Quality assurance of Gaver’s formula for multi-precision Laplace transform inversion in real case. (English) Zbl 1398.65361

Summary: We are concerned with Gaver’s formula, which is at the heart of a numerical algorithm, widely used in scientific and engineering applications, for computing approximations of inverse Laplace transform in multi-precision arithmetic systems. We demonstrate that, once parameters \(n\) (i.e. the number of terms of Gaver’s formula) and \(\delta\) (i.e. an upper bound on noise on data) are given, then the number of correct significant digits of computed values of the inverse function is bounded above by \(-\lceil\log_{10}(\delta)\rceil+1\). In case of noise free data this number is arbitrarily large, as it is bounded below by \(n\). We establish the requirement of the multi-precision system ensuring that the quality of numerical results is fulfilled. Experiments and comparisons validate the effectiveness of such approach.


65R32 Numerical methods for inverse problems for integral equations
44A10 Laplace transform
65Y04 Numerical algorithms for computer arithmetic, etc.
68W40 Analysis of algorithms
Full Text: DOI


[1] Piessens R, Huysmans R. Algorithm 619: automatic numerical inversion of the laplace transform. ACM Trans Math Softw. 1984;10(3):348-353. · Zbl 0546.65087
[2] Garbow S, Giunta G, Lyness NJ, et al. A fortran software package for the numerical inversion of a laplace transform based on weeks method. ACM Trans Math Softw. 1988;54:163-170. · Zbl 0642.65086
[3] Murli A, Rizzardi M. Talbots method for the laplace inversion problem. ACM Trans Math Softw. 1990;16:347-371. · Zbl 0900.65374
[4] 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 (TOMS). 1999;25:279-305. · Zbl 0962.65109
[5] D’Amore L, Campagna R, Mele V, et al. Relative an ansi c90 software package for the real laplace transform inversion. Numer Algorithm. 2013;63(1):187-211. · Zbl 1267.65202
[6] D’Amore L, Campagna R, Mele V, et al. Algorithm 946: reliadiff a c++ software package for real laplace transform inversion based on algorithmic differentiation. Trans Math Softw, ACM-TOMS. 2014;14–2:31:1–31:20. · Zbl 1371.65134
[7] Kryzhniy VV. On regularization of numerical inversion of laplace transforms. J Inverse Ill-Posed Probl. 2004;12(3):279-296. · Zbl 1059.65119
[8] Murli A, Cuomo S, D’Amore L, et al. A smoothing spline that approximates laplace transform functions only known on measurements on the real axis. Inverse Probl. 2012;28(2):025007.
[9] Provencher SW. A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput Phys Commun. 1982;12(3):213-227.
[10] Cohen A. Numerical methods for laplace transform inversion. New York (NY): Springer; 2007. · Zbl 1127.65094
[11] Gaver DJ. Observing stochastic processes and approximate transform inversion. Oper Res. 1966;14:444-459.
[12] D’Amore L, Murli A. Regularization of a fourier series method for the laplace transform inversion with real data. Inverse Probl. 2002;18(2):1185-1205. · Zbl 1005.65139
[13] Cuomo S, D’Amore L, Murli A, et al. Computation of the inverse laplace transform based on a collocation method which uses only real values. J Comput Appl Math. 2007;198(1):98-115. · Zbl 1105.65120
[14] Bellman R, Roth R. The laplace transform. Singapore: World Scientific; 1984.
[15] 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 Probl. 2007;23(2):7-23. · Zbl 1122.65128
[16] Tikhonov A. Solution of incorrectly formulated problems and the regularization method. Dokl Akad Nauk SSSR. 1963;151:501-504. · Zbl 0141.11001
[17] Tikhonov NA, Arsenin VY. Solutions of ill-posed problems. New York (NY): Wiston-Wiley; 1977.
[18] Groetsch CW. The theory of tikhonov regularization for fredholm equations of the first kind. Boston: Pitman; 1984.
[19] Garbow BS, Giunta G, Lyness JN, et al. Software for an implementation of weeks’ method for the inverse laplace transform. ACM Trans Math Softw (TOMS). 1988;14:163-170. · Zbl 0642.65086
[20] Widder DV. The laplace transform. Princeton (NJ): Princeton University Press; 1946.
[21] Stehfest H. Algorithm 368: numerical inversion of laplace transforms. Commun ACM. 1970;13(1):47-49.
[22] Abate J, Valkó P. Multi-precision laplace transform inversion. Int J Numer Methods Eng. 2004;60(5-7):979-993.
[23] Valkó P, Vajda S. Inversion of noise-free laplace transforms: towards a standardized set of test problems. Inverse Probl Eng. 2002;10(5):467-483.
[24] Fu Z, Chen W, Qin Q. Three boundary meshless methods for heat conduction analysis in nonlinear fgms with kirchhoff and laplace transformation. Adv Appl Math Mech. 2012;4(5):519-542. · Zbl 1262.65137
[25] Bieniasz LK. Modelling electroanalytical experiments by the integral equation method. Berlin Heidelberg: Springer; 2015. · Zbl 1348.92003
[26] Montella C. Lsv modelling of electrochemical systems through numerical inversion of laplace transforms. i- the gs-lsv algorithm. J Electroanal Chem. 2008;614:121-130.
[27] Montella JDC. New approach of electrochemical systems dynamics in the time-domain under small-signal conditions i. a family of algorithms based on numerical inversion of laplace transforms. J Electroanal Chem. 2008;623:29-40.
[28] Montella JDC. New approach of electrochemical systems dynamics in the time-domain under small-signal conditions ii. modelling the response of electrochemical systems by numerical inversion of laplace transforms. J Electroanal Chem. 2009;625:156-164.
[29] Guo S, Zhangn J, Li G, et al. Three-dimensional transient heat conduction analysis by laplace transformation and multiple reciprocity boundary face method. Eng Anal Boundary Elem. 2013;37:15-22. · Zbl 1351.80016
[30] Valkó PP, Joseph Abate J. Numerical inversion of 2-d laplace transforms applied to fractional diffusion equations. Appl Numer Math. 2005;53(1):73-88. · Zbl 1060.65681
[31] Fu ZJ, Chen W, Yang HT. Boundary particle method for laplace transformed time fractional diffusion equations. J Comput Phys. 2013;235:52-66. · Zbl 1291.76256
[32] Hassanzadeh H, Pooladi-Darvish M. Comparison of different numerical laplace inversion methods for engineering applications. Appl Math Comput. 2007;189:1966-1981. · Zbl 1243.65151
[33] Kuznetsov A. On the convergence of the gaver-stehfest algorithm. SIAM J Num An. 2013;51:2984-2998. · Zbl 1461.65258
[34] Abate J, Valkó P. Comparison of sequence accelerators for the gaver method of numerical laplace transform inversion. Comput Math Appl. 2004;48:629-636. · Zbl 1064.65152
[35] Davies B, Martin B. Numerical inversion of laplace transform. a survey and comparison of mathods. J Comput Phys. 1979;33(1):1-32. · Zbl 0416.65077
[36] Dahlquist G, Björck A. Numerical methods. London: Prentice-Hall International; 1974.
[37] Higham N. Accuracy and stability of numerical algorithms. Philadelphia (PA): SIAM; 2002.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.