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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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 210; no. 1; pp. 84 - 98
Main Authors: Campagna, R., D’Amore, L., Murli, A.
Format: Journal Article Conference Proceeding
Language:English
Published: Amsterdam Elsevier B.V 31.12.2007
Elsevier
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2006.10.077