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Giovedì 03 aprile 2025, ore 12.30, IAC-CNR
Federico Pichi
SISSA
Graph-based machine learning approaches for model order reduction
Abstract:
The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches. One drawback of the techniques based on convolutional autoencoders is the lack of geometrical consistency when dealing with complex domains defined on unstructured meshes. The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [1]. Our methodology extracts the latent space’s evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high generalizability in the low-data regime and the great speedup. Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [2]. This way, we overcome the limitations of the up- and down-sampling procedures by building a resolution-invariant GFN-ROM strategy capable of training and testing on different mesh sizes, resulting in a more lightweight and flexible architecture. Finally, we will show preliminary results concerning the time-extrapolation regime for dynamical systems in a Deep Operator Networks (DeepONet) framework, integrating the graph-based architectures with core-splitting tensor train decomposition and operator inference to learn the temporal evolution [3].
References
[1] Pichi, F., Moya, B. and Hesthaven, J.S. (2024) ‘A graph convolutional autoencoder approach to model order reduction for parametrized PDEs’, Journal of Computational Physics, https://doi.org/10.1016/j.jcp.2024.112762.
[2] Morrison, O.M., Pichi, F. and Hesthaven, J.S. (2024) ‘GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications’, Computer Methods in Applied Mechanics and Engineering, https://doi.org/10.1016/j.cma.2024.117458.
[3] Chen, Y., Pichi, F., Gao, Z., and Rozza, G. (2025) ‘Multi-fidelity reduced-order model based on graph convolutional autoencoder for parameterized time-dependent partial differential equations’, In preparation
Giovedì 27 marzo 2025, ore 11.00, Sala di Consiglio
Fabio Durastante
Dipartimento di Matematica, Università di Pisa
Enforcing Katz and PageRank Centrality Measures in Complex Networks
Abstract:
Centrality measures are fundamental in the study of complex networks, offering insights into the relative importance of nodes based on different connectivity patterns. In this talk, we address the problem of enforcing a prescribed Katz [1] or PageRank [2] centrality while maintaining the network’s original structure as much as possible [3]. Specifically, we seek the minimal perturbation of edge weights necessary to achieve the desired centrality values, ensuring that modifications are both efficient and targeted. We formulate this problem as a constrained Quadratic Programming (QP) optimization task and propose scalable numerical methods [4] to solve it. Our approach leverages the structure of the optimization problem to enhance computational efficiency, making it feasible for large-scale networks. By carefully selecting which edges to modify, we ensure that the fundamental topology of the network remains largely intact while achieving the prescribed centrality objectives. Applications of our methodology span various domains. Through extensive computational experiments, we demonstrate the effectiveness of our approach in both synthetic and real-world networks. We also provide an open-source implementation Github of our algorithms, allowing for reproducibility and further research in the field.
Bibliography
[1] Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39–43.
[2] Page , L., & Brin, S. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer Networks, 30(1–7), 107–117.
[3] Cipolla, S., Durastante, F., & Meini, B. (2024). Enforcing Katz and PageRank Centrality Measures in Complex Networks, arXiv 2409.02524.
[4] Cipolla, S., & Gondzio, J. (2023). Proximal stabilized interior point methods and low-frequency-update preconditioning techniques. J. Optim. Theory Appl., 197(3), 1061–1103
Giovedì 13 marzo 2025, ore 12.00, IAC-CNR
Philipp Öffner
University of Clausthal
Advances in Summation-by-Parts Operators and Their Applications in Numerical Methods for Compressible Flows
Abstract:
Summation-by-parts (SBP) operators, which discretely mimic the integration-by-parts principle, provide a systematic framework for constructing energy-stable numerical methods for energy-bounded initial boundary value problems (IBVPs). Initially developed for finite difference schemes with polynomial approximations to achieve energy-stable semi-discretizations, significant advancements have emerged over the past decade. These include generalized and hybrid SBP operators for arbitrary grids, SBP operators on point clouds, upwind SBP operators, and the broader class of function-space SBP operators. This talk provides an overview of recent developments in SBP operators and their applications in numerical methods for advection-dominated problems. Particular attention is given to upwind SBP operators and function-space SBP operators, highlighting advancements in their construction and demonstrating their versatility across various settings, including finite difference (FD), discontinuous Galerkin (DG), and mesh-free approaches
Giovedì 06 marzo 2025, ore 12.00, Sala di Consiglio
Michael Dumbser
Università di Trento
On well-balanced finite volume and discontinuous Galerkin schemes for the Einstein-Euler system of general relativity
Abstract:
(joint work with Olindo Zanotti, Ilya Peshkov, Elena Gaburro, Gabriella Puppo)
In this talk we present a new family of well-balanced discontinuous Galerkin (DG) finite element schemes with subcell finite volume (FV) limiter for the numerical solution of the Einstein–Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Einstein-Euler Z4 system, which is composed of 64 equations, is analysed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows.
We furthermore present a novel family of central WENO finite difference schemes for a new first order reformulation of the classical BSSNOK system.
Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching four main achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum (ρ = p = 0) up to t = 1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered untractable within the FO-Z4 formalism, (iv) we perform a stable long-time evolution of a rotating binary black hole merger based on the new CWENO schemes for first order reformulation of the BSSNOK system.
References
[1] M. Dumbser, O. Zanotti, E. Gaburro and I. Peshkov, A well-balanced discontinuous Galerkin method for the first–order Z4 formulation of the Einstein–Euler system, Journal of Computational Physics 504 (2024), 112875.
[2] M. Dumbser, O. Zanotti and G. Puppo, A monolithic first–order BSSNOK formulation of the Einstein–Euler equations and its solution with conservative finite difference CWENO schemes, Physical Review D, in preparation
Giovedì 20 febbraio 2025, ore 12.00, IAC-CNR
Andrea Bondesan
Università di Parma
Diffusion asymptotics of the Boltzmann multi-species equation through perturbation of hypocoercivity
Abstract:
I will discuss the rigorous derivation of hydrodynamic limits of the Boltzmann multi-species equation, for Mach and Knudsen numbers vanishing at the same rate. Solutions to the kinetic system are constructed as fluctuations around local non-equilibrium Maxwellian distributions, whose physical observables solve the limit macroscopic model of interest. A general hypocoercive formalism is developed to establish a uniform (with respect to the small diffusion parameter) Cauchy theory for this perturbative setting. I will apply the method to derive the Maxwell-Stefan cross-diffusion system and present numerical results confirming this convergence
Giovedì 06 febbraio 2025, ore 12.30, Aula-L
Francesca Ignoto
Istituto per le Applicazioni del Calcolo "M. Picone"-Consiglio Nazionale delle Ricerche
Macroscopic MFGs for traffic flow on networks with V2V connection?
Abstract:
In questo seminario si presenterà un problema aperto di passaggio al limite da scala micro a scala macro per un problema di Mean Field Game per il traffico veicolare su reti stradali. Il problema è caratterizzato dal fatto che, sebbene tutti i guidatori possano prevedere la dinamica degli altri e ottimizzare il loro percorso in base alla suddetta previsione, ogni guidatore ha una percezione limitata dell'esistenza degli altri veicoli, e questa percezione evolve nel tempo. Dopo una breve presentazione del problema, il seminario seguirà con una discussione aperta sui possibili approcci da adottare. E' preferibile la partecipazione in presenza. Il seminario sarà in italiano.
Martedì 21 gennaio 2025, ore 15.00, Sala di Consiglio
Elia Onofri
Istituto per le Applicazioni del Calcolo “M. Picone”-Consiglio Nazionale delle Ricerche
On the Modelling of Granuloma-Like Structures: An Overview from In-Vitro to In-Silico Modelling
Abstract:
In 2024, tuberculosis remains a formidable global health challenge, contributing to millions of deaths annually. Central to the pathology of this disease are granulomas – intricate structures formed by the immune system in response to Mycobacterium tuberculosis infection. While various animal models have been developed to elucidate the mechanisms of granuloma formation and development, none of them fully recapitulate the human disease. Additionally, the limited throughout of these models presents a significant obstacle to their application in drug development. In recent years, three-dimensional cell cultures, known as Granuloma-Like Structures (GLS), have emerged as a promising tool that balances the complexity of in vivo conditions with the practicality required for high-throughput drug screening. Despite these advantages, in vitro models still present challenges that hinder their use in large-scale testing. Issues such as reproducibility, time consumption, and the need for human interaction limit their scalability. Therefore, as part of the European Regimen Accelerator for Tuberculosis (ERA4TB) project, we are developing an in silico model designed to simulate and reproduce GLS. In this talk, we will present the scientific context of tuberculosis research and introduce our multiscale GLS model, which combines an agent-based microscopic simulation of immune-pathogen interactions at the cellular level with a PDE-based macroscopic simulation of cell signalling diffusion at the molecular level. In addition to describing the model, we will discuss its calibration against experimental data, highlighting both the challenges and innovative solutions related to data analysis and integration
Martedì 14 gennaio 2025, ore 15.00, Sala di Consiglio
Luca Saluzzi
Università di Roma La Sapienza
A Data-Driven Tensor-Train Gradient Cross for Hamilton-Jacobi-Bellman equations
Abstract:
Hamilton-Jacobi-Bellman (HJB) equation plays a central role in optimal control and differential games, enabling the computation of robust controls in feedback form. The main disadvantage for this approach depends on the so-called curse of dimensionality, since the HJB equation and the dynamical system live in the same, possibly high dimensional, space. In this talk, I will present a data-driven method for approximating high-dimensional HJB equations based on tensor decompositions. The approach presented in this talk is based on the knowledge of the value function and its gradient on sample points and on a tensor train decomposition of the value function. The collection of the data will be derived by two possible techniques: Pontryagin Maximum Principle and State-Dependent Riccati Equations. The numerical experiments will demonstrate an at most linear complexity in the dimension and a better stability in presence of noise. Moreover, I will present an application to an agent-based model and a comparison with Deep Learning techniques. Finally, time permitting, I will consider the coupling of the proposed method with Model Order Reduction techniques and their application to boundary feedback control for the Navier-Stokes equations.
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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