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Fabio V. Difonzo

Publications and source records attributed to Fabio V. Difonzo.

11 recordsLinked to original sources
arXiv2025

Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering

Computational fluid dynamics (CFD) has become a cornerstone of modern water engineering, providing quantitative tools for the analysis, prediction, and management of complex hydraulic systems across a wide range of spatial and temporal scales. This survey reviews the mathematical models and numerical methods that underpin CFD applications in water engineering, from depth-averaged formulations such as the shallow water equations to fully three-dimensional Navier-Stokes models, as well as selected alternative modeling approaches. We examine the historical development of these models, their mathematical structure, and the numerical discretization and solution strategies commonly employed in practice, including finite difference, finite volume, and finite element methods. Beyond core solver technology, the survey addresses practical modeling issues such as source-term treatment, wetting and drying, turbulence modeling, free-surface representation, and computational efficiency. The growing role of data integration is also discussed, encompassing data assimilation, uncertainty quantification, and emerging machine-learning-assisted approaches that complement physics-based solvers. To illustrate the impact of modeling and numerical choices on real-world applications, representative case studies from large-scale water management systems are reviewed. By integrating theory, numerical techniques, and applied perspectives, this survey provides a unified reference for researchers and practitioners seeking to understand both the foundational principles and contemporary challenges of CFD in water engineering.

physics.flu-dyn↗
arXiv2025

Physics Informed Neural Networks for Learning the Horizon Size in Bond-Based Peridynamic Models

This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one-dimensional model of a linear microelastic material. We explore different kernel functions, including V-shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one-sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.

math.NA↗
arXiv2024

Stochastic Langevin Differential Inclusions with Applications to Machine Learning

Stochastic differential equations of Langevin-diffusion form have received significant attention, thanks to their foundational role in both Bayesian sampling algorithms and optimization in machine learning. In the latter, they serve as a conceptual model of the stochastic gradient flow in training over-parameterized models. However, the literature typically assumes smoothness of the potential, whose gradient is the drift term. Nevertheless, there are many problems for which the potential function is not continuously differentiable, and hence the drift is not Lipschitz continuous everywhere. This is exemplified by robust losses and Rectified Linear Units in regression problems. In this paper, we show some foundational results regarding the flow and asymptotic properties of Langevin-type Stochastic Differential Inclusions under assumptions appropriate to the machine-learning settings. In particular, we show strong existence of the solution, as well as an asymptotic minimization of the canonical free-energy functional.

math.OC↗
arXiv2024

Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties

Letting $P$ be a convex polytope in $\mathbb{R}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in $\mathbb{R}^n$ with at most $n-d-1$ vertices, and provide results about continuity and differentiability for the corresponding set-valued maps.

math.MG↗
arXiv2024

Optimal Chaining of Vehicle Plans with Time Windows

For solving problems from the domain of Mobility-on-Demand (MoD), we often need to connect vehicle plans into plans spanning longer time, a process we call plan chaining. As we show in this work, chaining of the plans can be used to reduce the size of MoD providers' fleet (fleet-sizing problem) but also to reduce the total driven distance by providing high-quality vehicle dispatching solutions in MoD systems. Recently, a solution that uses this principle has been proposed to solve the fleet-sizing problem. The method does not consider the time flexibility of the plans. Instead, plans are fixed in time and cannot be delayed. However, time flexibility is an essential property of all vehicle problems with time windows. This work presents a new plan chaining formulation that considers delays as allowed by the time windows and a solution method for solving it. Moreover, we prove that the proposed plan chaining method is optimal, and we analyze its complexity. Finally, we list some practical applications and perform a demonstration for one of them: a new heuristic vehicle dispatching method for solving the static dial-a-ride problem. The demonstration results show that our proposed method provides a better solution than the two heuristic baselines for the majority of instances that cannot be solved optimally. At the same time, our method does not have the largest computational time requirements compared to the baselines. Therefore, we conclude that the proposed optimal chaining method provides not only theoretically sound results but is also practically applicable.

math.OC↗
arXiv2024

Nonnegative moment coordinates on finite element geometries

In this paper, we introduce new generalized barycentric coordinates (coined as {\em moment coordinates}) on nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $n$ vertices (nodes) in $\mathbb{R}^2$, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $n$, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.

math.NA↗
arXiv2024

A Randomized Runge-Kutta Method for time-irregular delay differential equations

In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions $f=f(t,x,z)$ that are Lipschitz continuous with respect to $x$ but only Hölder continuous with respect to $(t,z)$. We give a construction of the randomized two-stage Runge-Kutta scheme for DDEs and investigate its upper error bound in the $L^p(Ω)$-norm for $p\in [2,+\infty)$. Finally, we report on results of numerical experiments.

math.NA↗
arXiv2023

Convergence Analysis of a Spectral Numerical Method for a Peridynamic Formulation of Richards' Equation

We study the implementation of a Chebyshev spectral method with forward Euler integrator to investigate a peridynamic nonlocal formulation of Richards' equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.

math.NA↗
arXiv2023

A preliminary model for optimal control of moisture content in unsaturated soils

In this paper we introduce an optimal control approach to Richards' equation in an irrigation framework, aimed at minimizing water consumption while maximizing root water uptake. We first describe the physics of the nonlinear model under consideration, and then develop the first-order necessary optimality conditions of the associated boundary control problem. We show that our model provides a promising framework to support optimized irrigation strategies, thus facing water scarcity in irrigation. The characterization of the optimal control in terms of a suitable relation with the adjoint state of the optimality conditions is then used to develop numerical simulations on different hydrological settings, that supports the analytical findings of the paper.

math.OC↗