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Many applications are tackled using the Laplace Transform (LT) known on a countable number of real values [J. Electroanal. Chem. 608, 37–46 (2007), Int. J. solid Struct. 41, 3653–3674 (2004), Imaging 26, 1183–1196 (2008), J. Magn. Reson. 156, 213–221 (2002)]. The usual approach to solve the LT inverse problem relies on a regularization technique combined with information a priori both on the LT function and on its inverse (see for instance [http://s-provencher.com/pages/ contin.shtml]). We propose a fitting model enjoying LT properties: we define a generalized spline that interpolates the LT function values and mimics the asymptotic behavior of LT functions. Then, we prove existence and uniqueness of this model and, through a suitable error analysis, we give a priori approximation error bounds to confirm the reliability of this approach. Numerical results are presented.

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D'Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185-1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix. © 2006 Elsevier B.V. All rights reserved.

In the numerical methods for the Laplace transform inversion the errors quantification in computational processes is a crucial issue. In this paper, we propose two inversion methods based on smoothing splines combined with a procedure for the derivation of error bounds. In particular, we numerically study the impact of the fitting error amplification through the analysis of several sources of error and their propagation.

The understanding of the breadmaking process requires to understand the changes in water mobility of dough. The dough ingredients as well as the processing conditions determine the structure of baked products which in turn is responsible for their apparence, texture, taste and stability. The transition from wheat flour to dough is a complex process in which several transformations take place, including those associated with changes in water distribution [13]. The molecular mobility of water in foods can be studied with proton Nuclear Magnetic Resonance (1H NMR). In this study, the measured transverse relaxation times (T2) were considered to investigate the wheat dough development during mixing. The interactions of the flour polymers with water during mixing reduce water mobility and result in different molecular mobilities in dough. The molecular dynamics in heterogeneous systems are very complex. From a mathematical point of view the NMR relaxation decay is generally modelled by the linear superposition of a few exponential functions of the relaxation times. This could be a too rough model and the classical fitting approaches could fail to describe physical reality. A more appealing procedure consists in describing the NMR relaxation decay in integral form by the Laplace transform [2]. In this work a discrete Inverse Laplace Transform procedure is considered to obtain the relaxation times distribution of a dataset provided as case study.

Multi-exponential decaying data are very frequent in applications and a continuous description of this type of data allows the use of mathematical tools for data analysis such as the Laplace Transform (LT). In this work a numerical procedure for the Laplace Transform Inversion (LTI) of multi-exponential decaying data is proposed. It is based on a new fitting model, that is a smoothing exponential-polynomial spline with segments expressed in Bernstein-like bases. A numerical experiment concerning the application of a LTI method applied to our spline model highlights that it is very promising in the LTI of exponential decay data.

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 δ (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 −┌log10 (δ)┐ + 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.

Different methods for the numerical evaluations of the inverse Laplace and inverse of joint Laplace–Hankel integral transforms are applied to solve a wide range of initial-boundary value problems often arising in engineering and applied mathematics. The aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature. Most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half-space to concentrated impulse or those related to viscoelastic wave motion in layered media. These classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations, theoretical seismology, soil–structure-interaction in time domain and so on. Therefore, their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems. Some different methods are also investigated in detail and compared for the inversion of the joint Hankel–Laplace transforms, where more sophisticated integrand functions are encountered. It is shown that Durbin, Crump, D’Amore, Fixed-Talbot, Gaver–Whyn–Rho (GWR), and Direct Integration methods have excellent performance and produce good results when applied to the same problems. On the contrary, Gaver–Stehfest and Piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for “ simple ” transformed functions. Particularly the performance of GWR algorithm is very good even for transformed functions with infinite number of singularities, where the other methods fail. In addition, in case of double integral transforms, only the Fixed-Talbot, Durbin and Weeks methods are recommended.

A software package for numerical inversion of Laplace transforms computable everywhere on the real axis is described. Besides the function to invert the user has only to provide the numerical value (even if it is an approximate value) of the abscissa of convergence and the accuracy required for the inverse function. The software provides a controlled accuracy, i. e. it dynamically computes the so-called maximum attainable accuracy such that numerical results are provided within the greatest value between the user's required accuracy and the maximum attainable accuracy. This is done because the intrinsic ill posedness of the real inversion problem sometime may prevent to reach the desired accuracy. The method implemented is based on a Laguerre polynomial series expansion of the inverse function and belongs to the class of polynomial-type methods of inversion of the Laplace transform, formally characterized as Collocation methods (C-methods). © 2012 Springer Science+Business Media, LLC.