Weekly Seminar

Abstract

Multi-scale problems are omnipresent in environmental and industrial processes, posing a challenge to classical numerical solvers given that the propagation speeds of information span several orders of magnitude.

In hyperbolic systems, the absolute fastest wave speed remains finite, but in classical, explicitly integrated numerical schemes, it determines the time step that ensures stability.

This means in particular that it may lead to vanishing time increments in the presence of fast processes caused for instance by high pressure or strong magnetic fields.

Furthermore, it is well-known that upwind schemes introduce spurious numerical diffusion into the approximate solution a problem that can only be partially mitigated by mesh refinement.

Therefore, a common approach, which is also applied here, is to use implicit or semi-implicit time integrators combined with centered differences for spatial derivatives in implicitly treated systems, in order to obtain scale-independent artificial dissipation and stability under large time steps.

As the evolution of the modeled variables is described by a nonlinear flux function, treating it fully or partially implicitly involves solving nonlinear systems. Depending on the problem, these systems can be large, coupled, ill-posed systems, for which solvers such as the Newton method may converge very slowly or not at all.

To avoid nonlinear implicit systems, we apply Jin-Xin relaxation to the implicitly treated stiff flux terms.

This leads to a linear flux structure in the obtained relaxation model. Therein, the nonlinearity of the stiff flux terms is transferred to an algebraic relaxation source term.

By splitting away the relaxation source terms, for each variable, a decoupled wave-type equation can be written, yielding a predicted solution at temporarily frozen wave speeds.

The final solution is obtained by projecting onto the relaxation equilibrium manifold, taking the relaxation source terms into account, and thus obtaining a prediction-correction scheme for the original hyperbolic multi-scale problem.

This is joint work with Gabriella Puppo (La Sapienza Univ. Roma) and Angelo Iollo (IMB Bordeaux, France). 

Abstract

We present numerical methods for the numerical solution of an overdetermined symmetric hyperbolic and thermodynamically compatible (SHTC) model of compressible two-phase flows.

The model has the peculiar feature that it is endowed with two entropy inequalities, one for each phase, as primary evolution equations ensuring total entropy growth. The total energy conservation law is an extra conservation law and is obtained via suitable linear combination of all other equations based on the Godunov variables also referred to as mainfield.

The model describes a two-phase flows of heat conducting fluids modelled by a relaxation process, where in the stiff relaxation limit the SHTC model tends to an asymptotically reduced Baer–Nunziato-type (BN) limit with Fourier-type heat conduction.

Moreover, by doing so, a unique choice for the interface velocity and the interface pressure is obtained, quantities that are usually heuristically determined in the BN model.

In addition, additional lift forces in the SHTC model can be identified which are not standard in BN representation of two-phase flows.

A key feature of the hyperbolic thermodynamically compatible (HTC) finite volume scheme is that it directly evolves the entropies and the energy is conserved as consequence.

We show computational results for several benchmark problems in one and two space dimensions, comparing numerical results obtained for the asymptotically reduced BN limit system.

Abstract (D'Alessio)

In this talk, I will present two machine learning models for the automatic recognition of granulomas within images acquired from in vitro experiments, while addressing the challenge of inter-annotator variability. Such models constitute a valuable tool, as morphological characteristics are believed to be meaningful indicators of the efficacy of drugs targeting Mycobacterium tuberculosis and of the progression of the infection. We address the problem proposing two methods. First, we develop an ensemble model based on three YOLOv11 networks which simulate three different annotators. Second, we employ a hybrid architecture that integrates a YOLOv11 object detection model, serving as a Region Proposal Network (RPN), with a U-Net designed for the high-precision semantic segmentation of the identified Regions of Interest (ROIs). This work has been developed within the ERA4TB project and is based on my Master Thesis, under the supervision of Dr. Enrico Mastrostefano, and in collaboration with Davide Moretti.

Abstract (De Rosa)

The effective management and positioning of satellite constellations are critical to ensuring optimal performance and communication capabilities in space missions. This research focuses on developing Model Predictive Control (MPC) strategies for the precise positioning of satellite constellations, with an emphasis on optimizing thruster usage while maintaining desired orbital separations between satellites. Specifically, we generate a suitable control dynamics to position an arbitrary number of satellites at prescribed positions within the same orbital plane. Our approach involves formulating a control framework that anticipates future orbital states and constraints, allowing for autonomous, real-time adjustments to thruster commands. We aim to crank down fuel consumption during orbital maneuvers while still achieving positional requirements in short time. The findings highlight the advantages of using MPC in satellite constellation management, offering a robust solution for fuel-efficient operations without compromising positional accuracy. This work paves the way for future developments in autonomous satellite positioning and the management of larger constellations, contributing to advancements in space mission planning and execution.

Abstract

In this talk, we address a scalar conservation law with discontinuous gradient-dependent flux, where the discontinuity is determined by the sign of the derivative u_x of the solution; namely, the flux is given  by two different smooth functions f(u) or g(u), when u_x is positive or negative, respectively. This problem is motivated by traffic flow modeling, under the assumption that drivers exhibit different behaviors in accelerating or decelerating mode.

A vanishing viscosity approximation identifies two different situations according to the mutual positions of the graphs of f and g: a well-posed parabolic problem when f(u)<g(u) for all u, or an ill-posed problem if f(u)>g(u) for all u. We refer to these problems as the stable/unstable case respectively.

In the stable case, we prove that semigroup trajectories obtained as a limit of a suitable wave-front tracking algorithm coincide with the unique limits of vanishing viscosity approximations.

In the unstable case, examples show that infinitely many solutions can occur. For piecewise monotone initial data, we prove that a global solution of the Cauchy problem exists. The solution has a finite number of interfaces, where the flux switches between f and g; such structure allows us to provide a uniqueness criterion based on minimizing the number of interfaces.

Joint work with Alberto Bressan and Wen Shen (Penn State University).

Abstract

Embedded, or immersed, approaches aim to minimize the computational costs associated with generating body-fitted meshes by employing fixed, often Cartesian, meshes. However, this boundary treatment introduces a geometric error of the order of the mesh size, which, if not properly addressed, can compromise the global accuracy of a high-order discretization. High-order embedded methods are used to appropriately correct the boundary conditions imposed on an unfitted boundary, thereby compensating for the aforementioned geometric error and achieving high-order accuracy. In this seminar, a stability analysis of discontinuous Galerkin methods coupled with embedded methods is conducted for the linear advection equation through the eigenspectrum visualization of the high-order discretized operators. Numerical experiments are presented to validate the stability analysis.

Abstract

When an optimization problem involves the use of high-fidelity mathematical models (HFMMs), it is natural to expect a high computational cost, often incompatible with project timelines. For this reason, interpolation or approximation models, known as meta-models, are often used. These meta-models are usually simple algebraic relations, calibrated using a limited number of HFMM evaluations. As a result, the optimization quality problem shifts to the quality of the meta-model itself. In this presentation, we will examine how Machine Learning techniques can be used to obtain meta-models with a quantifiable level of accuracy.

Abstract

In this talk, I review recent efforts on the development of registration methods for parametric model order reduction (MOR), with emphasis on advection-dominated flows. In computer vision and pattern recognition, registration refers to the process of finding a parametric transformation that aligns two datasets; in model order reduction, registration methods seek a parametric bijection that tracks coherent structures (e.g., shocks, shear layers) of the solution field. The ultimate goal is to enhance performance of traditional linear compression methods (e.g., POD) and mesh adaptation techniques for the mapped solution field. We discuss the application of registration techniques to model reduction. First, we illustrate the combination of registration with projection-based reduced-order models and parametric mesh adaptation. Second, we discuss the application of registration to nonlinear interpolation. We present numerical results for two- and three-dimensional parametric compressible flows, to show the potential of the method.
Work in collaboration with Nicolas Barral, Angelo Iollo, Jon Labatut, and Ishak Tifouti (Inria).  

Abstract

We present kinetic type methods able to approximate compressible type flow, with or without viscous and thermal effects. Many numerical example illustrate the methods and show effectiveness. The work is strongly inspired from [1,2,3,4,5].

[1] R. Abgrall and D. Torlo. Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme. Communications in Mathematical Sciences, 20(2):297–326, 2022.
[2] R. Abgrall and F. Nassajian Mojarrad. An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic. Communications on Applied Mathematics and Computation, (0123456789), 2023.
[3] Gauthier Wissocq and Rémi Abgrall. A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds. I: One-dimensional case. J. Comput. Phys., 518:29, 2024. Id/No 113333.
[4] Gauthier Wissocq and Rémi Abgrall. A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds. II: Multi-dimensional case. J. Comput. Phys., 516:27, 2024. Id/No 113376.
[5] Gauthier Wissocq, Yongle Liu, and Rémi Abgrall. A positive- and bound-preserving vectorial lattice Boltzmann method in two dimensions. SIAM SISC, accepted, 2025. ArXiv 2411.15001.  

Abstract

In this paper, we explore the use of the Virtual Element Method concepts to solve scalar and system hyperbolic problems on general polygonal grids. The new schemes stem from the active flux approach [4], which combines the usage of point values at the element boundaries with an additional degree of freedom representing the average of the solution within each control volume. Along the lines of the family of residual distribution schemes introduced in [1, 3] that integrate the active flux technique, we devise novel third order accurate methods that rely on the VEM technology to discretize gradients of the numerical solution by means of a polynomial-free approximation, by adopting a virtual basis that is locally defined for each element. The obtained discretization is globally continuous, and for nonlinear problems it needs a stabilization which is provided by a monolithic convex limiting strategy extended from [2]. This is applied to both point and average values of the discrete solution. We show applications to scalar problems, as well as to the acoustics and Euler equations in two dimension. The accuracy and the robustness of the proposed schemes are assessed against a suite of benchmarks involving smooth solutions, shock waves and other discontinuities.

This is a joint work with W. Boscheri (Laboratoire de Mathematiques UMR 5127 CNRS, Université Savoie Mont Blanc, France), and Yongle Liu (Institute of Mathematics, University of Zurich, Switzerland)

[1] R. Abgrall. A combination of residual distribution and the active flux formulations or a new class of schemes that can combine several writings of the same hyperbolic problem: application to the 1d Euler equations. Commun. Appl. Math. Comput., 5(1):370–402, 2023.
[2] R. Abgrall, M. Jiao, Y. Liu, and K. Wu. Bound preserving Point-Average-Moment PolynomiAl-interpreted (PAMPA) scheme: one-dimensional case. submitted, 2024. Arxiv: 2410.14292.
[3] R. Abgrall, J. Lin, and Y. Liu. Active flux for triangular meshes for compressible flows problems. Beijing Journal of Pure and Applied Mathematics, 2025. in press, also Arxiv preprint 2312.11271.
[4] T.A. Eyman and P.L. Roe. Active flux. 49th AIAA Aerospace Science Meeting, 2011.

Abstract

Hyperbolic balance laws can have particular moving equilibria involving particular source terms or based on divergence-free structures. Maintaining at a discrete level such equilibria (either exactly or very accurately) can give strong advantages with respect to classical methods. The Global Flux technique has been introduced to preserve 1D equilibria with source integrating the source into the flux term and has shown an increased accuracy in combination with many discretizations (FV, FD, FEM, DG). The 2D extensions have shown promising results in the linear acoustics case for vortex-like solutions. We will see what are the limits extending this method to nonlinear and multidimensional problems.

Abstract

Stochastic Galerkin formulations to random hyperbolic equations are based on the idea that the functional dependence on the stochastic input is described a priori by a polynomial chaos expansion and a Galerkin projection is used to obtain deterministic evolution equations for the coefficients in the series. Then, all envolved mathematical operations, e.g. products and norms, must be adopted and applied to the variables in the governing equations. In general, results for hyperbolic systems are not available, since desired properties like hyperbolicity and the existence of entropies are not transferred to the intrusive formulation. To this end, auxiliary variables have been introduced to establish wellposedness results. For instance, entropy-entropy flux pairs can be obtained by an expansion in entropy variables, i.e. the gradient of the deterministic entropy. Roe variables, which include the square root of the density, preserve hyperbolicity for Euler equations. The drawback of introducing auxiliary variables is an additional computational overhead that arises from an optimization problem, which is required to calculate the auxiliary variables. These results exploit quadratic relationships that are expressed efficiently by the Galerkin product. Extensions of the classical polynomial chaos expansions to general nonlinearities, e.g. with Legendre or Hermite polynomials, however, are not straightforward. More general nonlinearities, which occur for instance in isentropic Euler, level-set equations or when including source terms in shallow water equations, can be expressed by wavelet families that are generated by Haar-type matrices. Those are widely used in signal processing, e.g. for the Hadamard, Walsh, Chebyshev matrices and for the discrete cosine transform. In this talk we give an overview on methods to circumvent the loss of hyperbolicity and discuss briefly numerical challenges.

Abstract

Originally developed to model reaction-diffusion systems, Gradient Random Walk methods are particle-based techniques that track the evolution of spatial derivatives of a solution. Inspired by vortex methods for the Navier-Stokes equations, these approaches emerged in the 1990s and attracted significant interest due to their advantageous properties: (i) they eliminate the need for a computational grid, (ii) they naturally adapt to solution features by concentrating particles where gradients are steep, and (iii) they offer a notable reduction in variance.

In this seminar, we revisit and extend these ideas by demonstrating how the Gradient Random Walk framework can be adapted to a wider class of partial differential equations. To achieve this goal, we first extend the classical Monte Carlo method to the relaxation approximation of systems of conservation laws, and subsequently introduce a novel particle dynamics approach based on the spatial derivatives of the solution. This methodology, when combined with an asymptotic-preserving splitting discretization, enables the development of a new class of gradient-based Monte Carlo methods tailored for hyperbolic systems of conservation laws.

Abstract: The mathematical modeling of various real life phenomena involving the interplay of formation and consumption processes leads to non-linear Production--Destruction Systems (PDS) of the form y'(t) = (P(y(t)) - D(y(t))) e, where P,D:R^N \to \mathbb{R}^{N\times N}, \qquad \bm{e}=[1,\dots,1]^\mathsf{T}\in \mathbb{R}^N. In many practical applications, the underlying physics naturally yields PDS that are both positive and fully conservative for which $\Omega=\{ [x_1,\dots,x_N]^\mathsf{T} : 0 \leq x_i \leq \bm{e}^\mathsf{T}\bm{y}(t_0),  \ i=1,\dots,N \} $ is a positively-invariant set and the conservation law $\bm{e}^\mathsf{T}(\bm{y}(t)-\bm{y}(t_0))=0, \ \forall t\geq t_0$ holds true. As a result, there is an increasing demand for numerical integrators that retain both the positivity of the solution and the system's linear invariant with no limitations on the discretization steplength. In this context, \textit{Modified Patankar} (MP) methods have proven particularly beneficial (see \cite{Buchard,Huang2019,Izgin2024,Offner} and references therein). The purpose of this contribution is twofold. First, we present the results of \cite{Izzo2025}, where a novel class of $k$-step, unconditionally positive and conservative \textit{Modified Patankar Linear Multistep} (MPLM) methods is introduced. A thorough theoretical analysis is conducted, with emphasis placed upon the conditions on the \textit{Patankar Weight Denominators} (PWDs) required to achieve arbitrarily high orders of convergence. Additionally, the \(\sigma\)-embedding technique is proposed as a practical tool for the effective computation of PWDs. A comparison with other well-established MP discretizations reveals the competitiveness of the proposed methods for the simulation of PDS. \\ The second part of the present work explores the findings of \cite{MPSL}, where the framework of a \textit{Controlled PDS} (CPDS) \bm{y}'(t) = \bigl(P(\bm{y}(t))\odot \mathcal{P}(\bm{\alpha}(t)) - D(\bm{y}(t))\odot \mathcal{D}(\bm{\alpha}(t))\bigr)\bm{e},\quad \bm{\alpha}: \mathbb{R}^+ \rightarrow A\subset\mathbb{R}^M, \; \qquad \; \mathcal{P},\mathcal{D}: A \rightarrow \mathbb{R}^{N\times N} is outlined with the aim of influencing the dynamics of a PDS without altering its inherent properties. A general finite horizon \textit{Optimal Control Problem} (OCP) is formulated and addressed via the dynamic programming approach. Specifically, the OCP is restated in terms of a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation, whose unique viscosity solution corresponds to the value function \cite{Crandall}. We then propose a parallel-in-space \textit{Modified Patankar Semi-Lagrangian} (MPSL) approximation scheme for the HJB equation \cite{Falcone} and design an uncondionally positive and conservative reconstruction procedure for the optimal control in feedback and open-loop forms. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.