High-Order Ghost Point Methods for Accurate Boundary Conditions in PDEs on Complex Domains

• Armando Coco (University of Catania)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall Solving partial differential equations (PDEs) in domains with complex geometries is central to many scientific and engineering applications, yet it poses significant computational challenges, particularly in the accurate and stable imposition of boundary conditions. This talk presents high-accuracy discretization strategies for boundary conditions within the framework of unfitted boundary methods. We introduce boundary discretization schemes based on the ghost point method. The approach extends the computational grid beyond the physical domain by introducing ghost points whose values are determined so as to enforce boundary conditions with high accuracy. In contrast to traditional techniques that assign ghost point values through local or one-sided extrapolation, our method adopts a coupled formulation in which ghost point values are solved simultaneously with neighboring interior and ghost points. This leads to an augmented linear system that significantly improves both accuracy and numerical stability. To efficiently solve the resulting systems, we develop a specialized multigrid solver tailored to the presence of curved and irregular boundaries. The effectiveness of the proposed approach is demonstrated through numerical experiments on elliptic PDEs and its applicability is further illustrated in incompressible flow simulations, including dynamic configurations such as oscillating bubbles. In addition, we present recent extensions of the method to hyperbolic equations, highlighting its challenges for time-dependent, transport-dominated problems.

On the convergence of Monte Carlo methods for Boltzmann models

• Giacomo Borghi (Heriot-Watt University)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In the talk I will review stochastic particle approximations of Boltzmann-like equations and present a novel quantitative error analysis. The convergence result is based on a coupling tailored for the collisional dynamics and on Wasserstein concentration inequalities. I will also discuss possible extension to binary Monte Carlo methods for mean-field models. Joint work with Lorenzo Pareschi.

UFlex: A Flexibile & Efficient Multiscale Neural Physics Simulator

• Pietro Sittoni (Gran Sasso Science Institute)

Room: Polaris building (Rectorate) - Auditorium U-Net--style architectures are widely adopted for modeling physical systems, as their multiscale structure enables efficient processing of high-resolution data while reflecting the hierarchical structure of many physical phenomena. The most successful U--Net-based neural physics simulators typically combine convolutions with transformers; however, when applied to regular grids, they depend on highly structured components that limit adaptability across different spatial and spatiotemporal dimensions. In this work, we revisit transformer-based U-Nets with the goal of maximizing flexibility without sacrificing multiscale efficiency. We introduce a U-Net composed entirely of transformer blocks operating on a one-dimensional latent sequence, making it easy to extend across spatial and spatiotemporal dimensions. Evaluated on seven challenging benchmarks (four 2D and three 3D), our model scales to resolutions of up to $512\times512$ in 2D and $256\times128\times256$ in 3D, while reducing training memory and accelerating training compared to state-of-the-art transformer baselines, all while achieving competitive or state-of-the-art predictive accuracy.

Robust control strategies for magnetically confined fusion plasma.

• Federica Ferrarese (University of Ferrara)

Room: Polaris building (Rectorate), Auditorium The study of the problem of plasma confinement in huge devices, such as for example Tokamaks and Stellarators, has attracted a lot of attention in recent years. Strong magnetic fields in these systems can lead to instabilities, resulting in vortex formation. Due to the extremely high temperatures in plasma fusion, physical materials cannot be used for confinement, necessitating the use of external magnetic fields to control plasma density. This approach involves studying the evolution of plasma, made up of numerous particles, using the Vlasov-Poisson equations. In the first part of the talk, the case without uncertainty is explored. Particle dynamics are simulated using the Particle-in-Cell (PIC) method, known for its ability to capture kinetic effects and self-consistent interactions. The goal is to derive an instantaneous feedback control that forces the plasma density to achieve a desired distribution. Various numerical experiments are presented to validate the results. In the second part, uncertainty and collisions are introduced into the system, leading to the development of a different control strategy. This method is designed to steer the plasma towards a desired configuration even in the presence of uncertainty. The presentation concludes by outlining a future perspective focused on developing a robust control strategy based on neural networks.

Higher-order ghost finite element methods for Navier-Stokes equations on moving domains

• Hridya Dilip (University of Catania)

Room: Polaris building (Rectorate), Auditorium In this talk, we present a new numerical technique for approximating solutions of the Navier–Stokes equations in arbitrarily moving domains. The proposed approach relies on a space discretization based on the ghost finite element method (ghost-FEM), which allows computations on unfitted meshes and avoids costly remeshing as the domain evolves in time. Time integration is performed using a high-order implicit-explicit (IMEX) schemes, ensuring robustness and efficiency for incompressible flows. A key contribution of this work is the extension of the finite element formulation to moving domains through an extrapolation strategy based on the Aslam technique. Originally introduced in the context of finite difference methods, this technique is here carefully adapted to the finite element framework to transport and extend the numerical solution across the evolving computational domain. This approach enables a consistent and accurate treatment of the solution in newly activated regions of the mesh while maintaining the stability properties of the underlying discretisation. The error introduced by the geometrical approximation of the moving domain is addressed using the shifted boundary method, which allows an accurate representation of boundary conditions on unfitted meshes. Dirichlet boundary conditions are imposed weakly by means of Nitsche’s method. The associated stabilisation parameter is chosen optimally by solving a generalised eigenvalue problem, ensuring stability and accuracy of the numerical scheme. We present a series of numerical experiments designed to validate the accuracy and convergence properties of the proposed method. The performance of the scheme is further assessed through comparisons with established benchmark problems involving moving boundaries. These results demonstrate the effectiveness of the method in handling complex domain motions while retaining high-order accuracy in both space and time.

Numerical optimal control of non-linear Fokker-Planck equations arising in social dynamics

• Elisa Calzola (University of Ferrara)

Room: Polaris building (Rectorate), Auditorium We present a second-order numerical scheme to address the solution of optimal control problems constrained by the evolution of nonlinear Fokker-Planck equations arising from socio-economic dynamics. To design an appropriate numerical scheme for control realization, a coupled forward-backward system is derived based on the associated optimality conditions. The forward equation, corresponding to the Fokker-Planck dynamics, is discretized using a structure preserving scheme able to capture steady states.The backward equation, modeled as a Hamilton-Jacobi-Bellman problem, is solved via a semi-Lagrangian scheme that supports large time steps while preserving stability. Coupling between the forward and backward problems is achieved through a gradient descent optimization strategy, ensuring convergence to the optimal control. Numerical experiments demonstrate second-order accuracy, computational efficiency, and effectiveness in controlling different examples across various scenarios in social dynamics.

A IMEX-based spectral scheme with adaptive time-stepping for the Vlasov-Poisson system in the quasi-neutral limit

• Bernardo Collufio (Gran Sasso Science Institute)

Room: Polaris building (Rectorate), Auditorium The choice of the time step is important not only for stability, but also for efficiency and consistency with the equations under consideration. In the context of kinetic equations, these aspects are particularly critical. On the one hand, efficiency must be optimized because of the high dimensionality of phase space, which may range from 2 to 6 depending on the problem, leading to a very large computational cost. On the other hand, the simulation of models, especially in Plasma Physics, often involves very fast dynamics that require an appropriate time discretization in order to be accurately resolved. The aim of this talk is to present recent results on time-step selection strategies applied to a numerical scheme for the Vlasov–Poisson equation in the quasi-neutral limit. The scheme is based on a Hermite spectral decomposition in velocity and a finite-difference discretization in space, and was originally proposed in 2025 by Blaustein, Dimarco, Filbet, and Vignal. It relies on a fully implicit, L-stable DIRK time discretization, yielding an Asymptotic-Preserving method in the quasi-neutral limit, at the expense of a significant computational cost per time step. A key issue is that, although the L-stable scheme can capture the oscillating behavior in time of the Vlasov–Poisson system (particularly those of the electric field and current density) in the asymptotic regime, its dissipative nature may eventually damp these oscillations if the time step is not properly chosen. In this work, we instead adopt an IMEX approach, treating the nonlinear part implicitly and the linear part explicitly. This substantially improves the efficiency while maintaining stability, although the Asymptotic-Preserving property is lost. Combined with this semi-implicit treatment, we aim to exploit the accuracy control performed by the time-step selection technique and use it as main ingredient for recovering the fast and ample oscillations in time.

Contour integral methods and model order reduction for parametric linear control systems

• Mattia Manucci (Karlsruhe Institute of Technology)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In this talk, I present a contour integral method (CIM) for the efficient computation of outputs from parametric linear systems in control form over specified time intervals and under a user-prescribed accuracy. The CIM framework utilises a quadrature rule to approximate the inverse Laplace transform, which is executed along a modified integration contour. First, I discuss the non-parametric case and show how an accurate and extremely fast approximation of the output function can be achieved, without using any reduction approach, just by a suitable construction of the integration contour and the precomputation of small-size matrices. Then I discuss the parametric case and demonstrate how this method integrates effectively with projection-based model order reduction (MOR), where the projection spaces are constructed via a greedy-type algorithm guided by an appropriately derived error estimate, which relies on a suitably defined adjoint problem, and adhering to Hermite interpolation conditions. This methodology significantly reduces computational expenses when evaluating the input-output relations for varied parameters across the parametric domain, as well as for a wide class of input functions and initial conditions sufficiently captured by a low-dimensional subspace. The natural selection of the frequency points and interpolation parameters, together with the ability to target a prescribed time interval for different initial conditions, represents a distinctive feature compared with state-of-the-art methodologies. Finally, the precision and efficacy of the approach are demonstrated by numerical experiments on a range of benchmark test problems for both non-parametric and parametric linear control systems.

Online estimation of time-dependent parameters in population models

• Elisa Iacomini (University of Ferrara)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall Many systems governed by evolutionary parametric partial differential equations involve parameters that evolve in time and must be inferred from partial, noisy observations. Accurately tracking these nonstationary parameters is essential for reliable prediction. We present a continuous data assimilation framework for time-dependent parameter estimation based on the ensemble Kalman filter. Unknown coefficients are incorporated into an augmented state, enabling their online adaptation as new data become available in a computationally efficient manner. The approach is illustrated mainly on age-structured epidemiological models, where surveillance data are assimilated to reconstruct evolving mortality and incidence trends.

Asymptotic Preserving scheme and Gradient-flow structure for Multiscale Poisson-Nernst-Planck system

• Clarissa Astuto (University of Catania)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In this talk, we further develop the multiscale model introduced in Astuto, Raudino and Russo (2023) by deriving a gradient-flow structure that incorporates the time-dependent boundary condition obtained therein. The model describes a drift-diffusion equation for ion transport in the presence of a trapping potential, focusing on surface traps whose attraction range, denoted by $\delta$, is much smaller than the characteristic length scale of the problem. The multiscale formulation is derived via an asymptotic expansion with respect to this small parameter, leading to an effective boundary condition that captures the ion adsorption process at the surface. Using the same asymptotic framework, we derive the associated free energy and show that, at steady state, the ionic concentration concentrates on the trap surface in the sense of measures, giving rise to a Dirac-type contribution. This observation naturally motivates the study of the time evolution of the concentration in the space of positive measures. Numerical results for the drift-diffusion and the multiscale models are compared in one spatial dimension. In the second part of the talk, we extend the multiscale approach to a Poisson-Nernst-Planck (MPNP) system, accounting for the correlated dynamics of positive and negative ions and for Coulomb interactions. In the regime of very small Debye lengths, the quasi-neutral limit reduces the system to an effective diffusion equation for a single carrier. While this simplification facilitates the analysis, it may fail to capture relevant behaviors close to the quasi-neutral regime. To overcome this limitation, we develop and validate an Asymptotic Preserving numerical scheme that remains stable as the Debye length tends to zero, without imposing restrictive time-step constraints.

Hierarchical structures in neural PDE solvers

• Emanuele Zangrando (Gran Sasso Science Institute)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In recent years, deep learning, and particularly operator learning, has emerged as a powerful paradigm for solving PDEs, driven by its promise of high-speed surrogate simulations once models are trained. In practice, however, many high-impact applications remain bottlenecked by the cost of generating large, high-fidelity training datasets and the substantial compute required to train expressive models, even with access to high-performance computing. In this talk, we present Neural-HSS, a parameter-efficient architecture motivated by recent insights into the structure of Green’s functions for elliptic PDEs. Neural-HSS leverages the Hierarchical Semi-Separable (HSS) matrix representation to encode this structure directly, yielding models that are markedly more data-efficient while maintaining strong approximation capability. We provide theoretical justification for its data-efficiency on a certain class of PDEs and demonstrate its performance empirically across a range of benchmark problems.

High-order hierarchical dynamic domain decomposition method for the Boltzmann equation

• Domenico Caparello (University of Ferrara)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In this work, we present a high-order hierarchical dynamic domain decomposition method for the Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga. This criterion is used to dynamically partition the two-dimensional spatial domain into two regimes: the Euler regime, and the kinetic regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, and the Boltzmann equation where strong non-equilibrium effects dominate, such as near shocks and boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver is considerably cheaper than the kinetic Boltzmann model. We implement a coupling mechanism between the two regimes capable of preserving the high-order accuracy of both Euler and kinetic solvers, and we use state-of-the-art numerical techniques. This combination enables robust and scalable simulations of multi-scale kinetic flows with complex geometries. Joint work with Lorenzo Pareschi (University of Ferrara & Heriot-Watt University), Thomas Rey (Université Côte-d'Azur) and Tommaso Tenna (Université Côte-d'Azur & University of Rome "La Sapienza").

Hyperbolic Serre-Green-Naghdi equations? Only semi-implicit schemes

• Emanuele Macca (University of Catania)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall The Serre-Green-Naghdi (SGN) equations provide a valuable framework for modelling fully nonlinear and weakly dispersive shallow-water flows. However, their elliptic formulation can considerably increase the computational cost compared to the Saint-Venant equations. To overcome this difficulty, hyperbolic models (hSGN) have been proposed that replace the elliptic operators with first-order hyperbolic formulations augmented by relaxation terms, which recover the original elliptic formulation in the stiff limit. Nevertheless, although explicit treatments of such models are relatively easy to implement, they suffer from severe time-step restrictions induced by the relaxation parameter, thereby making semi-implicit techniques necessary to achieve computational efficiency.

Splitting, stiffness and low-rank: a second-order Strang method for matrix differential equations

• Carmen Scalone (University of L'Aquila)

Room: Aurora building (Ex-ISEF) - Main Lecture Hall In this talk, we present a second-order Strang splitting approach for efficiently solving a class of stiff matrix differential equations with Sylvester-type structure. The key idea is to decompose the dynamics into a stiff linear component, which can be handled exactly using matrix exponentials, and a nonlinear component, which is treated by a second-order dynamical low-rank method. We discuss the motivation behind this splitting strategy and highlight its advantages for stiff problems. A central result is that, under suitable assumptions, the scheme retains second-order accuracy. Numerical experiments illustrate the accuracy, robustness, and computational efficiency of the method. This is a joint work with N. Guglielmi.