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Martedì 02 Luglio 2024, ore 15.00, Sala di Consiglio
Giulia Tatafiore
Sapienza University of Rome
Computational geometry algorithms applied to an efficient semi-Lagrangian scheme for Fokker-Planck equations on unstructured grids
Abstract:
Semi-Lagrangian schemes are characteristic-based methods for the numerical solution of hyperbolic partial differential equations (PDEs), which maintain stability under large Courant numbers.However, the need to locate the feet of the characteristics and the challenge of managing flux-deformed grid elements can significantly reduce efficiency in the case of unstructured grids. In this work, we present a semi-Lagrangian scheme for Fokker-Planck equations, wherein these issues are addressed through the application of an efficiently initialized version of the Barycentric walk algorithm and the Sutherland-Hodgman algorithm, respectively.
Martedì 25 Giugno 2024, ore 15.00, Sala di Consiglio
Valentina Coscetti
Sapienza University of Rome
A numerical scheme for the critical value approximation of eikonal Hamilton–Jacobi equations on networks
Abstract:
A numerical algorithm for the Mañé critical value approximation of eikonal Hamilton–Jacobi equations on networks is presented. The proposed method is based on the long time approximation of the corresponding evolutive problem, which is solved numerically with a semi-Lagrangian scheme. More specifically, some numerical tests in the case of networks in \( \mathbb{R}^2 \) are shown, to illustrate the convergence of the algorithm.
Martedì 18 Giugno 2024, ore 15.00, Sala di Consiglio
Tan Bui-Tanh
University of Texas at Austin
Learn2Solve: A Deep Learning Framework for Real-Time Solutions of forward, inverse, and UQ Problems
Abstract:
Digital models (DMs) are designed to be replicas of systems and processes. At the core of a digital model (DM) is a physical/mathematical model that captures the behavior of the real system across temporal and spatial scales. One of the key roles of DMs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, to calibrate parameters to match the actual process and to quantify the uncertainties. In this talk, we will present various (faster than) real-time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, (transonic and supersonic) Euler, and Navier-Stokes equations will be presented.
Martedì 11 Giugno 2024, ore 15.00, Sala di Consiglio
Yulong Xing
Ohio State University
Runge-Kutta Discontinuous Galerkin Methods Beyond the Method of Lines
Abstract:
In the common practice of the method-of-lines (MOL) approach for discretizing a time-dependent partial differential equation (PDE), one first applies spatial discretization to convert the PDE into an ordinary differential equation system. Subsequently, a time integrator is used to discretize the time variable. When a multi-stage Runge-Kutta (RK) method is used for time integration, by default, the same spatial operator is used at all RK stages. However, recent studies on perturbed RK methods indicate that not all RK stages are born equal – breaking the MOL structure and applying rough approximations at specific RK stages may not affect the overall accuracy of the numerical scheme. In this talk, we present two of our recent explorations on blending rough stage operators in RK discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. In our first work, we mix the DG operator with the local derivative operator, yielding an RKDG method featuring compact stencils and simple boundary treatment. In our second work, we mix the DG operators with polynomials of degrees k and k-1, and the resulting method may allow larger time step sizes and fewer floating-point operations per time step.
Martedì 04 Giugno 2024, ore 15.00, Sala di Consiglio
Antoine Tordeux
University of Wuppertal
Modelling vehicle and pedestrian dynamics with input-state-output port-Hamiltonian systems
Abstract:
Traffic flow and pedestrian crowds are complex phenomena characterised by different collective dynamics. Inspired by recent work in control engineering by Knorn et al. and Matei et al., we explore the application of input-state-output port-Hamiltonian systems for the modelling and analysis of pedestrian and vehicle dynamics. We first present a two-dimensional port-Hamiltonian pedestrian model, emphasizing the modelling components and their role in the dynamics. In particular, we investigate the emergence of collective behaviours such as lane formation in counterflow and stripe formation in crossflow through numerical simulations. Next, we analyse the dynamics of stochastic port-Hamiltonian road traffic models in one dimension. We focus on a quadratic interaction potential, where the system is a multidimensional Ornstein-Uhlenbeck process. The uncontrolled dynamics without input exhibit instability under stochastic perturbations. However, implementing an external speed control allows the system to stabilise and to converge weakly to Gaussian limit distributions. The convergence remains unconditional with constant speed control. Interestingly, a specific stability condition arises when the input control acts as a dynamic feedback depending on the distance ahead. This talk is based on joint work with Julia Ackermann, Matthias Ehrhardt, Thomas Kruse, Barbara Rüdiger, Claudia Totzeck and Baris Ugurcan.
Martedì 28 Maggio 2024, ore 15.00, Sala di Consiglio
Igor Voulis
Georg-August-Universität Göttingen
An adaptive stochastic Galerkin method for elliptic PDEs
Abstract:
We model the uncertainties in (random) coefficient functions of an elliptic partial differential equation by expanding these coefficients as function series with scalar random coefficients. This gives us a deterministic formulation of a random PDE. Due to the combination of stochastic and spatial unknowns, this gives us a high-dimensional elliptic PDE. We present an adaptive stochastic Galerkin method for solving this PDE and discuss the optimality of this method. The method combines a multilevel representation of stationary random fields with a residual-based spatial adaptive scheme. An optimal operator compression is used for the stochastic operator. A Bramble-Pasciak-Xu (BPX)-frame is used to obtain a residual estimate and to achieve appropriate error reduction in the iterative linear solver. The numerical results and in the wavelet-case a complete rigorous analysis show that the obtained scheme is optimal. This talk is based on joint work with M. Bachmayr, M. Eigel and H. Eisenmann.
Giovedì 23 Maggio 2024, ore 13.00, Sala di Consiglio
Vincent Liu
University of Melbourne
A Hamilton-Jacobi-Bellman Approach to Ellipsoidal Approximations of Reachable Sets
Abstract:
Society's ever-increasing integration of autonomous systems in day-to-day life has simultaneously brought forth concerns as to how their safety and reliability can be verified. To this end, reachable sets lend themselves well to this task. These sets describe collections of states that a dynamical system can reach in finite time, which can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse-of-dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this talk, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. This result arises from the construction of a local viscosity supersolution and subsolution of a Hamilton-Jacobi-Bellman equation for the corresponding reachability problem. The proposed ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making the approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.
Martedì 21 Maggio 2024, ore 15.00, Sala di Consiglio
Francesca Lourdes Ignoto
Sapienza University of Rome
Dissolution of Multiple Variable-in-Shape Drug Particles Using the Level-Set Method
Abstract:
The dynamics of variable-in-shape drug particles are fundamental to predict the dissolution of drugs in a fluid. In this talk we propose a new approach which consists of describing the dissolution process of a drug particle using the level-set method. In this way, it is possible to recover the particle surface at any time as the zero level set of the solution of an Hamilton-Jacobi equation. This approach extends a previous work of 2022, and it allows us to deal with particles with rectangular shape as well as to simulate the joint dissolution of multiple particles with different shape and size. Firstly, we introduce the problem and the basic functions that come into play in the dissolution process, then we present the new methodology and finally we discuss some numerical simulations in the 2D case.
Martedì 14 Maggio 2024, ore 15.00, Sala di Consiglio
Mariarosa Mazza
University of Rome Tor Vergata
Exploring numerical challenges in differential models with fractional derivatives
Abstract:
Fractional derivatives, a widely recognized mathematical tool, have gained considerable attention in recent decades owing to their non-local behavior, particularly suitable for capturing anomalous diffusivity. They find application in various real-world scenarios that range from the interaction between particles and fields within plasma to the dynamics of networks in human environments. While the presence of a fractional derivative in a differential model can result in a better physical description, it also poses significant challenges in numerical treatment. Specialized strategies, including discretization methods and numerical solvers, are required to address such challenges effectively. This presentation aims to offer insight into the topic, with a specific emphasis on the numerical linear algebra obstacles it entails.
Martedì 07 Maggio 2024, ore 15.00, Sala di Consiglio
Eitan Tadmor
Fondations Sciences Mathematiques de Paris and University of Maryland
Swarm-Based Gradient Descent Methods for Non-Convex Optimization
Abstract:
We discuss a novel class of swarm-based gradient descent (SBGD) methods for non-convex optimization. The swarm consists of agents, each is identified with position, x, and mass, m. There are three key aspects to the SBGD dynamics. (i) persistent transition of mass from agents at high to lower ground; (ii) mass-dependent marching in directions randomly aligned with gradient descent; and (iii) time stepping protocol which decreases with m. The interplay between positions and masses leads to dynamic distinction between `leaders’ and `explorers’: heavier agents lead the swarm near local minima with small time steps; lighter agents use larger time steps to explore the landscape in search of improved global minimum, by reducing the overall ‘loss’ of the swarm. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer.
Martedì 23 Aprile 2024, ore 15.00, Sala di Consiglio
Jules Berry
IRMAR INSA Rennes
Approximation of stable solutions to second order mean field game systems
Abstract:
We introduce a general framework for the study of numerical approximations of a certain class of solutions, called stable solutions, of second order mean-field game systems for which uniqueness of solutions is not guaranteed. To illustrate the approach, we focus on a very simple example of stationary second-order MFG system with local coupling and a quadratic Hamiltonian. We provide sufficient conditions for the stability of solutions and it turns out that stability is a generic property of the MFG. We then re-express the solutions of the system as zeros of a well chosen nonlinear map and establish the fact that stable solutions are regular points of this map. This fact is then used to study the approximation of solutions by finite elements and the local convergence of Newton's method in infinite dimension.
Martedì 16 Aprile 2024, ore 15.00, Sala di Consiglio
Giulia Villani
Sapienza University of Rome
Optimal control for orbital transfer of LEO satellites with Low-Thrust engines
Abstract:
The research project, in collaboration with Thales Alenia Space Italia SpA, aims to study, develop and numerically simulate innovative methods for optimizing the orbital transfer of small satellites in LEO (Low Earth Orbit) using Low-Thrust engines, to be deployed in constellations for Earth observation applications. We developed the controlled dynamics of a single satellite, using the Dynamic Programming approach, based on the characterization of the value function via the Hamilton-Jacobi-Bellman equation. We have built an algorithm that fits a specific physical problem of industrial interest, applying numerical techniques as the Policy Iteration to make the algorithm faster, and using a more suitable grid to save memory. The main challenge is to build a solid control model to satisfy mission objectives and requirements, e.g. the time needed to reach the target orbit, or the use of propellent.
Martedì 02 Aprile 2024, ore 15.00, Sala di Consiglio
Martin Fleurial
Sapienza University of Rome
Two Dimensional Models of Multi-Lane Traffic Flow with Lane Changing Conditions
Abstract:
The first part of the talk is dedicated to the derivation on an advection-diffusion equation in two dimensions from a system of one dimensional hyperbolic PDEs modeling the macroscopic behavior of multi-lane traffic flow, taking lane changes into account. In the second part of the talk, we introduce the microscopic model the latter system originates from, and propose a generalisation of this model to two continuous space dimensions. We discuss its properties and well-posedness.
Martedì 19 Marzo 2024, ore 14.30, Sala di Consiglio
Agnese Pacifico
Sapienza University of Rome
Control and identification of unknown PDEs
Abstract:
In this talk we address the control of Partial Differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. Given an estimated parameter configuration, we compute the corresponding control using the State-Dependent Riccati Equation (SDRE) approach. Subsequently, we observe the trajectory and estimate a new parameter configuration using Bayesian Linear Regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. The systems arising from the discretization of PDEs are high dimensional, therefore we also focus on the computational cost of the algorithm. The Proper Orthogonal Decomposition (POD), a Model Order Reduction technique, is applied to the system in order to reduce the computational cost of the control computation step, and this provides impressive speedups. We present numerical examples to show the accurateness of the proposed method.
Martedì 05 Marzo 2024, ore 15.00, Sala di Consiglio
Simone Chiocchetti
University of Cologne
Towards simple and affordable solutions for a unified first order hyperbolic model of continuum mechanics
Abstract:
The talk concerns the ongoing development of a non-standard model of continuum mechanics, originally due to Godunov, Peshkov, and Romenski (GPR), and its numerical approximation in Finite Volume and Discontinuous Galerkin methods. The main feature of the model is that it describes a general continuum, rather than a classic fluid or solid medium, with the difference between the two being specified only by a choice of parameters. In this framework, rather general closure laws can be implemented, including non-Newtonian rheologies, visco-elasto-plasticity, material damage and fractures, melting and solidification, and more. The model is cast in a first order hyperbolic form with stiff relaxation sources, which means that it requires no second order diffusive fluxes, and that it yields a theory in which all signals propagate with finite speed, including heat conduction. A clear drawback of the model is its complexity, in particular when applied to Newtonian viscous fluids and compared to the well established Navier-Stokes equations. Together with stiff sources, one has to also consider the presence of differential involutions and algebraic constraints, together with other nonlinearities and representation issues concerning the evolution of matrix-valued data. Here I outline my efforts towards closing the complexity gap and making the formalism more accessible, mainly focusing on the treatment of stiff sources, algebraic constraints, and on new resolution improvements involving the formulation and solution of a quaternion-valued PDE.
Martedì 13 Febbraio 2024, ore 15.00, Sala di Consiglio
Matteo Piu
Sapienza University of Rome
Investigating Multi-lane Traffic Flow Models: from Micro to Macro
Abstract:
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
Nana Liu
Shanghai Jiao Tong University
Analog quantum simulation of partial differential equations
Abstract:
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
Elisa Calzola
Università di Ferrara
Exponential integrators for mean-field selective optimal control problems
Abstract:
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
Stephan Gerster
University of Mainz
Haar-type stochastic Galerkin formulations for random hyperbolic systems
Abstract:
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
Diogo Gomes
KAUST
Functional Analytic Insights into Mean Field Game Theory
Abstract:
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.
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