Strong underrelaxation in Kaczmarz's method for inconsistent systems. Numer. Math.41, 83–92 (1983).
Gordon, R., Bender, R., Herman, G. T.: Algebraic reconstruction techniques (ART) for threedimensional electron microscopy and X-ray photography. J. theor. Biol.29, 471–481 (1970).
Groetsch, C. W.: Generalized Inverses of Linear Operators. Representation and Approximation. New York: M. Dekker 1977.
Kaczmarz, S.: Angenäherte Auflösung von Systemen linearer Gleichungen. Bull. Acad. Polon. Sci. Lett.A 35, 355–357 (1937).
Marti, J. T.: On the convergence of the discrete ART algorithm for the reconstruction of digital pictures from their projections. Computing21, 105–111 (1979).
Trummer, M. R.: Rekonstruktion von Bildern aus ihren Projektionen. Diplomarbeit, Seminar für Angewandte Mathematik, ETH Zürich (1979).
Trummer, M. R.: Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation. Computing26, 189–195 (1981).
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Trummer, M.R. A note on the ART of relaxation. Computing 33, 349–352 (1984). https://doi.org/10.1007/BF02242277
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DOI: https://doi.org/10.1007/BF02242277