NUMERICAL TRANSFORM INVERSION USING GAUSSIAN QUADRATURE

2005 ◽  
Vol 20 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Peter den Iseger
Keyword(s):  
Laplace Transform ◽  
Stochastic Models ◽  
A Priori ◽  
Gaussian Quadrature ◽  
Laplace Transforms ◽  
Numerical Inversion ◽  
Inversion Algorithm ◽  
Machine Precision ◽  
The Laplace Transform ◽  
Laplace Transform Inversion

Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. In this article we present a simple Laplace transform inversion algorithm that can compute the desired function values for a much larger class of Laplace transforms than the ones that can be inverted with the known methods in the literature. The algorithm can invert Laplace transforms of functions with discontinuities and singularities, even if we do not know the location of these discontinuities and singularities a priori. The algorithm only needs numerical values of the Laplace transform, is extremely fast, and the results are of almost machine precision. We also present a two-dimensional variant of the Laplace transform inversion algorithm. We illustrate the accuracy and robustness of the algorithms with various numerical examples.

scholarly journals The noise handling properties of the Talbot algorithm for numerically inverting the Laplace transform

2018 ◽  
Vol 13 ◽  
pp. 174830181879706 ◽  
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Noisy Data ◽  
Standard Test ◽  
Numerical Inversion ◽  
Test Functions ◽  
Inversion Algorithm ◽  
Inverse Transform ◽  
Exact Data ◽  
The Laplace Transform

This paper examines the noise handling properties of three of the most widely used algorithms for numerically inverting the Laplace transform. After examining the genesis of the algorithms, their error handling properties are evaluated through a series of standard test functions in which noise is added to the inverse transform. Comparisons are then made with the exact data. Our main finding is that the for “noisy data”, the Talbot inversion algorithm performs with greater accuracy when compared to the Fourier series and Stehfest numerical inversion schemes as they are outlined in this paper.

scholarly journals Non-Gaussian Quadrature Integral Transform Solution of Parabolic Models with a Finite Degree of Randomness

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1112
Author(s):  
María-Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Laplace Transform ◽  
Integral Transform ◽  
Gaussian Quadrature ◽  
Finite Degree ◽  
Laplace Transform Method ◽  
Transform Method ◽  
The Laplace Transform ◽  
Non Gaussian ◽  
Laplace Transform Inversion ◽  
Integral Transform Method

In this paper, we propose an integral transform method for the numerical solution of random mean square parabolic models, that makes manageable the computational complexity due to the storage of intermediate information when one applies iterative methods. By applying the random Laplace transform method combined with the use of Monte Carlo and numerical integration of the Laplace transform inversion, an easy expression of the approximating stochastic process allows the manageable computation of the statistical moments of the approximation.

On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise

1982 ◽  
Vol 383 (1784) ◽  
pp. 15-29 ◽  
Keyword(s):  
Laplace Transform ◽  
A Priori ◽  
Inverse Function ◽  
Singular Values ◽  
Relaxation Rates ◽  
Value Analysis ◽  
The Laplace Transform ◽  
Exponential Relaxation ◽  
Laplace Transform Inversion ◽  
Positive Support

The problem of numerical inversion of the Laplace transform is considered when the inverse function is of bounded, strictly positive support. The recent eigenvalue analysis of McWhirter and Pike for infinite support has been generalized by numerical calculations of singular values. A priori knowledge of the support is shown to lead to increased resolution in the inversion, and the number of exponentials that can be recovered in given levels of noise is calculated.

scholarly journals Numerical inversion of the Laplace transform

2005 ◽  
Vol 18 (3) ◽  
pp. 515-530 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Aleksandar Cvetkovic
Keyword(s):  
Laplace Transform ◽  
Numerical Investigation ◽  
Gaussian Quadrature ◽  
New Method ◽  
Quadrature Rules ◽  
Numerical Inversion ◽  
Short Account ◽  
The Laplace Transform ◽  
Real Poles

We give a short account on the methods for numerical inversion of the Laplace transform and also propose a new method. Our method is inspired and motivated from a problem of the evaluation of the M?ntz polynomials (see [1]), as well as the construction of the generalized Gaussian quadrature rules for the M?ntz systems (see [2]). As an illustration of our method we consider an example with 100 real poles distributed uniformly on of the proposed method. 1 2 100. A numerical investigation shows the efficiency.

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

1975 ◽  
Vol 65 (4) ◽  
pp. 927-935
Author(s):  
I. M. Longman ◽  
T. Beer
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
High Accuracy ◽  
Time Variable ◽  
Inversion Method ◽  
The Laplace Transform ◽  
Previously Treated ◽  
The Mean ◽  
Function Approximations ◽  
Laplace Transform Inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

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