arXiv subjects

Roberto Ferretti

Publications and source records attributed to Roberto Ferretti.

13 recordsLinked to original sources

arXiv2025

Numerical Hopf-Lax formulae for Hamilton-Jacobi equations on unstructured geometries

We consider a scheme of Semi-Lagrangian (SL) type for the numerical solution of Hamilton-Jacobi (HJ) equation on unstructured triangular grids. As it is well known, SL schemes are not well suited for unstructured grids, due to the cost of the point location phase; this drawback is augmented by the need for repeated minimization. In this work, we propose a scheme that works only on the basis of node values and connectivity of the grid. In a first version, we obtain a monotone scheme; then, applying a quadratic refinement to the numerical solution, we improve accuracy at the price of some extra computational cost. The scheme can be applied to both time-dependent and stationary HJ equations; in the latter case, we also study the construction of a fast policy iteration solver. We perform a theoretical analysis of the two versions, and validate them with an extensive set of examples, both in the time-dependent and in the stationary case.

math.NA↗

arXiv2024

A macroscopic pedestrian model with variable maximal density

In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd - usually a fixed model parameter - is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations. Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident a priori which relationship between density and flux will be actually observed, due to the time-varying maximal density. An a posteriori analysis shows that the observed fundamental diagram has an elongated "tail" in the congested region, thus resulting similar to the concave/concave fundamental diagram with a "double hump" observed in real crowds. The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available at https://gitlab.com/cristiani77/code_arxiv_2406.14649

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arXiv2022

An algorithmic guide for finite-dimensional optimal control problems

We survey the main numerical techniques for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider two classical examples, simple but significant enough to be enriched and generalized to other settings: Zermelo and Goddard problems. We provide sample of the codes used to solve them and make these codes available online. We discuss direct and indirect methods, Hamilton-Jacobi approach, ending with optimistic planning. The examples illustrate the pros and cons of each method, and we show how these approaches can be combined into powerful tools for the numerical solution of optimal control problems for ordinary differential equations.

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arXiv2021

Efficient implementation of characteristic-based schemes on unstructured triangular grids

Using characteristics to treat advection terms in time-dependent PDEs leads to a class of schemes, e.g., semi-Lagrangian and Lagrange-Galerkin schemes, which preserve stability under large Courant numbers, and may therefore be appealing in many practical situations. Unfortunately, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids, and thus prevent the use of large time-step schemes on complex geometries. In this paper, we perform an in-depth analysis of the main recipes available for characteristic location, and propose a technique to improve the efficiency of this phase, using additional information related to the advecting vector field. This results in a clear improvement of execution times in the unstructured case, thus extending the range of applicability of large time-step schemes.

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arXiv2020

Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations

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arXiv2019

Stochastic hybrid differential games and match race problems

We discuss the general framework of a stochastic two-player, hybrid differential game, and we apply it to the modelling of a "match race" between two sailing boats, namely a competition in which the goal of both players is to proceed in the windward direction, while trying to slow down the other player. We provide a convergent approximation scheme for the computation of the value function of the game, and we validate the approach on some typical racing scenarios.

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arXiv2018

Null controllability of parabolic equations with interior degeneracy and one-sided control

For $α\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\vert x \vert ^αu_x)_x\qquad (1<x<1),$$ which degenerates at the interior point $x=0$, for locally distributed controls acting only one side of the origin (that is, on some interval $(a,b)$ with $0<a<b<1$). Our main results guarantees that $P$ is null controllable if and only if it is weakly degenerate, that is, $α\in (0,1)$. So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case $α\in [1,2)$. Our approach is based on spectral analysis and the moment method. Indeed, we completely describe the eigenvalues and eigenfunctions of the associated stationary operator in terms of Bessel functions and their zeroes for both weakly and strongly degenerate problems. Hence, we obtain lower $L^2$ bounds for the eigenfunctions on the control region in the case $α\in [0,1)$ and deduce the lack of observability in the case of $α\in [1,2)$. We also provide numerical evidence to illustrate our theoretical results.

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arXiv2017

A Hybrid control approach to the route planning problem for sailing boats

We present an optimal hybrid control approach to the problem of stochastic route planning for sailing boats, especially in short course fleet races, in which minimum average time is an effective performance index. We show that the hybrid setting is a natural way of taking into account tacking/gybing maneuvers and other discrete control actions, and provide examples of increasing complexity to model the problem. Moreover, we carry out a numerical validation of the approach and show that results are in good agreement with theoretical and practical knowledge.

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arXiv2017

A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation

A numerical method for the two-dimensional, incompressible Navier--Stokes equations in vorticity--streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving unconditional stability without the need to solve linear systems beyond that required by the Poisson solver for the reconstruction of the streamfunction. A description of the discretization of Dirichlet boundary conditions for the semi-Lagrangian approach to diffusion terms is also presented. Numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method.

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arXiv2016

A semi-Lagrangian algorithm in policy space for hybrid optimal control problems

The mathematical framework of hybrid system is a recent and general tool to treat control systems involving control action of heterogeneous nature. In this paper, we construct and test a semi-Lagrangian numerical scheme for solving the Dynamic Programming equation of an infinite horizon optimal control problem for hybrid systems. In order to speed up convergence, we also propose an acceleration technique based on policy iteration. Finally, we validate the approach via some numerical tests in low dimension.

math.NA↗

arXiv2016

A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.

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arXiv2015

Blended numerical schemes for the advection equation and conservation laws

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations and nonlinear conservation laws. By coupling a macroscopic (Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of multiscale numerical method.

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arXiv2015

Flux form Semi-Lagrangian methods for parabolic problems

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and convergence analysis are proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems.

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