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.

On the recovery and resolution of exponential relaxation rates from experimental data II. The optimum choice of experimental sampling points for Laplace transform inversion

1984 ◽  
Vol 393 (1804) ◽  
pp. 51-65 ◽  
Keyword(s):  
Experimental Data ◽  
Laplace Transform ◽  
Upper And Lower Bounds ◽  
Relaxation Rates ◽  
Scientific Experiments ◽  
The Laplace Transform ◽  
Data Points ◽  
Exponential Relaxation ◽  
Ill Conditioning ◽  
Laplace Transform Inversion

In a previous paper (Bertero, Boccacci & Pike, Proc. R. Soc. Lond . A 383, 15 (1982)) we investigated the benefits of a priori knowledge of finite support of the solution in the inversion of the Laplace transform and determined the number of exponential components that could be recovered from given data in the presence of quantified amounts of noise. The support of the data was assumed to be unrestricted. In the present paper these calculations are extended to cover the case of a necessarily finite number of experimental data points. We consider two cases: uniform distribution and geometric distribution of data points. In both cases the condition number of singular value inversions is minimized with respect to the placing of the data points for fixed numbers of data samples. The results allow optimum placing of the experimental data points as a function of the ratio, γ , of given upper and lower bounds of the support of the solution. The number of exponentials recoverable is less than in the unrestricted case but for γ ≼ 8 we find that with 32 linearly spaced data points, optimally placed, this reduction is rather small. Geometrical sampling of experimental data reduces further the number of sample points required for inversion. For γ ≼ 8 we show that with 5 geometrically spaced data points, points, optimally placed, the ill-conditioning in the restoration of 2 to 4 exponential components is smaller than with 32 linearly spaced data samples. This has important implications for the design of a number of scientific experiments and instruments.

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.

On the recovery and resolution of exponential relaxation rates from experimental data. - III. The effect of sampling and truncation of data on the Laplace transform inversion

1985 ◽  
Vol 398 (1814) ◽  
pp. 23-44 ◽  
Keyword(s):  
Laplace Transform ◽  
Continuous Solution ◽  
Singular Values ◽  
Discrete Data ◽  
Continuous Data ◽  
Singular Function ◽  
Relaxation Rates ◽  
Singular Functions ◽  
The Laplace Transform ◽  
Data Points

The method of singular function expansions, which in previous papers in this series was used for the inversion of the Laplace transform for the cases, respectively, of continuous data and continuous solution and discrete data and discrete solution, is extended to cover the case of discrete data and continuous solution. Two discrete data points distributions are considered: uniform and geometric. For both we prove that the singular values and the singular functions (in the solution space) of the problem with discrete data and continuous solution converge to the singular values and singular functions of the problem with continuous data and continuous solution when the number of points tends to infinity and the distance between adjacents points tends to zero. Furthermore, we show by means of numerical computations that for geometrically sampled data it is possible to obtain even better approximations of the greatest singular values than in the discrete-to-discrete case using a similar number of data points. Excellent approximations of the continuous-to-continuous case singular functions are also obtained. Implementation of the inversion procedure gives continuous solutions with high computational 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.

scholarly journals Approximating the Laplace transform of the sum of dependent lognormals

2016 ◽  
Vol 48 (A) ◽  
pp. 203-215 ◽  
Author(s):  
Patrick J. Laub ◽  
Søren Asmussen ◽  
Jens L. Jensen ◽  
Leonardo Rojas-Nandayapa
Keyword(s):  
Monte Carlo ◽  
Laplace Transform ◽  
Taylor Expansion ◽  
Multivariate Normal ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
Error Factor ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Mean Vector

AbstractLet (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and letSn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form andI(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacinghθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙*is presented, and it is shown thatI(θ)→ 1 as θ→∞. A variety of numerical methods for evaluatingI(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density ofSn) are also given.

scholarly journals Classroom note: Fourier method for Laplace transform inversion

2001 ◽  
Vol 5 (3) ◽  
pp. 193-200 ◽  
Author(s):  
M. Iqbal
Keyword(s):  
Laplace Transform ◽  
Fourier Method ◽  
Test Functions ◽  
The Laplace Transform ◽  
Laplace Transform Inversion

A method is described for inverting the Laplace transform. The performance of the Fourier method is illustrated by the inversion of the test functions available in the literature. Results are shown in the tables.

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.

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