arXiv subjects

Simone Cacace

Publications and source records attributed to Simone Cacace.

At least 19 recordsLinked to original sources

arXiv2026

The Wolf and the Cello: Modelling and design of multiple resonance suppressors in large string instruments

The wolf note is an acoustic instability that occurs in large bowed string instruments when a strong body resonance interacts with the vibrating string, producing amplitude modulation and loss of tonal control. Various wolf suppressors - tuned mass dampers attached to the string or to the instrument body - are used in practice to mitigate this effect. In this paper, we propose a mathematical model describing the coupled dynamics of a string and a two-dimensional body equipped with one or two wolf suppressors. Both string and body include elastic (second-order) and stiffness (fourth-order) contributions and can be excited either by plucking or bowing. Three performance indicators are introduced: The first one perceives the wolf-tone appearance, the second one quantifies the attenuation of the notes possibly caused by the wolf suppressor, and the third one measures the acoustic fidelity (in terms of spectrum) with respect to the original instrument. The proposed numerical tests give insights about optimal tuning and placement of one or two suppressors, achieving effective wolf-note suppression while preserving as much as possible the global tonal balance.

math.DS↗

arXiv2025

Agent-Based Optimal Control for Image Processing

We investigate the use of multi-agent systems to solve classical image processing tasks, such as colour quantization and segmentation. We frame the task as an optimal control problem, where the objective is to steer the multi-agent dynamics to obtain colour clusters that segment the image. To do so, we balance the total variation of the colour field and fidelity to the original image. The solution is obtained resorting to primal-dual splitting and the method of multipliers. Numerical experiments, implemented in parallel with CUDA, demonstrate the efficacy of the approach and its potential for high-dimensional data.

math.OC↗

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↗

arXiv2025

Modified Patankar Semi-Lagrangian Scheme for the Optimal Control of Production-Destruction systems

In this manuscript, we present a comprehensive theoretical and numerical framework for the control of production-destruction differential systems. The general finite horizon optimal control problem is formulated and addressed through the dynamic programming approach. We develop a parallel in space conservative scheme for the corresponding backward-in-time Hamilton-Jacobi-Bellman equation. Furthermore, we provide a suitable reconstruction algorithm for optimal controls and trajectories. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.

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

math.DS↗

arXiv2023

Reliable optimal controls for SEIR models in epidemiology

We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin's maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.

math.OC↗

arXiv2022

Approximation of the value function for optimal control problems on stratified domains

In optimal control problems defined on stratified domains, the dynamics and the running cost may have discontinuities on a finite union of submanifolds of RN. In [8, 5], the corresponding value function is characterized as the unique viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying additional viscosity conditions on the submanifolds. In this paper, we consider a semi-Lagrangian approximation scheme for the previous problem. Relying on a classical stability argument in viscosity solution theory, we prove the convergence of the scheme to the value function. We also present HJSD, a free software we developed for the numerical solution of control problems on stratified domains in two and three dimensions, showing, in various examples, the particular phenomena that can arise with respect to the classical continuous framework.

math.OC↗

arXiv2021

A dynamic programming approach for controlled fractional SIS models

We investigate a susceptible-infected-susceptible (SIS) epidemic model based on the Caputo-Fabrizio operator. After performing an asymptotic analysis of the system, we study a related finite horizon optimal control problem with state constraints. We prove that the corresponding value function is a viscosity solution of a dynamic programming equation. We then turn to the asymptotic behavior of the value function, proving its convergence to the solution of a stationary problem, as the planning horizon tends to infinity. Finally, we present some numerical simulations providing a qualitative description of the optimal dynamics and the value functions involved.

math.AP↗

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.

math.NA↗

arXiv2021

Constrained reachability problems for a planar manipulator

We address an optimal reachability problem for a planar manipulator in a constrained environment. After introducing the optmization problem in full generality, we practically embed the geometry of the workspace in the problem, by considering some classes of obstacles. To this end, we present an analytical approximation of the distance function from the ellipse. We then apply our method to particular models of hyper-redundant and soft manipulators, by also presenting some numerical experiments.

math.OC↗

arXiv2020

A policy iteration method for Mean Field Games

The policy iteration method is a classical algorithm for solving optimal control problems. In this paper, we introduce a policy iteration method for Mean Field Games systems, and we study the convergence of this procedure to a solution of the problem. We also introduce suitable discretizations to numerically solve both stationary and evolutive problems. We show the convergence of the policy iteration method for the discrete problem and we study the performance of the proposed algorithm on some examples in dimension one and two.

math.AP↗

arXiv2020

A Mean Field Games model for finite mixtures of Bernoulli and Categorical distributions

Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.

stat.ML↗

arXiv2020

Optimal reachability and grasping for a soft manipulator

We investigate optimal reachability and grasping problems for a planar soft manipulator, from both a theoretical and numerical point of view. The underlying control model describes the evolution of the symmetry axis of the device, which is subject to inextensibility and curvature constraints, a bending moment and a curvature control. Optimal control strategies are characterized with tools coming from the optimal control theory of PDEs. We run some numerical tests in order to validate the model and to synthetize optimal control strategies.

math.OC↗

arXiv2019

A Mean Field Games approach to Cluster Analysis

In this paper, we develop a Mean Field Games approach to Cluster Analysis. We consider a finite mixture model, given by a convex combination of probability density functions, to describe the given data set. We interpret a data point as an agent of one of the populations represented by the components of the mixture model, and we introduce a corresponding optimal control problem. In this way, we obtain a multi-population Mean Field Games system which characterizes the parameters of the finite mixture model. Our method can be interpreted as a continuous version of the classical Expectation-Maximization algorithm.

math.NA↗

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.

math.NA↗

arXiv2018

Modeling and Optimal Control of an Octopus Tentacle

We present a control model for an octopus tentacle, based on the dynamics of an inextensible string with curvature constraints and curvature controls. We derive the equations of motion together with an appropriate set of boundary conditions, and we characterize the corresponding equilibria. The model results in a system of fourth-order evolutive nonlinear controlled PDEs, generalizing the classic Euler's dynamic elastica equation, that we approximate and solve numerically introducing a consistent finite difference scheme. We proceed investigating a reachability optimal control problem associated to our tentacle model. We first focus on the stationary case, by establishing a relation with the celebrated Dubins' car problem. Moreover, we propose an augmented Lagrangian method for its numerical solution. Finally, we address the evolutive case obtaining first order optimality conditions, then we numerically solve the optimality system by means of an adjoint-based gradient descent method.

math.OC↗

arXiv2018

A measure theoretic approach to traffic flow optimization on networks

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective isto minimise/maximise macroscopic quantities, such as traffic volume or average speed,controlling few agents, for example smart traffic lights and automated cars. The measuretheoretic approach allows to study in a same setting local and nonlocal drivers interactionsand to consider the control variables as additional measures interacting with the driversdistribution. We also propose a gradient descent adjoint-based optimization method, ob-tained by deriving first-order optimality conditions for the control problem, and we providesome numerical experiments in the case of smart traffic lights for a 2-1 junction.

math.OC↗

arXiv2018

An ergodic problem for Mean Field Games: qualitative properties and numerical simulations

This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian $\bar H$ and of the effective drift $\bar b$. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where $\nabla_P \bar H(P,\alpha)$ coincides or not with $\bar b(P, \alpha)$. On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of $\bar H$ and $\bar b$. In particular, we provide a numerical estimate of the discrepancy $\nabla_P \bar H(P,\alpha)-\bar b(P, \alpha)$.

math.AP↗