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The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as$$r(x)=\sum _{i=0}^n a_i(x) f_i\big /\sum _{j=0}^m b_j(x)$$r(x)=∑i=0nai(x)fi/∑j=0mbj(x)in terms of data values$$f_i$$fiand some functions$$a_i$$aiand$$b_j$$bjfor$$i=0,\ldots ,n$$i=0,…,nand$$j=0,\dots ,m$$j=0,⋯,mwith a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows withn, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless ofn. We support our theoretical results with numerical experiments.

The Laplace transform inversion is a well-known ill-conditioned problem and many numerical schemes in literature have investigated how to solve it. In this paper, we revise the Gaver-Stehfest method by using polyharmonic splines augmented with polynomials to approximate the Laplace transform into the numerical inversion formula. Theoretical accuracy bounds for the fitting model are given. Discussions on the effectiveness of the inversion algorithm are produced and confirmed by numerical experiments about approximation errors and inversion results. Comparisons with an existing model are also presented.

Well hydraulics concerns the process of flow toward/from wells, which occurs because of a vertical extraction or injection in the aquifer. The paper shows preliminary tests performed on the high performance computing (HPC) ReCaS infrastructure to compute simulations on the fluctuation of the water table in heterogeneous porous formation based on a stochastic parametrization of the transmissivity field of the aquifer for the horizontal flow around the pumping well. From a computational point of view, the more expensive aspects concern the storage of the samples and the numerical solution of the related boundary value problem. The numerical simulations were carried out by using packages/libraries included in the SCoPE toolkit v.3 available at the HPC ReCaS infrastructure.

La concessione di contributi per la riparazione dei danni causati dal sisma del 06 Aprile 2009 in Abruzzo, nei centri storici del Comune dell’Aquila e degli altri Comuni del Cratere, nella fase conseguente alla cessazione dello stato di emergenza (agosto 2012), è disciplinata da un modello parametrico. Il contributo per la riparazione del danno ed il miglioramento sismico dell’edificato è basato su un’analisi preliminare del livello di danno e vulnerabilità riferito alla consistenza degli edifici, considerando anche la presenza di elementi di pregio storico-artistico, architettonico e paesaggistico. Il modello implementato dall’Ufficio Speciale per la Ricostruzione dei Comuni del Cratere (Modello Integrato per i Comuni del Cratere - MIC) prevede la compilazione online di una scheda che consente di definire il contributo parametrico concedibile per l’intervento di riparazione e miglioramento sismico e la contestuale consegna del progetto esecutivo dei lavori. Qualora l’importo dell’intervento rientri nei limiti del contributo concedibile definito su base parametrica il progetto è oggetto di un’istruttoria semplificata che permette di liquidare il contributo stesso in tempi ridotti. Il modello parametrico presenta peculiarità che lo differenziano dai modelli parametrici adottati nei terremoti passati e in parte anche dal modello parametrico adottato dal Comune dell’Aquila. Nella memoria sono illustrate sinteticamente le considerazioni alla base del modello stesso e le sue caratteristiche principali.

Gli interventi di riparazione dei danni causati dal sisma del 06 Aprile 2009 in Abruzzo nei centri storici dei comuni del cosiddetto “cratere” sono disciplinati dal Modello Integrato per i Comuni del Cratere (MIC), che prevede la consegna contestuale di una scheda parametrica e del progetto esecutivo dell’intervento. Il modello parametrico adottato contiene apposite linee guida per l’identificazione degli elementi di pregio e la progettazione degli interventi nei centri storici, volte a valorizzare e preservare il costruito storico, ponendo dei limiti all’esecuzione di interventi riconosciuti dalla letteratura scientifica come particolarmente invasivi se non addirittura dannosi. Sono state individuate a tal fine 12 categorie di incremento del contributo collegate ad altrettante tipologie di interventi, che guidano i progettisti nella scelta degli interventi più efficaci in relazione alle caratteristiche del costruito. Nella memoria è effettuata una prima disamina dei costi di intervento dei progetti presentati con il modello MIC oltre ad alcune considerazioni sull’efficacia degli incrementi concessi.

In recent years, some strategies (e.g., server consolidation by means of virtualisation techniques) helped the managers of large Information Technology (IT) infrastructures to limit, when possible, the use of hardware resources in order to provide reliable services and to reduce the Total Cost of Ownership (TCO) of such infrastructures. Moreover, with the advent of Cloud computing, a resource usage rationalisation can be pursued also for the users applications, if this is compatible with the Quality of Service (QoS) which must be guaranteed. In this perspective, modern datacenters are “elastic”, i.e., able to shrink or enlarge the number of local physical or virtual resources from private/public Clouds. Moreover, many of large computing environments are integrated in distributed computing environment as the grid and cloud infrastructures. In this document, we report some advances in the realisation of a utility, we named Adaptive Scheduling Controller (ASC) which, interacting with the datacenter resource manager, allows an effective and efficient usage of resources, also by means of users jobs classification. Here, we focus both on some data mining algorithms which allows to classify the users activity and on the mathematical formalisation of the functional used by ASC to find the most suitable configuration for the datacenter’s resource manager. The presented case study concerns the SCoPE infrastructure, which has a twofold role: local computing resources provider for the University of Naples Federico II and remote resources provider for both the Italian Grid Infrastructure (IGI) and the European Grid Infrastructure (EGI) Federated Cloud.

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.

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 this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($$\lambda$$ λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the $$\lambda$$ λ -greedy algorithm enables us to define target-data-independent interpolation nodes.

Carrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer’s heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase.

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.

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.

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.

In Food Science and Technology applications complex phenomena that involve macroscopic measurements are generally challenging to be represented in a formal (mathematical) way. In this paper we propose to model the evolution of some morphology descriptors of bread making process by adopting a well-known methodology: the Particle Filtering. The main idea is to describe the volume variations, related to the yeast content in a bread dough, with a stochastic differential model to forecast the dynamics of leavening and baking bread processes, when some samples are known in several time instants. Numerical experiments confirm that the proposed approach is able to accurately predict values of leavening and baking function. Finally, we highlight that for Food Science and Technology applications an interesting feature of the proposed scheme is its ability to forecast variable states also when few instant samples are available.

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 deployment, management and total cost of ownership of large computing environments always involve huge investments. These systems, once in production, have to meet the needs of users belonging to large and heterogeneous communities whose demands are often conflicting. Only an efficient and effective use of these systems can pay back the investment carried out. The infrastructure needs to be sustained also by means of the building of an "appropriate" community of users (the "catchment area"), to be, firstly, recruited and then driven to a conscious use of resources. Based on the experience we gained at the University of Naples Federico II, in the activities related to "support users and applications" on the SCoPE system, we "enjoyed" to write our "recipe". Since 2008, also in the framework of ReCaS, that recipe represents our way to recruit new users and to consolidate the "catchment area" in terms of both number and awareness in the use of the infrastructure.

The minimisation of the total cost of ownership is hard to be faced by the owners of large scale computing systems, without affecting negatively the quality of service for the users. Modern datacenters, often included in distributed environments, appear to be ``elastic'', i.e., they are able to shrink or enlarge the number of local physical or virtual resources, also by recruiting them from private/public clouds. This increases the degree of dynamicity, making the infrastructure management more and more complex. Here, we report some advances in the realisation of an adaptive scheduling controller (ASC) which, by interacting with the datacenter resource manager, allows an effective and an efficient usage of resources. In particular, we focus on the mathematical formalisation of the ASC's kernel that allows to dynamically configure, in a suitable way, the datacenter resources manager. The described formalisation is based on a probabilistic approach that, starting from both a hystorical resources usage and on the actual users request of the datacenter resources, identifies a suitable probability distribution for queue time with the aim to perform a short term forecasting. The case study is the SCoPE datacenter at the University of Naples Federico II.

L'iniziativa SCoPE@Scuola nasce con l'obiettivo principale di stimolare la curiosità degli studenti delle scuole superiori sulle tematiche del supercalcolo quale strumento indispensabile per la soluzione dei problemi-sfida della scienza e della tecnologia. Le tematiche connesse con il supercalcolo coinvolgono, infatti, un ampio spazio di conoscenze e competenze: dalla modellizzazione matematica dei problemi alla progettazione di algoritmi, dall'implementazione del software alla progettazione e gestione di sistemi informatici complessi. L'iniziativa, partita alla fine del 2014, ha coinvolto tre scuole, è durata sei mesi (articolandosi in più incontri e tipi di attività) e si è conclusa, per questa prima edizione, alla fine del mese di maggio 2015.

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.

In many applications, the definition of fitting models that mimic the behaviour of experimental data is a challenging issue. In this paper a data-driven approach to represent (multi)exponential decay data is presented. We propose a fitting model based on smoothing splines defined by means of a differential operator. To solve the linear system involved in the smoothing exponential-polynomial spline definition, the main idea is to define B-spline like functions for the spline space, that are locally represented by Bernstein-like bases through Hermite interpolation conditions