Next Monday March 10, 2025:
FAU DCN-AvH Mini-Workshop on Analysis, Numerics, Control and Machine Learning
Organized by: FAU DCN-AvH, Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship at FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)
Session 01.
Title: Model order reduction for parametric dynamical systems
Speaker: Giulia Sambataro
Affiliation: Postdoctoral researcher at FAU DCN-AvH
Abstract. The numerical approximation of partial differential equations (PDEs) plays a crucial role in various fields, including engineering, mechanics and physics, for design and assessment. To accurately account for uncertainty in parameter values, we must solve the numerical model for a wide range of relevant parameters; an efficient numerical solution of this type of problem is even more challenging in a real time context. Model order reduction (MOR) methods have the purpose to overcome the computational obstacle of numerical simulations to large-scale dynamical systems: MOR has proved, during the last few decades, to be successful in providing rapid and accurate surrogate models under parameter variations. I will show linear and nonlinear solution manifold approximations and their validity for different applications in radioactive waste management and in contact mechanics. In the first part, I will present a component-based model order reduction formulation for parametrized PDEs based on overlapping domain partitions. This formulation—which is the limit of a classic overlapping Schwarz algorithm—reads as an optimization statement that penalizes the jump at the components’ interfaces subject to the approximate satisfation of the PDE in each local subdomain. The well posedness of the mathematical formulation is discussed in [1] for linear coercive problems. I will present numerical investigations for a neo-Hookean nonlinear mechanics problem and for an unstationary thermo-hydro mechanical system in radioactive waste management. The validity of the proposed method is also shown with respect to the standard overlapping Schwarz method. In the second part, I will present the adaptation of recent nonlinear model order reduction approaches to predict the solutions to time-dependent parametrized variational inequalities, in particular to discrete contact problems for crowd motion (cf. [2]). The solution set of this class of problem is characterized by a slow decaying Kolmogorov n-width, which limits the effectiveness of linear compression methods such as the reduced basis technique. I will describe the employment of supervised machine-learning as a postprocessing step of the approximation—by a standard reduced basis model— to provide accurate estimates of particles velocities and contact forces. This work represents a preliminary step towards the investigation of more efficient nonlinear reduced order modeling approaches for a wider class of contact problems.
[1] A. Iollo, G. Sambataro, and T. Taddei (2023). A one-shot overlapping Schwarz method for component- based model reduction: application to nonlinear elasticity. In: Computer Methods in Applied Mechanics and Engineering 404, p. 115786. doi:10.1016/j.cma.2022.115786.
[2] V. Ehrlacher and G. Sambataro (2025) A nonlinear reduced-order model for parametrized variational inequalities: application to crowd motion. Preprint. https://hal.science/hal-04936941
Session 02.
Title: Coagulation with collisional fragmentation and applications in wave turbulence kinetics
Speaker: Arijit Das
Affiliation: Postdoctoral researcher at FAU DCN-AvH
Abstract. In this talk, we examine a nonlinear, hyperbolic population balance equation that simultaneously accounts for both coagulation and collisional fragmentation events. We adopt both analytical and numerical approaches to study this problem. First, we discuss the well-posedness and qualitative properties of solutions, addressing key mathematical aspects. Next, we explore numerical methods for approximating solutions, focusing on the development of numerical schemes and analyzing weak convergence across different grid types. Finally, we highlight applications to wave turbulence kinetics, particularly in the context of the three-wave turbulence equation, which can be approximated by a fully nonlinear collisional type equation. Through numerical simulations, we investigate energy cascading phenomena in three-wave kinetics, providing insights into wave turbulence dynamics.
Session 03.
Title: Control of a Lotka-Volterra System with Weak Competition
Speaker: Maicon Sônego
Affiliation: Humboldt Research Fellow at FAU DCN-AvH
Abstract. In this talk, we explore the controllability of a Lotka-Volterra system modeling weak competition between two species. Using boundary-constrained controls, we establish conditions under which the system can be guided towards different target states. Specifically, we demonstrate finite-time controllability towards coexistence states, whenever they exist, and asymptotic controllability towards single-species dominance or total extinction, depending on domain size, diffusion, and competition rates. Additionally, we identify scenarios where controllability is not achievable and construct barrier solutions that prevent the system from reaching certain states. Our findings provide key insights into the interplay between competition, diffusion, and spatial domain in species dynamics under constrained control strategies.
Session 04.
Title: Boundary and Interior Control in a Diffusive Lotka-Volterra Model
Speaker: João Fernandes Barreira
Affiliation: Postdoctoral researcher (CAPES/DAAD Scholarship) at FAU DCN-AvH
Abstract. In this work, we explore the controllability properties of a generalized diffusive Lotka-Volterra competition model for two species, incorporating both boundary controls and an interior multiplicative control. The system is analyzed within a smooth, bounded N-dimensional domain under various ecological scenarios, with a focus on equilibria representing single-species survival and coexistence states. The research is motivated by real-world applications, where constraints on both the controls and the system state introduce significant complexity.The main results show that a combination of interior multiplicative control and boundary controls enables asymptotic controllability toward single-species survival states and finite-time controllability toward a specific heterogeneous coexistence state.
WHEN
Monday March 10, 2025 at 11:00H
WHERE
Friedrich-Alexander-Universität Erlangen-Nürnberg
Room 01.150-128 Seminarraum. Felix-Klein building
Cauerstraße 11, 91058 Erlangen
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