Martedì 13 Febbraio 2024, ore 15.00, Sala di Consiglio
Investigating Multi-lane Traffic Flow Models: from Micro to Macro
This talk is devoted to the modeling and stability of multi-lane traffic flow in both microscopic and macroscopic frameworks. Firstly, we explore the dynamics of lane changing in microscopic variables, presenting in particular a second-order microscopic hybrid model called the "Bando-Follow-the-Leader" model, in which simple lane changing conditions are proposed. Afterwards, we describe the derivation of novel first and second-order macroscopic multi-lane models that are obtained without postulating ad hoc micro-to-macro scalings. Furthermore, we investigate the equilibria for such models and establish conditions for their stability. Finally, we discuss some numerical tests.
Giovedì 25 Gennaio 2024, ore 16.00, Sala di Consiglio
Shanghai Jiao Tong University
Analog quantum simulation of partial differential equations
Quantum simulators were originally proposed to be helpful for simulating one partial differential equation (PDE) in particular – Schrodinger’s equation. If quantum simulators can be useful for simulating Schrodinger’s equation, it is hoped that they may also be helpful for simulating other PDEs. As with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality (costs scale exponentially in the dimension D of the PDE), which a quantum treatment might in certain cases be able to mitigate. To enable simulation of PDEs on quantum devices that obey Schrodinger’s equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger’s equations. In this talk, I will introduce the notion of Schrodingerisation: a procedure for transforming non-Schrodinger PDEs into a Schrodinger-form. This simple methodology can be used directly on analog or continuous quantum degrees of freedom – called qumodes, and not only on qubits. This continuous representation can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. I show how this method can also be applied to both autonomous and non-autonomous linear PDEs, certain nonlinear PDEs, nonlinear ODEs and also linear PDEs with random coefficients, which is important in uncertainty quantification. This formulation makes it more amenable to more near-term quantum simulation methods and enables simulation of PDEs that are not possible with qubit-based formulations in the near-term.
Martedì 07 novembre 2023, ore 15.00, Sala di Consiglio
Università di Ferrara
Exponential integrators for mean-field selective optimal control problems
We consider mean-field optimal control problems with selective action of the control, where the constraint is a continuity equation involving a non-local term and diffusion. First order optimality conditions are formally derived in a general framework, accounting for boundary conditions. Hence, the optimality system is used to construct a reduced gradient method, where we introduce a novel algorithm for the numerical realization of the forward and the backward equations, based on exponential integrators. We illustrate extensive numerical experiments on different control problems for collective motion in the context of opinion formation and pedestrian dynamics.
Martedì 26 settembre 2023, ore 15.00, Sala di Consiglio
University of Mainz
Haar-type stochastic Galerkin formulations for random hyperbolic systems
The idea to represent stochastic processes by orthogonal polynomials has been employed in uncertainty quantification and inverse problems. This approach is known as stochastic Galerkin formulation with a generalized polynomial chaos (gPC) expansion. The gPC expansions of the stochastic input are substituted into the governing equations. Then, they are projected by a Galerkin method to obtain deterministic evolution equations for the gPC coefficients. Applications of this procedure have been proven successful for diffusion and kinetic equa- tions. So far, results for general hyperbolic systems are not available. A problem is posed by the fact that the deterministic Jacobian of the projected system differs from the random Jacobian of the original system and hence hyperbolicity is not guaranteed. Applications to hyperbolic conservation laws are in general limited to linear and scalar hyperbolic equations. We analyze the loss of hyperbolicity for isentropic Euler equations. In particular, hy- perbolicity depends on the choice of gPC expansion. In particular, the dependency on a random input is described by Haar-type wavelet systems. Theoretical results are illus- trated numerically by CWENO-type reconstructions combined with a numerical entropy indicator that allow also for higher-order discretizations of balance laws.
Venerdì 22 settembre 2023, ore 11.00, Sala di Consiglio
Functional Analytic Insights into Mean Field Game Theory
Monotonicity conditions are crucial in Mean Field Game (MFG) theory, highlighted by the uniqueness results of Larry and Lions. This talk introduces a functional analytic framework to understand MFGs that satisfy monotonicity conditions. By leveraging ideas introduced in Hessian-Riemannian flows from optimization, we propose regularized versions of MFGs and construct contracting flows that can be used for numerical approximation. Our findings present a consolidated view of our prior works and give a different perspective on this class of problems.