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Regularization Processes for Real Functions and Ill-posed Toeplitz Problems

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Recent Advances in Operator Theory and its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 160))

Abstract

Most preconditioners for Toeplitz systems An(f) arising in the discretization of ill-posed problems give rise to instability and noise amplification. Indeed, since these preconditioners are constructed from linear approximation processes of the generating function f, they inherit the ill-posedness of the problem.

Here we first identify a novel set of approximation processes which regularizes the inversion of real functions. Then, such processes are used as a basic tool for the computation of preconditioners endowed with regularizing properties. We show that these preconditioners provide fast convergence and noise control of iterative methods for discrete ill-posed Toeplitz systems.

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References

  1. O. Axelsson and G. Lindskög, The rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 52, 1986, pp. 499–523.

    Google Scholar 

  2. D. Bertaccini, Reliable preconditioned iterative linear solvers for some integrators, Numerical Linear Algebra Appl., 8, 2001, pp. 111–125.

    MathSciNet  MATH  Google Scholar 

  3. D. Bertaccini, The spectrum of circulant-like preconditioners for some general linear multistep formulas for linear boundary value problems, SIAM J. Numer. Anal., 40, 2002, pp. 1798–1822.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Bertaccini and M.K. Ng, Block {ω }-circulant preconditioners for the systems of differential equations, Calcolo, 40, 2003, pp. 71–90.

    MathSciNet  Google Scholar 

  5. M. Bertero and P. Boccacci, Introduction to Inverse Problem in Imaging, Institute of Physics Publishing, London, 1998.

    Google Scholar 

  6. C. Biamino, Applicazioni del metodo di Landweber per la ricostruzione di immagini, Graduate Thesis in Mathematics, Dipartimento di Matematica, Università di Genova, 2003.

    Google Scholar 

  7. D.A. Bini and F. Di Benedetto, A new preconditioner for the parallel solution of positive definite Toeplitz systems, in Proc. 2nd SPAA, Crete, Greece, July 1990, ACM Press, New York, pp. 220–223.

    Google Scholar 

  8. D.A. Bini and P. Favati, On a matrix algebra related to the discrete Hartley transform, SIAM J. Matrix Anal. Appl., 14, 1993, pp. 500–507.

    Article  MathSciNet  Google Scholar 

  9. D.A. Bini, P. Favati and O. Menchi, A family of modified regularizing circulant preconditioners for image reconstruction problems, Computers & Mathematics with Applications, 48, 2004, pp. 755–768.

    Article  MathSciNet  Google Scholar 

  10. R.H. Chan and M.K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38, 1996, pp. 427–482.

    Article  MathSciNet  Google Scholar 

  11. R.H. Chan and M. Yeung, Circulant preconditioners constructed from kernels, SIAM J. Numer. Anal., 30, 1993, pp. 1193–1207.

    MathSciNet  Google Scholar 

  12. T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comp., 9, 1988, pp. 766–771.

    MATH  Google Scholar 

  13. P. Davis, Circulant matrices, John Wiley & Sons, 1979.

    Google Scholar 

  14. F. Di Benedetto and S. Serra Capizzano, A unifying approach to abstract matrix algebra preconditioning, Numer. Math., 82-1, 1999, pp. 57–90.

    Google Scholar 

  15. H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

    Google Scholar 

  16. C. Estatico, A class of filtering superoptimal preconditioners for highly ill-conditioned linear systems, BIT, 42, 2002, pp. 753–778.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Estatico, Classes of regularization preconditioners for image processing, Proc. SPIE, 2003 — Advanced Signal Processing: Algorithms, Architectures, and Implementations XIII, Ed. F.T. Luk, Vol. 5205, pagg. 336–347, 2003, Bellingham, WA, USA.

    Google Scholar 

  18. C. Estatico, ”A classification scheme for regularizing preconditioners, with application to Toeplitz systems“, Linear Algebra Appl., 397, 2005, pp. 107–131.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Estatico, Regularized fast deblurring for the Large Binocular Telescope, Tech. Rep. N. 490, Dipartimento di Matematica, Università di Genova, 2003

    Google Scholar 

  20. R. Gray, Toeplitz and Circulant matrices: a review, http: www-isl.stanford.edu/~gray/toeplitz.pdf, 2000.

    Google Scholar 

  21. U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, Second edition, Chelsea, New York, 1984.

    Google Scholar 

  22. C.W. Groetsch, Generalized inverses of linear operators: representation and approximation, Pure and Applied Mathematics, 37, Marcel Dekker, New York, 1977.

    Google Scholar 

  23. M. Hanke, J.G. Nagy and R. Plemmons, Preconditioned iterative regularization for ill-posed problems, Numerical Linear Algebra and Scientific Computing, L. Reichel, A. Ruttan and R.S. Varga, eds., Berlin, de Gruyter, 1993, pp. 141–163.

    Google Scholar 

  24. J. Kamm and J.G. Nagy, Kronecker product and SVD approximations in image restoration, Linear Algebra Appl., 284, 1998, pp. 177–192.

    Article  MathSciNet  Google Scholar 

  25. M. Kilmer, Cauchy-like Preconditioners for Two-Dimensional Ill-Posed Problems, SIAM J. Matrix Anal. Appl., 20,No. 3, 1999, pp. 777–799.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Kilmer and D. O’Learly, Pivoted Cauchy-like preconditioners for regularized solution of ill-posed problems, SIAM J. Sci. Comp., 21, 1999, pp. 88–110.

    Article  Google Scholar 

  27. J.G. Nagy, M.K. Ng and L. Perrone, Kronecker product approximation for image restoration with reflexive boundary conditions, SIAM J. Matrix Anal. Appl., 25, 2004, pp. 829–841.

    Article  MathSciNet  Google Scholar 

  28. S. Serra Capizzano, Toeplitz preconditioners constructed from linear approximation processes, SIAM J. Matrix Anal. Appl., 20-2, 1998, pp. 446–465.

    MathSciNet  Google Scholar 

  29. G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74, 1986, pp. 171–176.

    MATH  Google Scholar 

  30. E. Tyrtyshnikov, A unifying approach to some old and new theorems in distribution and clustering, Linear Algebra Appl., 232, 1996, pp. 1–43.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Tyrtyshnikov and N. Zamarashkin, Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationship, Linear Algebra Appl., 270, 1997, pp. 15–27.

    MathSciNet  Google Scholar 

  32. N. Trefethen, Approximation theory and numerical linear algebra, J. Mason and M. Cox, eds., Chapman and Hall, London, 1990, pp. 336–360.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146, Genova, Italy

    Claudio Estatico

Authors
  1. Claudio Estatico
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Israel

    Israel Gohberg

  2. Beer-Sheva

    D. Alpay  & P. A. Fuhrmann  & 

  3. Haifa

    J. Arazy  & L. E. Lerer  & 

  4. Tel Aviv

    A. Atzmon  & A. Ben-Artzi  & 

  5. Blacksburg

    J. A. Ball

  6. Bloomington

    H. Bercovici

  7. Chemnitz

    A. Böttcher  & G. Heinig  & 

  8. Athens, USA

    K. Clancey

  9. Buffalo

    L. A. Coburn

  10. Waterloo, Ontario

    K. R. Davidson

  11. College Station

    R. G. Douglas

  12. Groningen

    A. Dijksma

  13. Rehovot

    H. Dym

  14. Mainz

    B. Gramsch

  15. La Jolla

    J. A. Helton

  16. Amsterdam

    M. A. Kaashoek

  17. Argonne

    H. G. Kaper

  18. Tokyo

    S. T. Kuroda

  19. Calgary

    P. Lancaster

  20. Columbus

    B. Mityagin

  21. Manhattan, Kansas

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  22. Williamsburg

    L. Rodman  & I. M. Spitkovsky  & 

  23. Charlottesville

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  24. Berkeley

    D. E. Sarason  & D. Voiculescu  & 

  25. Providence

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  26. Marburg

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  27. Leiden

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  28. Santa Cruz

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  29. Nashville

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  30. Rennes

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  31. Department of Mathematics, FEW Vrije Universiteit, De Boelelaan 1081A, 1081 HV, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  32. Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    Sebastiano Seatzu  & Cornelis van der Mee  & 

Additional information

This work is dedicated to Prof. Israel Gohberg, on the occasion of his 75th birthday.

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Estatico, C. (2005). Regularization Processes for Real Functions and Ill-posed Toeplitz Problems. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_7

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