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Lanczos-Based Exponential Filtering for Discrete Ill-Posed Problems

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Abstract

We describe regularizing iterative methods for the solution of large ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The regularization is specified by a filter function of Gaussian type. A parameter μ determines the amount of regularization applied. The iterative methods are based on a truncated Lanczos decomposition and the filter function is approximated by a linear combination of Lanczos polynomials. A suitable value of the regularization parameter is determined by an L-curve criterion. Computed examples that illustrate the performance of the methods are presented.

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References

  1. D. Calvetti, B. Lewis and L. Reichel, Smooth or abrupt: A comparison of regularization methods, in: Advanced Signal Processing Algorithms, Architectures and Implementations VIII, ed. F.T. Luk, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), Vol. 3461 (The International Society for Optical Engineering, Bellingham, WA, 1998) pp. 286–295.

    Google Scholar 

  2. D. Calvetti, B. Lewis and L. Reichel, An L-curve for the MINRES method, in: Advanced Signal Processing Algorithms, Architectures, and Implementations X, ed. F.T. Luk, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), Vol. 4116 (The International Society for Optical Engineering, Bellingham, WA, 2000) pp. 385–395.

    Google Scholar 

  3. D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for ill-posed problems, in: Advanced Signal Processing Algorithms, ed. F.T. Luk, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), Vol. 2563 (The International Society for Optical Engineering, Bellingham, WA, 1995) pp. 338–347.

    Google Scholar 

  4. D. Calvetti, L. Reichel and Q. Zhang, Iterative exponential filtering for large discrete ill-posed problems, Numer. Math. 83 (1999) 535–556.

    Google Scholar 

  5. D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for large linear discrete ill-posed problems, Applied and Computational Control, Signals and Circuits 1 (1999) 313–367.

    Google Scholar 

  6. V.L. Druskin and L. Knizhnerman, Two polynomial methods of calculating functions of a symmetric matrix, USSR Comput. Math. Math. Phys. 29(6) (1989) 112–121.

    Google Scholar 

  7. G.H. Golub and G. Meurant, Matrices, moments and quadrature, in: Numerical Analysis 1993, eds. D.F. Griffins and G.A. Watson (Longman, Essex, UK, 1994) pp. 105–156.

    Google Scholar 

  8. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins Univ. Press, Baltimore, 1996).

    Google Scholar 

  9. M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems (Longman, Harlow, 1995).

    Google Scholar 

  10. M. Hanke and P.C. Hansen, Regularization methods for large-scale problems, Surv. Math. Ind. 3 (1993) 253–315.

    Google Scholar 

  11. M. Hanke and J.G. Nagy, Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques, Inverse Problems 12 (1996) 157–173.

    Google Scholar 

  12. P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580.

    Google Scholar 

  13. P.C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms 6 (1994) 1–35.

    Google Scholar 

  14. P.C. Hansen, Rank Deficient and Discrete Ill-Posed Problems (SIAM, Philadelphia, 1998).

    Google Scholar 

  15. P.C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in: Advances in Biomedicine, Vol. 4, ed. P. Johnston (WIT Press, Southampton, UK, 2000) pp. 119–142.

    Google Scholar 

  16. A.K. Louis, A unified approach to regularization methods for linear ill-posed problems, Inverse Problems 15 (1999) 489–498.

    Google Scholar 

  17. D.L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9 (1962) 84–97.

    Google Scholar 

  18. C.B. Shaw, Jr., Improvements of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl. 37 (1972) 83–112.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

    D. Calvetti

  2. Department of Mathematics and Computer Science, Kent State University, Kent, OH, 44242, USA

    L. Reichel

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  1. D. Calvetti
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  2. L. Reichel
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Calvetti, D., Reichel, L. Lanczos-Based Exponential Filtering for Discrete Ill-Posed Problems. Numerical Algorithms 29, 45–65 (2002). https://doi.org/10.1023/A:1014899604567

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  • DOI: https://doi.org/10.1023/A:1014899604567