arXiv subjects

Felice Iavernaro

Publications and source records attributed to Felice Iavernaro.

At least 19 recordsLinked to original sources

arXiv2026

Maximum Entropy Least Squares Solutions of Overdetermined Linear Systems

We investigate the theoretical foundations of a recently introduced entropy-based formulation of weighted least squares for the approximation of overdetermined linear systems, motivated by robust data fitting in the presence of sparse gross errors. The weight vector is interpreted as a discrete probability distribution and is determined by maximizing Shannon entropy under normalization and a prescribed mean squared error (MSE) constraint. Unlike classical ordinary least squares, where the error level is an output of the minimization process, here the MSE value plays the role of a control parameter, and entropy selects the least biased weight distribution achieving the prescribed accuracy. The resulting optimization problem is nonconvex due to the nonlinear coupling between the weights and the solution induced by the residual constraint. We analyze the associated optimality system and characterize stationary points through first- and second-order conditions. We prove the existence and local uniqueness of a smooth branch of entropy-maximizing configurations emanating from the ordinary least squares solution and establish its global continuation under suitable nondegeneracy conditions. Furthermore, we investigate the asymptotic regime as the prescribed MSE tends to zero and show that, under appropriate assumptions, the limiting configuration concentrates on a largest subset of data consistent with the linear model, thus suppressing the influence of outliers. Two numerical experiments illustrate the theoretical findings and confirm the robustness properties of the method.

math.NA↗

arXiv2025

A Multi-Order Extension of Fractional HBVMs (FHBVMs)

The efficient numerical solution of fractional differential equations has been recently tackled through the definition of Fractional HBVMs (FHBVMs), a class of Runge-Kutta type methods. Corresponding Matlab (c) codes have been also made available on the internet, proving to be very competitive w.r.t. existing ones. However, so far, FHBVMs have been given for solving systems of fractional differential equations with the same order of fractional derivative, whereas the numerical solution of multi-order problems (i.e., problems in which different orders of fractional derivatives occur) has not been handled, yet. Due to their relevance in applications, in this paper we propose an extension of FHBVMs for addressing fractional multi-order problems, providing full details for such an approach. A corresponding Matlab (c) code, handling the case of two different fractional orders, is also made available, proving very effective for numerically solving these problems.

math.NA↗

arXiv2025

High-order Gauss-Legendre methods admit a composition representation and a conjugate-symplectic counterpart

One of the most classical pairs of symplectic and conjugate-symplectic schemes is given by the Midpoint method (the Gauss-Runge-Kutta method of order 2) and the Trapezoidal rule. These can be interpreted as compositions of the Implicit and Explicit Euler methods, taken in direct and reverse order, respectively. This naturally raises the question of whether a similar composition structure exists for higher-order Gauss-Legendre methods. In this paper, we provide a positive answer by first examining the fourth-order case and then outlining a generalization to higher orders.

math.NA↗

arXiv2025

On the use of the principle of maximum entropy to improve the robustness of bivariate spline least-squares approximation

We consider fitting a bivariate spline regression model to data using a weighted least-squares cost function, with weights that sum to one to form a discrete probability distribution. By applying the principle of maximum entropy, the weight distribution is determined by maximizing the associated entropy function. This approach, previously applied successfully to polynomials and spline curves, enhances the robustness of the regression model by automatically detecting and down-weighting anomalous data during the fitting process. To demonstrate the effectiveness of the method, we present applications to two image processing problems and further illustrate its potential through two synthetic examples. Unlike the standard ordinary least-squares method, the maximum entropy formulation leads to a nonlinear algebraic system whose solvability requires careful theoretical analysis. We provide preliminary results in this direction and discuss the computational implications of solving the associated constrained optimization problem, which calls for dedicated iterative algorithms. These aspects suggest natural directions for further research on both the theoretical and algorithmic fronts.

stat.ME↗

arXiv2025

Analysis and implementation of collocation methods for fractional differential equations

Recently, the class of Runge-Kutta type methods named Fractional HBVMs (FHBVMs) has been introduced for the numerical solution of initial value problems of fractional differential equations, and a corresponding Matlab software has been released. Though an error analysis has already been given, a corresponding linear stability analysis is still lacking. We here provide such an analysis, together with some improvements concerning the mesh selection. This latter has been implemented into a new version of the code, which is available on the web.

math.NA↗

arXiv2025

Does the fluid-static equilibrium of a self-gravitating isothermal sphere of van der Waals' gas present multiple solutions?

We take up the investigation we left in the future-work stack in Giordano \textit{et al.} [``Fluid statics of a self-gravitational isothermal sphere of van der Waals' gas,'' Phys. Fluids \textbf{36}, 056127 (2024)], in which we pointed out the obvious necessity to inquire about the existence or absence of values of the characteristic numbers \itm{\alphay} and \itm{\betay} in correspondence to which the perfect-gas model's self gravitational effects, namely, upper boundedness of the gravitational number, spiraling behavior of peripheral density, oscillating behavior of central density, and the existence of multiple solutions corresponding to the same value of the gravitational number, appear also for the van der Waals' model. The development of our investigation brings to the conversion of our M$_{2}$ scheme based on a second-order differential equation into an equivalent system of two first-order differential equations that incorporates Milne's homology invariant variables. The converted scheme \fomt\ turns out to be much more efficacious than the M$_{2}$ scheme in terms of numerical calculations' easiness and richness of results. We use the perfect-gas model as benchmark to test the \fomt\ scheme; we re-derive familiar results and put them in a more general and rational perspective that paves the way to deal with the van der Waals' gas model. We introduce variable transformations that turn out to be the key to study (almost) analytically the monotonicity of the peripheral density with respect to variations of the gravitational number. The study brings to the proof that the gravitational number is not constrained by upper boundedness, the peripheral density does not spiral, and the central density does not oscillate for any couple of values assumed by the characteristic numbers $α$ and $β$; however, multiple solutions ...

astro-ph.GA↗

arXiv2024

A shooting-Newton procedure for solving fractional terminal value problems

In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness.

math.NA↗

arXiv2024

Numerical solution of FDE-IVPs by using Fractional HBVMs: the fhbvm code

In this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab code fhbvm, which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.

math.NA↗

arXiv2023

Fluid statics of a self-gravitating isothermal sphere of van der Waals' gas

We subject to scrutiny the physical consistency of adopting the perfect-gas thermodynamic model within self-gravitation circumstances by studying the fluid statics of a self-gravitating isothermal sphere with the van der Waals' thermodynamic model, whose equation of state features well-known terms that account for molecular attraction and size. The governing equations are formulated for any thermodynamic model with two intensive degrees of freedom, applied with the van der Waals' model and solved numerically in nondimensional form by finite-difference algorithms. After a brief summary of thermodynamic characteristics possessed by the van der Waals' model, and relevant to the present study, we proceed to the description of the results in terms of comparative graphs illustrating radial profiles of density, pressure and gravitational field. We complement them with graphs that compare the dependence of central and wall densities on gravitational number for both perfect-gas and van der Waals' models and that attest dramatically and unequivocally how the presence of molecular-attraction and -size terms removes questionable fluid-statics results systematically found accompanying the perfect-gas model in standard treatments. We also describe, within a very brief and preliminary digression, how the sanitising action of the mentioned terms affects the thermodynamics of the isothermal sphere by providing evidence of how the gravitational correction to entropy corresponding to the van der Waals' model makes sure that there is no risk of gravothermal catastrophes, negative specific heats, and thermal instabilities. Furthermore, we investigate the phenomenology related to self-gravitationally induced both liquid-gas phase equilibria and metastable-gas states and we describe how they arise naturally and self-consistently from the governing equations. We conclude with a summary of the main results ...

physics.flu-dyn↗

arXiv2023

An entropy-based approach for a robust least squares spline approximation

We consider the weighted least squares spline approximation of a noisy dataset. By interpreting the weights as a probability distribution, we maximize the associated entropy subject to the constraint that the mean squared error is prescribed to a desired (small) value. Acting on this error yields a robust regression method that automatically detects and removes outliers from the data during the fitting procedure, by assigning them a very small weight. We discuss the use of both spline functions and spline curves. A number of numerical illustrations have been included to disclose the potentialities of the maximal-entropy approach in different application fields.

math.NA↗

arXiv2023

A probabilistic approach to the twin prime and cousin prime conjectures

We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$ and $p_n$ being two consecutive primes, and evaluates the probability $q_n$ that such an interval contains a twin prime and a cousin prime. Combining Merten's third theorem with the properties of the binomial distribution, we show that $q_n$ approaches $1$ as $n \to \infty$. A study of the convergence properties of the sequence $\{q_n\}$ allows us to propose a new, more stringent conjecture concerning the existence of infinitely many twin and cousin primes. In accord with the Hardy-Littlewood conjecture, it is also shown that twin and cousin primes share the same asymptotic distribution.

math.NT↗

arXiv2022

A new framework for polynomial approximation to differential equations

In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is further extended to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

math.NA↗

arXiv2022

A didactically motivated reexamination of a particle's quantum mechanics with square-well potentials

We address two questions regarding square-well potentials from a didactic perspective. The first question concerns whether or not the justification of the standard a priori omission of the potential's vertical segments in the analysis of the eigenvalue problem is licit. The detour we follow to find out the answer considers a trapezoidal potential, includes the solution, analytical and numerical, of the corresponding eigenvalue problem and then analyzes the behavior of that solution in the limit when the slope of the trapezoidal potential's ramps becomes vertical. The second question, obviously linked to the first one, pertains whether or not eigenfunction's and its first derivative's continuity at the potential's jump points is justified as a priori assumption to kick-off the solution process, as it is standardly accepted in textbook approaches to the potential's eigenvalue problem.

quant-ph↗

arXiv2022

(Spectral) Chebyshev collocation methods for solving differential equations

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.

math.NA↗

arXiv2022

Continuous-Stage Runge-Kutta approximation to Differential Problems

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge-Kutta methods, which is here recalled and revisited for general differential problems.

math.NA↗

arXiv2022

Arbitrary high-order methods for one-sided direct event location in discontinuous differential problems with nonlinear event function

In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of energy-conserving methods for Poisson systems. The actual implementation of the methods is fully discussed, with a particular emphasis to the problem at hand. Some numerical tests are reported, to assess the theoretical findings.

math.NA↗

arXiv2022

Arbitrarily high-order energy-conserving methods for Poisson problems

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings.

math.NA↗

arXiv2021

Maximal-entropy driven determination of weights in least-square approximation

We exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one single problem in least-square approximation. We provide evidence of the method's capabilities and performance through its application to representative test cases by working with polynomials as a first step. We conclude by formulating suggestions for future work to improve the version of the method we present in this paper.

math.NA↗