Events
Sensitivity of centrality measures in networks: the role of localization and spectral gaps
The main goal of the talk is to present some results on the effect of edge modifications on centrality measures, especially eigenvector centrality, from the point of view of localization. The latter property is related to sparsity and the size of the spectral gap. For simplicity, the focus is on the undirected case. In addition, I will describe an approach to finding gaps in the spectra of large symmetric matrices by means of randomized trace estimators. This part of the talk is based on work with Michele Rinelli and Igor Simunec.
Learning Value Functions with High-order Information
This talk is devoted to recent results on the approximation of high-dimensional HJB PDEs with supervised learning. We will discuss supervised learning of high- dimensional value functions using high-order data, a.k.a. Sobolev training. We expand our previous work on gradientaugmented learning for value functions to incorporate second-order (Hessian) information. Unlike gradient information that is available directly from PMP solvers, the Hessian values is recovered by solving a non-autonomous Riccati equation depending on the optimal trajectory. Hence, it is crucial to understand the dimension/nonlinearity regime where the cost of obtaining second-order information is justified in term of data efficiency.
A Journey Across Kinetic Theory and Numerical Methods
Kinetic theory provides a powerful mathematical framework for describing the collective behavior of large systems of interacting agents. Originally developed in the context of rarefied gas dynamics, kinetic models have progressively expanded their scope to include a wide range of applications such as plasma physics, granular media, traffic flow, collective behavior, and socio-economic systems. The increasing complexity of these models, often characterized by multiscale interactions, high dimensionality, and nonlinear dynamics, has posed significant challenges for both their theoretical analysis and their numerical approximation. This workshop has the aim to bring together researchers working at the intersection of kinetic theory, numerical analysis, and applications to complex systems. Particular attention will be devoted to the development of efficient and reliable numerical methods for kinetic and mean-field models, including multiscale schemes, structure-preserving discretizations, uncertainty quantification, and data-driven approaches. By fostering dialogue between mathematicians, computational scientists, and researchers from related disciplines, the meeting seeks to highlight recent advances in the field and stimulate new collaborations. During this event we will honor Lorenzo Pareschi's 60th birthday.
Pedestrian Models with Congestion Effects
We study the validity of the dissipative Aw-Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while indu- cing a steering behavior. Furthermore, we introduce a semi-implicit, structure- preserving, and asymptotic-preserving numerical scheme that can handle the numerical solution of the model efficiently. We perform the first numerical si- mulations of the dissipative Aw-Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical expe- riments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.
Can neural networks solve high dimensional optimal feedback control problems?
Deep Reinforcement Learning has established itself as a standard method for solving nonlinear optimal feedback control problems. In this method, the optimal value function (and in some variants also the optimal feedback law) is stored using a deep neural network. Hence, the applicability of this approach to high-dimensional problems crucially relies on the network's ability to store a high-dimensional function. It is known that for general high-dimensional functions, neural networks suffer from the same exponential growth of the number of coefficients as traditional grid based methods, the so-called curse of dimensionality. In this talk, we use methods from distributed optimal control to describe optimal control problems in which this problem does not occur. The talk is based on joint work with Mario Sperl, Dante Kalise, and Luca Saluzzi.
A multi-fidelity approach to control beam-plasma instabilities without feedback
Maintaining the stability and shape of a plasma is a crucial task in many technological applications ranging from beam shaping to fusion energy. Even if a device operates within the MHD (i.e. fluid) stability limits, kinetic effects can lead to so-called microinstabilities that can have severe adverse effects on plasma confinement. Finding effective controls for such instabilities is challenging for two main reasons. First, kinetic equations are, due to the high-dimensional phase space, much more expensive numerically than fluid models. Second, the dynamics of such instabilities can be extremely fast (pico to microseconds) and measuring the distribution function is difficult. This makes feedback based control difficult. Here we study a bump-on-tail instability that is caused by a beam injected into the plasma to heat it (as is commonly done in fusion devices). It turns out that the time-dependent beam intensity can have a significant effect on the stability of the system. Moreover, the frequency at which the beam intensity changes can be much slower than the characteristic frequency of the plasma system. Finding optimal beam profiles is a PDE constraint optimization problem with a very rough landscape. We use global optimization techniques to find good solutions. In order to reduce computational cost, we also explore the use of dynamical low-rank methods. This naturally gives us a multi-fidelity strategy where we can, in an almost continuous fashion, trade off computational cost and accuracy.
New Challenges in the Kinetic Theory of Complex Fluids
In this talk, I will present a unified kinetic framework for ordered fluids - continuum systems whose elements possess internal structure. The key idea is to systematically enlarge the phase space by incorporating the appropriate generalized angular momenta associated with Capriz’s order-parameter manifold. This construction leads to a uniquely determined mesoscopic model for any continuum with microstructure. To make the theory concrete, I will showcase the particular example of liquids composed of calamitic (rodlike) molecules. Using Noether’s theorem, I will show how the symmetries of microscopic interactions determine the conserved quantities that govern mesoscopic dynamics. Our kinetic equation is derived under the assumption that microscopic interactions are weak with respect to the mechanisms responsible for ordering, and I will discuss how this kinetic viewpoint clarifies the emergence of mesoscopic behavior. Finally, I will outline the conditions under which an H-theorem holds for ordered fluids and identify the class of Vlasov potentials that can give rise to collective phenomena.
Models for mixtures of gases, from the kinetic to the macroscopic setting
Macroscopic models for mixtures of gases are often derived from heuristic considerations. This is the case in particular of two phase flows. This talk will describe how to derive macroscopic models for mixtures from kinetic equations. The MHD equations for plasmas can be derived from a BGK mixture model, under suitable scalings, as has already been shown, from different perspectives, in several works. But an analogous approach can be used to derive macroscopic two phase flow models of Baer Nunziato type. This instead is, as far as we know, a novel approach.
ADAMUS kick-off meeting
The meeting will be an opportunity to present the project’s main objectives, research plan, and expected outcomes, as well as to discuss the roles of the different partners and the schedule of upcoming activities. We look forward to seeing you and starting this collaboration together.