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3. On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Author

Auricchio, Ferdinando, Belardo, Maria Roberta, Fabiani, Gianluca, Calabrò, Francesco, and Pascaner, Ariel F.

Subjects
Mathematics - Numerical Analysis, Computer Science - Artificial Intelligence, 65D05
Abstract

In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. The focus is then on the accuracy of the interpolation outside of the given sampling interpolation nodes when they are the equispaced, the Chebychev, and the randomly selected ones. The study is motivated by the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when growing the number of interpolation nodes, we raise the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge's function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training.

Published
2023

4. Effectiveness of a new commercial ocular spray containing Biosecur in reducing conjunctival microbial flora as surgical prophylaxis in patients undergoing cataract surgery

Author

Toro, Mario Damiano, Concilio, Marina, Laezza, Maria Paola, Mazzantini, Diletta, Ghelardi, Emilia, Gelso, Aldo, Sbordone, Mario, Orrico, Ada, De Angelis, Vincenzo, Calabrò, Francesco, De Rosa, Pasquale, Mazzeo, Salvatore, Iula, Dora, Perillo, Francesco, Rossi, Claudia, and Costagliola, Ciro

Published
2024
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5. The effect of time discretization on the solution of parabolic PDEs with ANNs

Author

Calabrò, Francesco, Cuomo, Salvatore, di Serafino, Daniela, Izzo, Giuseppe, and Messina, Eleonora

Subjects
Mathematics - Numerical Analysis
Abstract

We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $\theta$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties of the methods. The results of numerical experiments confirm that ELM-based solution techniques combined with BDF methods can provide high-accuracy solutions of parabolic PDEs., Comment: 21 pages

Published
2022

7. Numerical Solution of Stiff ODEs with Physics-Informed RPNNs

Author

Galaris, Evangelos, Fabiani, Gianluca, Calabrò, Francesco, di Serafino, Daniela, and Siettos, Constantinos

Subjects
Mathematics - Numerical Analysis, Computer Science - Machine Learning, 65L04, 68T07, 65D12, 60B20
Abstract

We propose a numerical method based on physics-informed Random Projection Neural Networks for the solution of Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) with a focus on stiff problems. We address an Extreme Learning Machine with a single hidden layer with radial basis functions having as widths uniformly distributed random variables, while the values of the weights between the input and the hidden layer are set equal to one. The numerical solution of the IVPs is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights by the Gauss-Newton method, using a simple adaptive scheme for adjusting the time interval of integration. To assess its performance, we apply the proposed method for the solution of four benchmark stiff IVPs, namely the Prothero-Robinson, van der Pol, ROBER and HIRES problems. Our method is compared with an adaptive Runge-Kutta method based on the Dormand-Prince pair, and a variable-step variable-order multistep solver based on numerical differentiation formulas, as implemented in the \texttt{ode45} and \texttt{ode15s} MATLAB functions, respectively. We show that the proposed scheme yields good approximation accuracy, thus outperforming \texttt{ode45} and \texttt{ode15s}, especially in the cases where steep gradients arise. Furthermore, the computational times of our approach are comparable with those of the two MATLAB solvers for practical purposes.

Published
2021

9. Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines

Author

Fabiani, Gianluca, Calabrò, Francesco, Russo, Lucia, and Siettos, Constantinos

Subjects
Mathematics - Numerical Analysis, Mathematics - Analysis of PDEs, Mathematics - Dynamical Systems, 65-XX, 65Nxx, 65-XX, 65P30
Abstract

We address a new numerical scheme based on a class of machine learning methods, the so-called Extreme Learning Machines with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous mixed boundary conditions, and, (b) the one and two-dimensional Liouville-Bratu-Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of accuracy and size of the grid) of the proposed numerical machine learning scheme is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element methods (FEM). We show that the proposed ELM numerical method outperforms both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids.

Published
2021
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10. Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

Author

Calabrò, Francesco, Fabiani, Gianluca, and Siettos, Constantinos

Subjects
Mathematics - Numerical Analysis
Abstract

We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a single hidden layer with sigmoidal functions and fixed, random, internal weights and biases can be used to compute accurately a collocation solution. The choice to fix internal weights and bias leads to the so-called Extreme Learning Machine network. We discuss how to determine the range for both internal weights and biases in order to obtain a good underlining approximating space, and we explore the required number of collocation points. We demonstrate the efficiency of the proposed method with several one-dimensional diffusion-advection-reaction problems that exhibit steep behaviors, such as boundary layers. The boundary conditions are imposed directly as collocation equations. We point out that there is no need of training the network, as the proposed numerical approach results to a linear problem that can be easily solved using least-squares. Numerical results show that the proposed method achieves a good accuracy. Finally, we compare the proposed method with finite differences and point out the significant improvements in terms of computational cost, thus avoiding the time-consuming training phase.

Published
2020
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11. Evaluation Aspects in the Strategic Planning of a Reticular DMO for the Promotion of Inner Areas of Calabria

Author

Calabrò, Francesco, Lorè, Immacolata, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Rocha, Ana Maria A. C., editor, Garau, Chiara, editor, Scorza, Francesco, editor, Karaca, Yeliz, editor, and Torre, Carmelo M., editor

Published
2023
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12. The Strategic Planning for the Promotion of Cultural Tourism in a Wide Area of Calabria: The Armeni Valley

Author

Calabrò, Francesco, Lorè, Immacolata, Viglianisi, Angela, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Rocha, Ana Maria A. C., editor, Garau, Chiara, editor, Scorza, Francesco, editor, Karaca, Yeliz, editor, and Torre, Carmelo M., editor

Published
2023
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16. On the Roads of Bruttium Between Italìa and Kalavrìa. Rediscovering Ancient and New Cultural Landscapes

Author

Mormino, Mauro, Monaca, Mariangela, Calabrò, Francesco, Castelli, Emanuele, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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17. The Journey of Ulysses: A Cultural Itinerary Among the Shores of the Mediterranean for the Promotion of Dialogue and Sustainable Development as Tools for Peace and Territorial Growth

Author

Alberotanza, Roberta, Calabrò, Francesco, Monaca, Mariangela, Tramontana, Carmela, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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18. Economic Feasibility of a Project for the Reuse of the Old Hospital of Nicotera as a Center for Eating Disorders and for the Enhancement of the Mediterranean Diet

Author

Santucci, Luca, Cassalia, Giuseppina, Calabrò, Francesco, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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19. The Cultural Routes of the Council of Europe and the Journey of Ulysses: Shared Values and Good Practices

Author

Alberotanza, Roberta, Calabrò, Francesco, Monaca, Mariangela, Tramontana, Carmela, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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20. The Management Models of a Tourist Destination in Italy

Author

Viglianisi, Angela, Calabrò, Francesco, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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21. New Technologies for Accessibility and Enhancement of Cultural Heritage Sites. The Archaeological Areas of Locride

Author

Calabrò, Francesco, Cassalia, Giuseppina, Fragomeni, Paolo, Lorè, Immacolata, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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22. A Project of Enhancement and Integrated Management: The Cultural Heritage Agency of Locride

Author

Calabrò, Francesco, Cassalia, Giuseppina, Lorè, Immacolata, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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23. Accessible Culture: Guidelines to a Cultural Accessibility Strategic Plan (C.A.S.P.) for MArRC Museum

Author

Cassalia, Giuseppina, Ventura, Claudia, Bagnato, Francesco, Calabrò, Francesco, Malacrino, Carmelo, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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24. Economic Feasibility of an Integrated Program for the Enhancement of the Byzantine Heritage in the Aspromonte National Park. The Case of Staiti

Author

Spatari, Giovanna, Lorè, Immacolata, Viglianisi, Angela, Calabrò, Francesco, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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25. The Valuation of Unused Public Buildings in Support of Culture-Led Inner Areas’ Small Villages NRRP Strategies: An Application Model in Southern Italy

Author

Cassalia, Giuseppina, Calabrò, Francesco, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Calabrò, Francesco, editor, Della Spina, Lucia, editor, and Piñeira Mantiñán, María José, editor

Published
2022
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33. Measures to Face Population Decline of Small Villages: Sustainable and Integrated Energy Strategies for the Internal Areas

Author

Calabrò, Francesco, Massimo, Domenico Enrico, Musolino, Mariangela, Pisello, Anna Laura, Editorial Board Member, Hawkes, Dean, Editorial Board Member, Bougdah, Hocine, Editorial Board Member, Rosso, Federica, Editorial Board Member, Abdalla, Hassan, Editorial Board Member, Boemi, Sofia-Natalia, Editorial Board Member, Mohareb, Nabil, Editorial Board Member, Mesbah Elkaffas, Saleh, Editorial Board Member, Bozonnet, Emmanuel, Editorial Board Member, Pignatta, Gloria, Editorial Board Member, Mahgoub, Yasser, Editorial Board Member, De Bonis, Luciano, Editorial Board Member, Kostopoulou, Stella, Editorial Board Member, Pradhan, Biswajeet, Editorial Board Member, Abdul Mannan, Md., Editorial Board Member, Alalouch, Chaham, Editorial Board Member, O. Gawad, Iman, Editorial Board Member, Nayyar, Anand, Editorial Board Member, Amer, Mourad, Series Editor, Naddeo, Vincenzo, editor, Choo, Kwang-Ho, editor, and Ksibi, Mohamed, editor

Published
2022
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36. Inclusive Strategic Programming: Methodological Aspects of the Case Study of the Jonian Valleys of Peloritani (Sicily, Italy)

Author

Bombino, Giuseppe, Calabrò, Francesco, Cassalia, Giuseppina, Errante, Lidia, Vinci, Viviana, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Misra, Sanjay, editor, Garau, Chiara, editor, Blečić, Ivan, editor, Taniar, David, editor, Apduhan, Bernady O., editor, Rocha, Ana Maria A. C., editor, Tarantino, Eufemia, editor, and Torre, Carmelo Maria, editor

Published
2021
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37. The Knowledge Phase of the Strategic Programming: The Case Study of the Jonian Valleys of Peloritani (Sicily, Italy)

Author

Bombino, Giuseppe, Calabrò, Francesco, Cassalia, Giuseppina, Errante, Lidia, Vinci, Viviana, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Misra, Sanjay, editor, Garau, Chiara, editor, Blečić, Ivan, editor, Taniar, David, editor, Apduhan, Bernady O., editor, Rocha, Ana Maria A. C., editor, Tarantino, Eufemia, editor, and Torre, Carmelo Maria, editor

Published
2021
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43. How to get high order without loosing efficiency for the resolution of systems of nonlinear equations: A short review of Shamanskii's m method

Author

Calabrò, Francesco and Polsinelli, Andrea

Subjects
Mathematics - Numerical Analysis, Mathematics - Optimization and Control, 65H10, 41A25, 65Y04
Abstract

We present relations between some recently proposed methods for the solution of a nonlinear system of equations. In particular, we review the Shamanskii's m method, that is an iterative method derived from Newton's method that converge with order m+1. We discuss efficient implementation of this method via matrix factorization and some relevant properties. We believe that recent developments in the research of solutions of systems of equations did not take sufficiently into account this method. The hope, with this paper, is to encourage the entire community to remember this simple method and use it for comparison when new methods are introduced. This work is dedicated to Prof. Elvira Russo: a very special teacher.

Published
2016

44. Fast formation of isogeometric Galerkin matrices by weighted quadrature

Author

Calabrò, Francesco, Sangalli, Giancarlo, and Tani, Mattia

Subjects
Mathematics - Numerical Analysis
Abstract

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for $p$ large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree $k$-refinement in isogeometric analysis.

Published
2016
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46. Quadrature Rules in the Isogeometric Galerkin Method: State of the Art and an Introduction to Weighted Quadrature

Author

Calabrò, Francesco, Loli, Gabriele, Sangalli, Giancarlo, Tani, Mattia, Patrizio, Giorgio, Editor-in-Chief, Canuto, Claudio, Series Editor, Coletti, Giulianella, Series Editor, Gentili, Graziano, Series Editor, Malchiodi, Andrea, Series Editor, Marcellini, Paolo, Series Editor, Mezzetti, Emilia, Series Editor, Moscariello, Gioconda, Series Editor, Ruggeri, Tommaso, Series Editor, Giannelli, Carlotta, editor, and Speleers, Hendrik, editor

Published
2019
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49. Decision Support Model for Conservation, Reuse and Valorization of the Historic Cultural Heritage

Author

Della Spina, Lucia, Calabrò, Francesco, Hutchison, David, Series Editor, Kanade, Takeo, Series Editor, Kittler, Josef, Series Editor, Kleinberg, Jon M., Series Editor, Mattern, Friedemann, Series Editor, Mitchell, John C., Series Editor, Naor, Moni, Series Editor, Pandu Rangan, C., Series Editor, Steffen, Bernhard, Series Editor, Terzopoulos, Demetri, Series Editor, Tygar, Doug, Series Editor, Weikum, Gerhard, Series Editor, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Misra, Sanjay, editor, Stankova, Elena, editor, Torre, Carmelo M., editor, Rocha, Ana Maria A.C., editor, Taniar, David, editor, Apduhan, Bernady O., editor, Tarantino, Eufemia, editor, and Ryu, Yeonseung, editor

Published
2018
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