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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 aims of the present work were to evaluate the impact of ultraviolet treatment of re-milled semolina on structural development of dough during mixing and leavening and to define a mathematical modelling for leavened doughs. The re-milled semolina was treated with UVC irradiation for 5, 15, 30 and 120 min. The UVC treatment caused some changes of moisture content, colour and dough farinographic indices. The leavening kinetic was investigated by recording over time the variation of the dough volume by means of Image Analysis procedure. The mathematical procedure in this study was to model the evolution of a macroscopic morphology descriptor (volume) by a data-driven approach; the Particle Filter (PF) methodology incorporates the direct observations of the evolutionary dynamics in order to predict the data evolution. A specific differential problem is calibrated on the volume data of semolina doughs. An interesting feature of the proposed scheme is its ability to forecast variable states even when little information is available. More in detail, the volume variations are represented with a stochastic differential model and the dynamics of dough leavening are described by an evolution-observation scheme. The approach suggests the possibility of overcoming the costly data managing and image acquisitions.

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the considered dynamics, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo-Levi spaces. The performances of the method are then tested on functions which are common in the framework of the Laplace transform inversion. Comparisons with Support Vector Regression (SVR) are also carried out and show that the proposed method is effective particularly since it does not require to train complex architecture constructions.

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.

An elliptic partial differential equation with a singular forcing term, describing a steady state flow determined by a pulse-like extraction at a constant volumetric rate, is approximated by a radial basis function approach which takes advantage of decomposing the original domain. The discretization error of such scheme is numerically estimated and we also face up to instability issues. This produces an effective tool for real applications, as confirmed by comparisons with classical grid-based approaches.

With the aim of generalizing P-splines, we here define a special type of penalized splines, called HP-splines, where polynomial splines are replaced by the richer class of hyperbolic-polynomial splines and a suitably tailored discrete penalty term is used. Hyperbolic-polynomial splines, important in several applications, are a natural generalization of polynomial splines consisting of piecewise-defined functions with segments spanned by ‘atoms’ of type x r e α x where r = 0 , … , l and α ∈ R . HP-splines, that reduce to P-splines for α = 0 , are more suitable to data with an exponential trend which is frequent in applications.

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.

Numerous environmental/engineering applications relying upon the theory of diffusion phenomena into chaotic environments have recently stimulated the interest toward the numerical solution of parabolic partial differential equations (PDEs). In the present paper, we outline a formulation of the mathematical problem underlying a quite general diffusion mechanism in the natural environments, and we shortly emphasize some remarks concerning the applicability of the (straightforward) finite difference method. An illustration example is also presented.

Minimizing variational models by means of (un)constrained optimization algorithms is a well-known approach for dealing with the image denoising problem. In this paper, we propose a modification of the widely explored TV-ROF model named H-TV-ROF, in which a penalty term based on higher order derivatives is added. A Split Bregman iterative scheme is used to solve the proposed model and its convergence is proved. The performance of the new algorithm is analized and compared with TV-ROF on a set of numerical experiments.

The algorithms based on the Bregman iterative regularization are known for efficiently solving convex constraint optimization problems. In this paper, we introduce a second order derivative scheme for the class of Bregman algorithms. Its properties of convergence and stability are investigated by means of numerical evidences. Moreover, we apply the proposed scheme to an isotropic Total Variation (TV) problem arising out of the Magnetic Resonance Image (MRI) denoising. Experimental results confirm that our algorithm has good performance in terms of denoising quality, effectiveness and robustness.

The computation of the European options price in a Black-Scholes market, characterized by the presence of no arbitrage condition, is an important applicative problem. In this paper we are interested in highlighting some numerical issues related to this problem. The proposed procedure is mainly divided into three parts: the test of the lognomality of the risk asset, the estimation of the volatility of the underlying and, finally, the determination of the price. As concerns the first point, we propose the adoption of the the Shapiro-Wilk test, in the second one we suggest to estimate the volatility by the sample standard deviation and in the third point we apply the Black-Scholes formula and we introduce an approximation for a Normal function value by means of a quadrature formula.

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.

The scientific and application-oriented interest in the Laplace transform and its inversion is testified by more than 1000 publications in the last century. Most of the inversion algorithms available in the literature assume that the Laplace transform function is available everywhere. Unfortunately, such an assumption is not fulfilled in the applications of the Laplace transform. Very often, one only has a finite set of data and one wants to recover an estimate of the inverse Laplace function from that. We propose a fitting model of data. More precisely, given a finite set of measurements on the real axis, arising from an unknown Laplace transform function, we construct a dth degree generalized polynomial smoothing spline, where d = 2m 1, such that internally to the data interval it is a dth degree polynomial complete smoothing spline minimizing a regularization functional, and outside the data interval, it mimics the Laplace transform asymptotic behavior, i.e. it is a rational or an exponential function (the end behavior model), and at the boundaries of the data set it joins with regularity up to order m 1, with the end behavior model. We analyze in detail the generalized polynomial smoothing spline of degree d = 3. This choice was motivated by the (ill)conditioning of the numerical computation which strongly depends on the degree of the complete spline. We prove existence and uniqueness of this spline. We derive the approximation error and give a priori and computable bounds of it on the whole real axis. In such a way, the generalized polynomial smoothing spline may be used in any real inversion algorithm to compute an approximation of the inverse Laplace function. Experimental results concerning Laplace transform approximation, numerical inversion of the generalized polynomial smoothing spline and comparisons with the exponential smoothing spline conclude the work. © 2012 IOP Publishing Ltd.