184 research outputs found
Fast Collocation methods for Volterra Integral equations of convolution type
No full textIn this paper we present fast discrete collocation methods forVolterra integral equations of Hammerstein type, where the Laplace
transform of the kernel is known a priori. To compute the numerical solution over N time steps, the constructed methods require
O(N log(N )) operations, O(log(N )) memory and preserve the order of accuracy of the corresponding exact collocation methods.
The numerical experiments confirm the expected accuracy and the computational cost
Fast Runge-Kutta methods for nonlinear convolution systems of Volterra Integral equations
No full textIn this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra
integral equations of Hammerstein type are constructed. The coefficients of the methods
are expressed in terms of the values of the Laplace transform of the kernel. These
methods have been suitably constructed in order to be implemented in an efficient way,
thus leading to a very low computational cost both in time and in space. The order of
convergence of the constructed methods is studied. The numerical experiments confirm
the expected accuracy and computational cost
Efficient numerical methods for Volterra integral equations of Hammerstein type
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications.
The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution.
The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs.
It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N2t) operations and O(Nt) memory space.
In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations.
The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods.
We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations.
We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero.
The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods.
The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel.
We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution.
It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour.
For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE.
Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term.
In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
High performance parallel numerical methods for Volterra equations with weakly singular kernels
No full textNon-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations
Asymptotic periodicity of nonlinear discrete Volterra equations and applications
No full textSufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations
Il Risorgimento italiano visto da Parigi: la figura del brigante postunitario in "Le Monde illustré"
No full textAbstract in inglese
The essay illustrates the French dimension of the way of seeing the brigand in the second half of the 19th century, through writing and graphic organisation strategies typical of a new press capable of having an impact on public opinion. In particular, the newspaper examined, «Le Monde illustré» (of which the author analyses the 714 issues between 1857 and 1870) is the epitome of both the 'political' emptying of the French brigand from romantic hero to humorous character, and of the representation of the Italian brigand as a figure assimilable to backward realities not only in Europe, but also in North Africa (Algeria, Morocco) and America (Mexico). The author argues that the national-popular declination, between the burlesque and the macchiettistico, of the ambiguous figure of the brigand, endowed with an extraordinary semantic ductility, prevents the transalpine newspaper from understanding the complexity of the Italian Risorgimento even in the most tragic phase of the political-war conflict.Abstract in italiano
Il saggio illustra la dimensione francese del modo di vedere il brigante nel secondo Ottocento, attraverso strategie di scrittura e di organizzazione grafica proprie di una nuova stampa capace di avere impatto sull’opinione pubblica nazionale. In particolare, il giornale preso in esame, «Le Monde illustré» (di cui si analizzano i 714 fascicoli compresi tra il 1857 e il 1870) si fa l’epitome sia dello svuotamento “politico” del brigante francese da eroe romantico a personaggio umoristico, sia della rappresentazione del brigante italiano come figura assimilabile a realtà arretrate non solo in Europa, ma anche nel Nord Africa (Algeria, Marocco) e in America (Messico). L’autrice sostiene che la declinazione nazional-popolare, tra il burlesco e il macchiettistico, della figura ambigua del brigante, dotata di una straordinaria duttilità semantica, impedisce al giornale d’Oltralpe di comprendere la complessità del Risorgimento italiano anche nella fase più tragica del conflitto politico-bellico
Capitale sociale
No full textil libro passa in rassegna le origini e le definizioni del capitale sociale negli scritti di Bourdieu, Loury, Coleman e altri autori, distinguendo quattro fonti di capitale sociale ed esaminando le loro dinamiche. Le applicazioni del concetto nella letteratura sociologica enfatizzano il suo ruolo nel controllo sociale, nel supporto familiare e nei benefici mediati dalle reti extra-familiari. Illustrerò esempi di ciascuna di queste funzioni positive. Per ottenere un quadro equilibrato delle forze in gioco, meritano attenzione anche le conseguenze negative degli stessi processi. Analizzerò quindi le quattro applicazioni e le illustrerò con esempi rilevanti. Scritti recenti sul capitale sociale hanno esteso il concetto da risorsa individuale a caratteristica delle comunità e persino delle nazioni. Le pagine finali descrivono questa estensione concettuale e ne esaminano i limiti. Affermerò inoltre che il capitale sociale, come sintesi delle conseguenze positive della socialità, ha un posto definito nella teoria sociologica; tuttavia, un’estensione eccessiva del concetto può compromettere il suo valore euristico.The book is the translation of the critical essay of Alejandro Portes. It describes the origins and tha main definitions of Social Capital starting from the contributions of Bourdieu, Loury, Coleman and other scholars.
The author distinguishes four sources of social capital and analizes their dynamics.
In the sociological literature, the applications of the concept enphasize the social capital role in the social control, in the family support and in the benefits coming from the extra familiar networks.
In the book there will be shown not only examples of these positive functions, but also the negative effects of the same processes
EFFICIENT NUMERICAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS OF HANMERSTEIN TYPE
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications. The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution. The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs. It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N^2t) operations and O(Nt) memory space. In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations. The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods. We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations. We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero. The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods. The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel. We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution. It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour. For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE. Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term. In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
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