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Martedì 07 gennaio 2025, ore 15.00, Sala di Consiglio
Alessio Oliviero
Università di Roma La Sapienza
A TV-flow scheme for first order ergodic mean field games
Abstract:
In this talk, we introduce a class of first order ergodic mean field games (MFG) arising from ecology, and a numerical method for the approximation of their solution. The algorithms we propose are inspired by best response dynamics in evolutionary game theory, the minimisation of the highest income of a game, and De Giorgi's minimising movement schemes. Unlike other non-standard schemes for MFG, this kind of approximation does not assume any variational structure of the system. To conclude, we present and compare various numerical simulations both in 1D and 2D domains.
Giovedì 12 dicembre 2024, ore 15.30, Sala di Consiglio
Nicholas Corbin
University of California San Diego
Computing Solutions to the Polynomial-Polynomial Regulator Problem
Abstract:
We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. In this talk, I will present some of our recent work developing scalable numerical methods for computing these polynomial feedback laws. Their performance will be illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control, and then on a semidiscretization of a partial differential equation.
Martedì 10 dicembre 2024, ore 15.00, Sala di Consiglio
Monica Nonino
University of Vienna
Towards an Arbitrary Lagrangian Eulerian MOR framework for advection dominated problems
Abstract:
Advection dominated problems represent still nowadays a great challenge for the Model Order Reduction community, because of their intrinsic difficult nature. In this talk we will focus on hyperbolic problems with self-similar solutions. I will present a MOR approach for transport dominated problems, in the non-parametrized and in the parametrized setting, with a particular focus on the SOD problem in 1D, the DMR problem and the triple point problem in 2D. The approach is based on the definition of suitable deformation maps from the physical domain into itself: these maps are obtained by means of an optimization procedure. Once the map is found, a standard POD on the modified snapshots is performed. For the online phase, an Artificial Neural Network approach is used to compute the coefficients of the online solution. The whole procedure represents a first step towards an ALE approach, and is applied to problems where the solution presents multiple travelling discontinuities (shocks, rarefactions), whose location in the physical domain is unknown. Promising results are shown, to highlight the good performance of the whole methodology.
Martedì 19 novembre 2024, ore 15.00, Aula Tullio Levi-Civita
Emiliano Cristiani
IAC-CNR
Microscopic and macroscopic models for pedestrian flow with variable maximal density
Abstract:
In this paper we deal with pedestrian modeling, aiming at simulating crowd behavior in normal and emergency scenarios, including highly congested mass events. We will present two models: the first one is an agent-based, continuous-in-space, discrete-in-time, nondifferential model, where pedestrians have finite size and are compressible to a certain extent. The model also takes into account the pushing behavior appearing at extremely high densities. The second one is a macroscopic (fluid dynamics) model characterized by the fact that the maximal density reachable by the crowd – usually a fixed model parameter – is instead a state variable. 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. Interestingly, both models are able to reproduce the concave/concave fundamental diagram with a "double hump" (i.e. with a second peak) which shows up in the experimental literature when high-density crowds are observed.
Martedì 05 novembre 2024, ore 15.00, Sala di Consiglio
Maria Strazzullo
Politecnico di Torino
Dynamical Low-Rank Approximation for Nonlinear Feedback Control
Abstract:
Effective feedback control is essential for optimizing dynamical systems by minimizing a predefined cost function, thereby stabilizing the system toward a desired state. Despite its proven effectiveness, the applicability of feedback control is often limited by the high dimensionality of state spaces, especially in parametric settings. To address these challenges, we apply Riccati-based Dynamical Low-Rank Approximation (R-DLRA). In practice, the standard DLRA basis is enriched with information related to the solution of the State-Dependent Riccati Equations (SDREs), yielding efficient, accurate solutions for high-dimensional feedback control problems. To solve the SDRE solutions, we propose a Cascade Newton-Kleinman (C-NK) algorithm, which leverages prior parametric and time knowledge of the Riccati solution, to improve the convergence of Newton-based methods applied to SDREs across different parameters and time instances. Our approach significantly accelerates the solution process for infinite horizon optimal control by constructing a low-dimensional, compact representation of the evolving system, thereby enhancing both accuracy and real-time control across multiple parametric instances. The proposed R-DLRA approach demonstrates faster and more accurate performance than the full-order model, when compared to the standard DLRA, global Proper Orthogonal Decomposition (POD), and Riccati-based POD.
Martedì 28 ottobre 2024, ore 15.00, Sala di Consiglio
Giacomo Albi
Università di Verona
Feedback stabilization strategies for magnetically confined fusion plasma
Abstract:
The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated need the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this talk, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides feedback on the equation of motion based on an instantaneous prediction of the discretized system. This permits to directly embed the minimization of a given cost functional into the particle interactions of the corresponding Vlasov model. Furthermore, we will show that such control strategy can be conveniently extended to plasma dynamics in presence of uncertainties which severely affect this process due to erroneous measurements and missing information. The numerical results demonstrate the validity of our control approach and the capability of an external magnetic field, even if in a simplified setting, to lead the plasma far from the boundaries.
Martedì 15 ottobre 2024, ore 15.00, Sala di Consiglio
Adriano Festa
Politecnico di Torino
A network model for urban planning
Abstract:
In this seminar we present a mathematical model to describe the evolution of a city, which is determined by the interaction of two large populations of agents, workers and firms. The map of the city is described by a network with the edges representing at the same time residential areas and communication routes. The two populations compete for space while interacting through the labour market. The resulting model is described by a two population Mean-Field Game system coupled with an Optimal Transport problem. We prove existence and uniqueness of the solution and we provide some numerical tools to develop several numerical simulations. This is a joint work with Fabio Camilli (Sapienza Roma) and Luciano Marzufero (Libera Università di Bolzano).
Giovedì 05 settembre 2024, ore 14.00, Sala di Consiglio
Andreas Meister
University of Kassel
Modified Patankar-Runge-Kutta Methods: Introduction, Analysis and Numerical Applications
Abstract:
Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production-destruction systems. They adapt explicit Runge-Kutta schemes in a way to ensure positivity and conservation irrespective of the time step size. We introduce a general definition of MPRK schemes and present a thorough investigation of necessary as well as sufficient conditions to derive first, second and third order accurate MPRK schemes. The theoretical results will be confirmed by numerical experiments in which MPRK schemes are applied to solve non-stiff and stiff systems of ordinary differential equations. Furthermore, we present an investigation of MPRK schemes in the context of convection-diffusion-reaction equations with source terms of production-destruction type.
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