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Trasformata inversa di laplace per funzioni positive e problema dei momenti

  • Marco Frontini
  • Aldo Tagliani
Article

Abstract

Si propone un nuovo metodo per l'inversione numerica della transformata di Laplace per funzioni positive, tipicamente densità di probabilità. La funzione approssimante è ottenuta ricorrendo al principio di Massima Entropia, assumendo come informazione limitata i momenti della funzione da ricostruire, a loro volta legati da ben note relazioni alle derivate della funzione transformata. Dopo aver richiamato alcuni risultati teorici noti in letteratura concernenti l'esistenza dell'approssimante e la convergenza in entropia (e di conseguenza in normaL 1), si presentano alcuni risultati teorici originali sul condizionamento del problema e alcuni risultati numerici che illustrano la bontà dell'approssimante.

Summary

A new method for the numerical Laplace transform inversion of positive functions, tipically probability density functions, is proposed. The approximating function is obtained resorting to the maxientropic approach, by assuming its moments as partial available information. Some known theoretical results concerning the existence and entropy convergence are reviewed. New theoretical results about the conditioning of the problem and some numerical simulations are illustrated.

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Bibliografia

  1. [1]
    Bellman R., Kabala R. E., Lockett J. A.,Numerical inversion of the Laplace transform, New York American Elsevier, 1966.Google Scholar
  2. [2]
    Davis B., Martin B.,Numerical inversion of the Laplace transform; a survey and comparison of methods, J. of Comput. Physics, 33, 1979, 1–32.CrossRefGoogle Scholar
  3. [3]
    Frontini M., Tagliani A.,Maximum entropy in the finite Stieltjes and Hamburger moment problem, J. Math. Physics, 35, 1994, 6748–56.CrossRefGoogle Scholar
  4. [4]
    Frontini M., Tagliani A.,Entropy-convergence in Stieltjes and Hamburger moment problem, J. Applied Mathematics and Computation., 1996, in corso di pubblicazione.Google Scholar
  5. [5]
    Kullback S.,Information Theory and Statistics, Dover, New York, 1962.Google Scholar
  6. [6]
    Kapur J. N.,Maximum-Entropy Models in Science and Engineering, John Wiley Sons, 1989.Google Scholar
  7. [7]
    Jaynes E. T.,Where do we stand on maximum entropy, in R.D. Levine and M. Tribus Eds.,The maximum entropy formalism, MIT Press Cambridge MA, 1978, 15–118Google Scholar
  8. [8]
    Lyness J. N.,Differentiation formulas for analytic functions, Math. of Computation, 22, 1968, 352–62.CrossRefGoogle Scholar
  9. [9]
    Piessens R., Branders M.,Numerical inversion of the Laplace transform using generalized Laguerre polynomials, Proc. IEE, 118, 1971, 1517–22.Google Scholar
  10. [10]
    Polya G., Szego G.,Problems and Theorems in Analysis, 1, Springer-Verlag, 1972.Google Scholar
  11. [11]
    Shohat J. A., Tamarkin J. D.,The problem of moments, American Mathematical Society, Providence, RI, 1963.Google Scholar
  12. [12]
    Sobczyk K., Trebicki J.,Maximum entropy principle in stochastic dynamics, Probabilistic Engineering Mechanics, 5, 1990, 102–110.CrossRefGoogle Scholar
  13. [13]
    Weeks W. T., Numerical inversion of the Laplace transform using Laguerre functions, Jour. ACM, 13, 1966, 419–29.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Marco Frontini
    • 1
  • Aldo Tagliani
    • 1
  1. 1.Politecnico di MilanoMilanoItaly

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