Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Minimax entropy solutions of ill-posed problems

Author: Fred Greensite
Journal: Quart. Appl. Math. 67 (2009), 137-161
MSC (2000): Primary 47A52, 45Q05; Secondary 65J20, 34A55
DOI: https://doi.org/10.1090/S0033-569X-09-01120-7
Published electronically: January 8, 2009
MathSciNet review: 2497601
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Abstract: Convergent methodology for ill-posed problems is typically equivalent to application of an operator dependent on a single parameter derived from the noise level and the data (a regularization parameter or terminal iteration number). In the context of a given problem discretized for purposes of numerical analysis, these methods can be viewed as resulting from imposed prior constraints bearing the same amount of information content. We identify a new convergent method for the treatment of certain multivariate ill-posed problems, which imposes constraints of a much lower information content (i.e., having much lower bias), based on the operator's dependence on many data-derived parameters. The associated marked performance improvements that are possible are illustrated with solution estimates for a Lyapunov equation structured by an ill-conditioned matrix. The methodology can be understood in terms of a Minimax Entropy Principle, which emerges from the Maximum Entropy Principle in some multivariate settings.

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Additional Information

Fred Greensite
Affiliation: Department of Radiological Sciences, University of California, Orange, California 92868
Email: fred.greensite@uci.edu

DOI: https://doi.org/10.1090/S0033-569X-09-01120-7
Keywords: Inverse problems, ill-posed problems, Sylvester equation
Received by editor(s): August 7, 2007
Published electronically: January 8, 2009
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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