Abstract
Kaczmarz’s method and variants thereof have been used effectively in the numerical resolution of linear algebraic equations arising from tomography and other areas of reconstruction from projections. The method is applied for example, to the system of equations arising from sampling and full discretization of the Radon transform. In contrast, Cimmino’s method which has universal convergence properties similar (in theory) to Kaczmarz’s method, does not seem to have as extensively studied or advocated in practice. In this paper, we develop continuous and/or semicontinuous analogues of the iterative methods of Cimmino and Kaczmarz for linear operator equations on infinite dimensional function spaces. The formulation of these algorithms hinges upon expressing the operator equation in terms of a family of hyperplanes in an appropriate function space. We identify and analyze two wide classes of linear operators for which this is possible. The semicontinuous analogue which involves a finite number of these hyperplanes is studied in particular in the framework of moment discretization (rather than full discretization) which is germane to problems of integral and operator equations with discrete data or sampling. Convergence properties of variants of Cimmino’s method are established; continuous and semicontinuous analogues are developed in a manner that permits convergence analysis in a similar manner. Several examples are given.
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Nashed, M.Z. (1981). Continuous and Semicontinuous Analogues of Iterative Methods of Cimmino and Kaczmarz with Applications to the Inverse Radon Transform. In: Herman, G.T., Natterer, F. (eds) Mathematical Aspects of Computerized Tomography. Lecture Notes in Medical Informatics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93157-4_14
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DOI: https://doi.org/10.1007/978-3-642-93157-4_14
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