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Remarks on numerical algorithms for computing the inverse Laplace transform. (English) Zbl 1302.65145

Summary: Algorithms for computing the inverse Laplace transform that consist essentially in choosing a series expansion for the original function are particularly effective in many cases and are widely used. The main purpose of this paper is to review these algorithms in the context of regularization. We relate this viewpoint to the design of reliable algorithms destined to be run on finite precision arithmetic systems.

MSC:

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65R30 Numerical methods for ill-posed problems for integral equations
65R32 Numerical methods for inverse problems for integral equations
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[1] Bellman, R., Kalaba, R., Lockett, J.A.: Numerical inversion of the Laplace transform: applications to biology, economics, engineering, and physics. Elsevier, Amsterdam (1966) · Zbl 0147.14003
[2] Bertero, M; Pike, ER, Exponential-sampling method for Laplace and other dilationally invariant transforms: I. singular-system analysis, Inverse Problems, 7, 21-41, (1991) · Zbl 0721.65086
[3] Cuomo, S; D’Amore, L; Murli, A; Rizzardi, M, Computation of the inverse Laplace transform based on a collocation method which uses only real values, J. Comput. Appl. Math., 198, 98-115, (2007) · Zbl 1105.65120
[4] Cuomo, S; D’Amore, L; Murli, A; Rizzardi, M; Bischof, CH (ed.); Bücker, HM (ed.); Hovland, PD (ed.); Naumann, U (ed.); Utke, J (ed.), A modification of weeks’ method for numerical inversion of the Laplace transform in the real case based on automatic differentiation, 45-54, (2008), Berlin · Zbl 1154.65379
[5] Cullum, J, Numerical differentiation and regularization, SIAM, J. Num. Anal., 8, 254-265, (1971) · Zbl 0224.65005
[6] Cohen, A.M.: Numerical methods for Laplace transform inversion. Springer, Berlin (2007) · Zbl 1127.65094
[7] D’Amore, L; Laccetti, G; Murli, A, An implementation of a Fourier series method for the numerical inversion of the Laplace transform, ACM Trans. Math. Softw., 25, 279-305, (1999) · Zbl 0962.65109
[8] D’Amore, L; Laccetti, G; Murli, A, Algorithm 796: a Fortran software package for the numerical inversion of the Laplace transform based on a Fourier series method, ACM Trans. Math. Softw., 25, 306-315, (1999) · Zbl 0962.65110
[9] D’Amore, L; Murli, A; Rizzardi, M, An extension of the henrici formula for Laplace transform inversion, Inverse Problems, 16, 1441-1456, (2000) · Zbl 0976.44002
[10] D’Amore, L; Murli, A, Regularization of a Fourier series based method for the Laplace transform inversion in the real case, Inverse Problems, 18, 1185-1205, (2002) · Zbl 1005.65139
[11] D’Amore, L; Campagna, R; Murli, A, An efficient algorithm for regularization of Laplace transform inversion in real case, J. Comput. Appl. Math., 210, 84-98, (2007) · Zbl 1144.65078
[12] D’Amore, L; Campagna, R; Galletti, A; Marcellino, L; Murli, A, A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis, Inverse Problems, 28, 025007, (2012) · Zbl 1257.65074
[13] D’Amore, L; Campagna, R; Mele, V; Murli, A, Relative. an ansi C90 software package for the real Laplace transform inversion, Num. Algor., 63, 187-211, (2013) · Zbl 1267.65202
[14] D’Amore, L; Mele, V; Murli, A, Performance analysis of the Taylor expansion coefficients computation as implemented by the software package TADIFF, J. Num. Anal. Indus. Appl. Math. (JNAIAM), 8, 1-12, (2013) · Zbl 1432.65004
[15] D’Amore, L., Campagna, R., Mele, V., Murli A. Reliadiff.: A C++ software package for Laplace transform Inversion, ACM Transaction on Mathematical Software, (in press).
[16] Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer Academic Publisher, Netherlands (2000) · Zbl 0859.65054
[17] Davies, B; Martin, D, Numerical inversion of Laplace transform. A survey and comparison of methods, J. Comp. Phys., 33, 1-32, (1979) · Zbl 0416.65077
[18] Demmel, J, The probability that a numerical analysis problem is difficult, Math. Comp., 50, 449-480, (1998) · Zbl 0657.65066
[19] Garbow, S; Giunta, G; Lyness, NJ; Murli, A, Algorithm 662: a Fortran software package for the numerical inversion of a Laplace transform based on week’s method, ACM Trans. Math. Softw., 54, 163-170, (1988) · Zbl 0642.65086
[20] Giunta, G; Murli, A; Brezinski, C (ed.), An algorithm for inverting the Laplace transform using real and real sampled function values, 589-592, (1989), Missouri
[21] Groetsch, C.W.: The theory of tikhonov regularization for fredholm equations of the first kind. Research notes in mathematics. Pitman Advanced publishing Program, Edinburgh (1990)
[22] Hanke, M; Scherzer, O, Inverse problems light: numerical differentiation, Amer. Math. Month., 108, 512-521, (2001) · Zbl 1002.65029
[23] Hansen, P.C.: Rank-deficient and discrete ill posed problems. SIAM, New Delhi (1998) · Zbl 0890.65037
[24] Higham, N.: Accuracy and stability of numerical algorithms. SIAM, New Delhi (1996) · Zbl 0847.65010
[25] Kryzhniy, VV, Direct regularization of the inversion of real-valued Laplace transforms, Inverse Problems, 19, 573-583, (2003) · Zbl 1024.65125
[26] Mcwirther, JC; Pike, ER, On the numerical inversion of Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A, 11, 1729-1745, (1978) · Zbl 0404.65060
[27] Murli, A; Cuomo, S; D’Amore, L; Galletti, A, Numerical regularization of a real inversion formula based on the Laplace transform’s eigenfunction expansion of the inverse function, Inverse Problems, 23, 713-731, (2007) · Zbl 1122.65128
[28] Murli, A; Rizzardi, M, Algorithm 682: talbot’s method of the Laplace inversion problems, ACM Trans. Math. Softw. (TOMS), 16, 158-168, (1990) · Zbl 0900.65374
[29] Piessens, R, A new numerical method for the inversion of the Laplace tranform, J. Inst. Math. Appl., 10, 185-192, (1972) · Zbl 0246.65035
[30] Piessens, R, Algorithm 113: inversion of the Laplace transform, Algor. Suppl. Comp. J., 25, 278-282, (1982) · Zbl 0477.65089
[31] Spinelli, RA, Numerical inversion of a Laplace transform, SIAM J. Numer. Anal., 3, n.4, (1966) · Zbl 0166.13302
[32] Trefethen, N.: The definition of numerical analysis. SIAM News, Bangkok (1992) · Zbl 0798.15005
[33] Weidemann, J, Algorithms for parameter selection in the weeks method for inverting the Laplace transform, SIAM J. Sci. Comput., 21, 118-128, (1999) · Zbl 0944.65137
[34] Wilkinson, J.H.: Rounding errors in algebraic processes. Dover Publications, New York (1994) · Zbl 0868.65027
[35] Tikhonov, A.N., Arsenine, A.N.: Methodes de resolution de problemes mal poses. Editions MIR, Moscov (1974)
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