research-article

Algorithm 946: ReLIADiff—A C++ Software Package for Real Laplace Transform Inversion based on Algorithmic Differentiation

Authors:

Rosanna Campagna,

Almerico MurliAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 40, Issue 4

Article No.: 31, Pages 1 - 20

https://doi.org/10.1145/2616971

Published: 08 July 2014 Publication History

Abstract

Algorithm 662 of the ACM TOMS library is a software package, based on the Weeks method, which is used for calculating function values of the inverse Laplace transform. The software requires transform values at arbitrary points in the complex plane. We developed a software package, called ReLIADiff, which is a modification of Algorithm 662 using transform values at arbitrary points on real axis. ReLIADiff, implemented in C++, relies on TADIFF software package designed for Algorithmic Differentiation. In this article, we present ReLIADiff focusing on its design principles, performance, and use.

Supplementary Material

ZIP File (946.zip)

Software for ReLIADiff - A C++ Software Package for Real Laplace Transform Inversion based on Algorithmic Differentiation

Download
5.45 MB

a31-damore-apndx.pdf (damore.zip)

Supplemental movie, appendix, image and software files for, Algorithm 946: ReLIADiff—A C++ Software Package for Real Laplace Transform Inversion based on Algorithmic Differentiation

Download
402.81 KB

References

[1]

J. Abate and P. Valko. 2002. Uniform Resource Locators (URL). Wolfram Information Center. http://library. wolfram.com/infocenter/MathSource/4738/.

[2]

J. Abate and W. Whitt. 1996. On the Laguerre method for numerically inverting Laplace transforms. J. Comput. 8, 4, 413--427.

[3]

R. S. Anderssen and F. R. de Hoog. 1984. Finite difference methods for the numerical differentiation of non-exact data. Computing 33, 259--267.

[4]

R. Bellman, R. Kalaba, and J. A. Lockett. 1966. Numerical inversion of the Laplace transform: Applications to biology, economics, engineering, and physics. Elsevier.

[5]

C. Bendtsen and O. Stauning. 1997. TADIFF, A flexible C++ package for automatic differentiation using Taylor series expansion. http://citeseer.ist.psu.edu/bendtsen97tadiff.html.

[6]

R. Campagna, L. D'Amore, and A. Murli. 2007. An efficient algorithm for regularization of Laplace transform inversion in real case. J. Computat. Appl. Math. 210, 84--98.

Digital Library

[7]

A. M. Cohen. 2007. Numerical Methods for Laplace Transform Inversion. Springer.

[8]

J. Cullum. 1971. Numerical differentiation and regularization. SIAM J. Numer. Anal. 8, 254--265.

Digital Library

[9]

S. Cuomo, L. D'Amore, A. Murli, and M. Rizzardi. 2007. Computation of the inverse Laplace transform based on a collocation method which uses only real values. J. Computat. Appl. Math. 198, 98--115.

Digital Library

[10]

S. Cuomo, L. D'Amore, A. Murli, and M. Rizzardi. 2008. A modification of Weeks' method for numerical inversion of the Laplace transform in the real case based on automatic differentiation. In Advances in Automatic Differentiation, Christian H. Bischof, H. Martin Bücker, Paul D. Hovland, Uwe Naumann, and J. Utke, Eds., (Springer). 45--54.

[11]

L. D'Amore and A. Murli. 2002. Regularization of a Fourier series based method for real inversion of Laplace transform. Inverse Prob. 18, 1185--1205.

[12]

L. D'Amore, R. Campagna, A. Galletti, L. Marcellino, and A. Murli. 2012. A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis. Inver. Prob. 28.

[13]

L. D'Amore, R. Campagna, V. Mele, and A. Murli. 2013a. ReLaTIve. An ANSI C90 software package for the real Laplace transform inversion. Numer. Algor. 63, 187--211.

Digital Library

[14]

L. D'Amore, G. Laccetti, and A. Murli. 1999a. An implementation of a Fourier series method for the numerical inversion of the Laplace transform. ACM Trans. Math. Soft. 25, 279--305.

Digital Library

[15]

L. D'Amore, G. Laccetti, and A. Murli. 1999b. Algorithm 796: A Fortran software package for the numerical inversion of the Laplace transform based on a Fourier series method. ACM Trans. Math. Soft. 25, 306--315.

Digital Library

[16]

L. D'Amore, V. Mele, and A. Murli. 2013b. Performance analysis of the Taylor expansion coefficients computation as implemented by the software package TADIFF. J. Numer. Anal. Indust. Appl. Math. 8, 1--2, 1--12.

[17]

B. Davies and D. Martin. 1979. Numerical inversion of Laplace transform. A survey and comparison of methods. J. Comput. Phys. 33, 1--32.

[18]

D. G. Duffy. 1993. On the numerical inversion of Laplace transforms: Comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Softw. 19, 3, 333--359.

Digital Library

[19]

W. Fair. Jr. 2008. Numerical Laplace transforms and inverse transforms in C&num;, http://www.codeproject.com/KB/recipes/LaplaceTransforms.aspx&quest;msg=3150794.

[20]

P. F. Faure and S. Rodts. 2008. Proton NMR relaxation as a probe for setting cement pastes. Magn. Reson. Imaging 26, 8, 1183--1196.

[21]

S. Garbow, G. Giunta, N. J. Lyness, and A. Murli. 1988. 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.

Digital Library

[22]

G. Giunta, G. Laccetti, and M. Rizzardi. 1988. More on Weeks method for the numerical inversion of the Laplace trasform. Numer. Mat. 193, 193--200.

Digital Library

[23]

G. Giunta and A. Murli. 1989. An algorithm for inverting the Laplace transform using real and real sampled function values. In Proceedings of the IMACS Symposium on Numerical and Applied Mathematics, C. Brezinski, Ed., 589--592.

[24]

G. Giunta, A. Murli, and G. Schmid. 1995. Error analysis of Rjabov algorithm for inverting Laplace transforms. Ricerche Matemat. 44, 1, 207--219.

[25]

A. Griewank and K. Kulshreshtha. 2012. On the numerical stability of algorithmic differentiation. Computing 94, 125--149.

Digital Library

[26]

C. W. Groetsch. 1990. The theory of Tikhonov regularization for Fredholm equations of the first kind. Research Notes in Mathematics, Pitman Advanced Publishing Program.

[27]

M. Hanke and O. Scherzer. 2001. Inverse problems light: Numerical differentiation. Amer. Math. Monthly 108, 6, 512--521.

[28]

P. Henrici. 1966. Elements of Numerical Analysis. Wiley.

[29]

N. Higham. 1996. Accuracy and Stability of Numerical Algorithms. SIAM.

Digital Library

[30]

V. V. Kryzhniy. 2004. On regularization of numerical inversion of Laplace transforms J. Inverse Ill-Posed Prob. 12, 3, 279--296.

[31]

Y. Y. Lin, N. H. Ge, and L. P. Hwang. 1993. Multiexponential analysis of relaxation decays based on linear, prediction and singular-value decomposition. J. Magn. Reson. A 105, 6571.

[32]

B. Magyari, I. Ioan, and P. P. Valko. 2010. Stokes first problem for micropolar fluids. Fluid Dynam. Res. 42, 1--15.

[33]

W. Magnus, F. Oberhettinger, and R. P. Soni. 1966. Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, New York.

[34]

C. Montella, R. Michel, and J. P. Diard. 2007. Numerical inversion of Laplace transforms.: A useful tool for evaluation of chemical diffusion coefficients in ion-insertion electrodes investigated by Pitt. J. Electroanal. Chem. 608, 1, 15, 37--46.

[35]

A. Murli, S. Cuomo, L. D'Amore, and A. Galletti. 2007. Numerical regularization of a real inversion formula based on the Laplace transform's eigenfunction expansion of the inverse function. Inv. Prob. 23, 713--731.

[36]

R. Piessens. 1972. A new numerical method for the inversion of the Laplace transform. J. Inst. Math. Appl. 10, 185--192.

[37]

R. Piessens. 1982. Algorithm 113: Inversion of the Laplace transform. Computer J. (Algor. Supplement) 25, 2, 278--282.

[38]

A. G. Ramm and A. B. Smirnova. 2001. On stable numerical differentiation. Math. Computat. 70, 235, 1131--1153.

Digital Library

[39]

H. Stehfest. 1970. Algorithm 368: Numerical inversion of Laplace transform Commun. ACM 13, 47--49.

Digital Library

[40]

G. Szego. 1939. Orthogonal polynomials. In American Mathematical Society Colloquium Publications, Vol. XXIII.

[41]

P. Valko and S. Vaida. 2002. Inversion of noise-free Laplace transforms: Towards a standardized set of test problems. Inverse Problems Eng. 10, 467--483.

[42]

L. Van Der Meerd, S. M. Melnikov, F. J. Vergeldt, E. G. Novikov, and H. Van As. 2002. Modelling of self-diffusion and relaxation time NMR in multicompartment systems with cylindrical geometry. J. Magn. Reson. 156, 2, 213--221.

[43]

W. Weeks. 1966. Numerical inversion of the Laplace transform using Laguerre functions. J. ACM 13, 419--429.

Digital Library

[44]

J. Weidemann. 1999. Algorithms for parameter selection in the Weeks method for inverting the Laplace transform. SIAM J. Sci. Comput. 21, 1, 118--128.

Digital Library

[45]

X. Zhao. 2004. An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate. Int. J. Solids Structures 41, 13, 3653--3674.

Cited By

Mele VLaccetti G(2023)Algorithm and Software Overhead: A Theoretical Approach to Performance PortabilityParallel Processing and Applied Mathematics10.1007/978-3-031-30445-3_8(89-100)Online publication date: 27-Apr-2023
https://doi.org/10.1007/978-3-031-30445-3_8
Ali ASuwan IAbdeljawad TAbdullah (2022)Numerical simulation of time partial fractional diffusion model by Laplace transformAIMS Mathematics10.3934/math.20221597:2(2878-2890)Online publication date: 2022
https://doi.org/10.3934/math.2022159
D’Amore LCacciapuoti RMele V(2020)Ab-initio Functional Decomposition of Kalman Filter: A Feasibility Analysis on Constrained Least Squares ProblemsParallel Processing and Applied Mathematics10.1007/978-3-030-43222-5_7(75-92)Online publication date: 19-Mar-2020
https://doi.org/10.1007/978-3-030-43222-5_7
Show More Cited By

Index Terms

Algorithm 946: ReLIADiff—A C++ Software Package for Real Laplace Transform Inversion based on Algorithmic Differentiation
1. Mathematics of computing
  1. Mathematical analysis
    1. Integral equations
    2. Numerical analysis
      1. Numerical differentiation
  2. Mathematical software
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures
    1. Error-correcting codes

Recommendations

Algorithm 796: a Fortran software package for the numerical inversion of the Laplace transform based on a Fourier series method

A software package for the numerical inversion of a Laplace Transform function is described. Besides function values of F (z) for complex and real z, the user has only to provide the numerical value of the Laplace convergence abscissa σ₀ or, failing this,...
Algorithm 486: Numerical inversion of Laplace transform [D5]

This work forms part of a thesis presented in Grenoble in March 1972. Improvements made to the Dubner and Abate algorithm for numerical inversion of the Laplace transform [1] have led to results which compare favorably with theirs and those of Bellmann [...
A Frame Decomposition of the Funk-Radon Transform
Scale Space and Variational Methods in Computer Vision
Abstract
The Funk-Radon transform assigns to a function defined on the unit sphere its integrals along all great circles of the sphere. In this paper, we consider a frame decomposition of the Funk-Radon transform, which is a flexible alternative to the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 40, Issue 4

June 2014

154 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2639949

Editor:
Michael A. Heroux
Sandia National Laboratories

Issue’s Table of Contents

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 July 2014

Accepted: 01 November 2013

Revised: 01 July 2013

Received: 01 October 2012

Published in TOMS Volume 40, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Badges

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
551
Total Downloads

Downloads (Last 12 months)7
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Mele VLaccetti G(2023)Algorithm and Software Overhead: A Theoretical Approach to Performance PortabilityParallel Processing and Applied Mathematics10.1007/978-3-031-30445-3_8(89-100)Online publication date: 27-Apr-2023
https://doi.org/10.1007/978-3-031-30445-3_8
Ali ASuwan IAbdeljawad TAbdullah (2022)Numerical simulation of time partial fractional diffusion model by Laplace transformAIMS Mathematics10.3934/math.20221597:2(2878-2890)Online publication date: 2022
https://doi.org/10.3934/math.2022159
D’Amore LCacciapuoti RMele V(2020)Ab-initio Functional Decomposition of Kalman Filter: A Feasibility Analysis on Constrained Least Squares ProblemsParallel Processing and Applied Mathematics10.1007/978-3-030-43222-5_7(75-92)Online publication date: 19-Mar-2020
https://doi.org/10.1007/978-3-030-43222-5_7
Di Luccio DKosta SCastiglione AMaratea AMontella R(2020)Vessel to shore data movement through the Internet of Floating Things: A microservice platform at the edgeConcurrency and Computation: Practice and Experience10.1002/cpe.598833:4Online publication date: 7-Sep-2020
https://doi.org/10.1002/cpe.5988
D’Amore LMele VCampagna R(2017)Quality assurance of Gaver’s formula for multi-precision Laplace transform inversion in real caseInverse Problems in Science and Engineering10.1080/17415977.2017.132296326:4(553-580)Online publication date: 8-May-2017
https://doi.org/10.1080/17415977.2017.1322963

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents