An efficient algorithm for regularization of Laplace transform inversion in real case

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The m...

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Veröffentlicht in:	Journal of computational and applied mathematics Jg. 210; H. 1; S. 84 - 98
Hauptverfasser:	Campagna, R., D’Amore, L., Murli, A.
Format:	Journal Article Tagungsbericht
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier B.V 31.12.2007 Elsevier
Schlagworte:	Algebra Algebraic geometry Automatic stopping rule Combinatorics Combinatorics. Ordered structures Designs and configurations Exact sciences and technology Fourier analysis GMRES Ill posed problem Laplace transform inversion Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Regularization Sciences and techniques of general use 65R32 65F10 65F22 GMRES Ill posed problem Regularization Automatic stopping rule Laplace transform inversion Numerical linear algebra Singular value Eigenvalue Fourier analysis Iteration Fourier series Algorithm Inverse problem Direct method Regularization method 65F22; 65F10; 65R32 Numerical analysis Linear system Numerical computation Laplace transform inversion; III posed problem; Regularization; GMRES; Automatic stopping rule Matrix inversion Applied mathematics Condition number Laplace transformation Preconditioning
ISSN:	0377-0427, 1879-1778
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Abstract	We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.
AbstractList	We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D'Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185-1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix. We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.
Author	Murli, A. D’Amore, L. Campagna, R.
Author_xml	– sequence: 1 givenname: R. surname: Campagna fullname: Campagna, R. email: rosanna.campagna@dma.unina.it – sequence: 2 givenname: L. surname: D’Amore fullname: D’Amore, L. email: luisa.damore@dma.unina.it – sequence: 3 givenname: A. surname: Murli fullname: Murli, A. email: almerico.murli@dma.unina.it
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Cites_doi	10.1007/s002110100339 10.1137/S106482759630526X 10.1088/0266-5611/18/4/315 10.1007/BF02149761 10.1016/S1570-579X(01)80025-2
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Keywords	65R32 65F10 65F22 GMRES Ill posed problem Regularization Automatic stopping rule Laplace transform inversion Numerical linear algebra Singular value Eigenvalue Fourier analysis Iteration Fourier series Algorithm Inverse problem Direct method Regularization method 65F22; 65F10; 65R32 Numerical analysis Linear system Numerical computation Laplace transform inversion; III posed problem; Regularization; GMRES; Automatic stopping rule Matrix inversion Applied mathematics Condition number Laplace transformation Preconditioning
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References	D’Amore, Murli (bib4) 2002; 18 P. Hansen, Regularization Tools. A Matlab package for analysis and solution of discrete ill-posed problems, Manual and Tutorial, Technical Report, Department of Mathematical Modelling, Technical University of Denmark, 1998, Numer. Algorithms 6 (1994) 1–35. Calvetti, Lewis, Reichel (bib2) 2002; 91 T. Chan, E. Chow, Y. Saad, M.C. Yeung, Preserving symmetry in preconditioned Krylov subspace methods, Technical Report, 1996. Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., 2000. Golub, van Loan (bib5) 1996 Banoczi, Chiu, Cho, Ipsen (bib1) 1999; 20 10.1016/j.cam.2006.10.077_bib6 10.1016/j.cam.2006.10.077_bib7 10.1016/j.cam.2006.10.077_bib3 Golub (10.1016/j.cam.2006.10.077_bib5) 1996 Calvetti (10.1016/j.cam.2006.10.077_bib2) 2002; 91 D’Amore (10.1016/j.cam.2006.10.077_bib4) 2002; 18 Banoczi (10.1016/j.cam.2006.10.077_bib1) 1999; 20
References_xml	– volume: 18 start-page: 1185 year: 2002 end-page: 1205 ident: bib4 article-title: Regularization of a Fourier series method for the Laplace transform inversion with real data publication-title: Inverse Problems – reference: T. Chan, E. Chow, Y. Saad, M.C. Yeung, Preserving symmetry in preconditioned Krylov subspace methods, Technical Report, 1996. – volume: 91 start-page: 605 year: 2002 end-page: 625 ident: bib2 article-title: On the regularizing properties of the GMRES method publication-title: Numer. Math. – reference: Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., 2000. – volume: 20 start-page: 203 year: 1999 end-page: 227 ident: bib1 article-title: The lack of influence of the right-hand side on the accuracy of linear system solution publication-title: SIAM J. Sci. Comput. – reference: P. Hansen, Regularization Tools. A Matlab package for analysis and solution of discrete ill-posed problems, Manual and Tutorial, Technical Report, Department of Mathematical Modelling, Technical University of Denmark, 1998, Numer. Algorithms 6 (1994) 1–35. – year: 1996 ident: bib5 article-title: Matrix Computations – volume: 91 start-page: 605 year: 2002 ident: 10.1016/j.cam.2006.10.077_bib2 article-title: On the regularizing properties of the GMRES method publication-title: Numer. Math. doi: 10.1007/s002110100339 – ident: 10.1016/j.cam.2006.10.077_bib3 – year: 1996 ident: 10.1016/j.cam.2006.10.077_bib5 – volume: 20 start-page: 203 issue: 1 year: 1999 ident: 10.1016/j.cam.2006.10.077_bib1 article-title: The lack of influence of the right-hand side on the accuracy of linear system solution publication-title: SIAM J. Sci. Comput. doi: 10.1137/S106482759630526X – volume: 18 start-page: 1185 year: 2002 ident: 10.1016/j.cam.2006.10.077_bib4 article-title: Regularization of a Fourier series method for the Laplace transform inversion with real data publication-title: Inverse Problems doi: 10.1088/0266-5611/18/4/315 – ident: 10.1016/j.cam.2006.10.077_bib6 doi: 10.1007/BF02149761 – ident: 10.1016/j.cam.2006.10.077_bib7 doi: 10.1016/S1570-579X(01)80025-2
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SubjectTerms	Algebra Algebraic geometry Automatic stopping rule Combinatorics Combinatorics. Ordered structures Designs and configurations Exact sciences and technology Fourier analysis GMRES Ill posed problem Laplace transform inversion Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Regularization Sciences and techniques of general use
Title	An efficient algorithm for regularization of Laplace transform inversion in real case
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