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 | References | Similar Articles | Additional Information

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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  • 1. A. Bouhamidi and K. Jbilou, Sylvester Tikhonov-regularization methods in image restoration, J. Comput. Appl. Math. 206 (2007), 86-98. MR 2333837
  • 2. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. MR 1408680 (97k:65145)
  • 3. J. N. Franklin, Well-posed stochatic extensions of ill-posed problems, J. Math. Anal. Appl. 31 (1970), 682-716. MR 0267654 (42:2556)
  • 4. Z. Gajic and M. T. J. Quereshi, Lyapunov Matrix Equation in System Stability and Control, Academic Press, San Diego, 1995. MR 1343974 (96g:93001)
  • 5. M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT 36 (1996), 287-301. MR 1432249 (97j:65098)
  • 6. P. C. Hansen, The discrete Picard condition for discrete ill-posed problems, BIT 30 (1990), 658-672. MR 1082808 (91m:65119)
  • 7. P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review 34 (1992), 561-580. MR 1193012 (93k:65035)
  • 8. E. T. Jaynes, Probability Theory, Cambridge University Press, Cambridge, United Kingdom, 2003. MR 1992316 (2004g:62006)
  • 9. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 1984. MR 836136 (87d:60001)
  • 10. A. Tarantola, Inverse Problem Theory, SIAM, Philadelphia, 2005. MR 2130010 (2007b:62011)
  • 11. C. F. van Loan, Generalizing the singular value decomposition, SIAM J. Numer. Anal. 11 (1976), 76-83. MR 0411152 (53:14891)
  • 12. J. M. Varah, Pitfalls in the numerical solution of linear ill-posed problems, SIAM J. Sci. Stat. Comput. 4 (1983), 164-176. MR 697171 (84g:65052)
  • 13. C. Vogel, Non-convergence of the L-curve regularization parameter selection method, Inverse Problems 12, (1996) 535-547. MR 1402108 (97k:65149)
  • 14. G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numer. Anal. 14 (1977), 651-667. MR 0471299 (57:11036)
  • 15. S. C. Zhu, Y. N. Wu, and D. Mumford, Minimax entropy principle and its application to texture modeling, Neural Computation 9 (1997), 1627-1660.

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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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