DATAHYKING is a Marie Curie Doctoral Network funded by the European commission under Grant Agreement No. 101072546. The DATAHYKING Doctoral Network aims at training a new generation of modeling and simulation experts to develop virtual experimentation tools and workflows that can reliably and efficiently exploit the potential of mathematical modeling and simulation of interacting particle systems.
To this end, we create a data-driven simulation framework for kinetic models of interacting particle systems, and define a common methodology for these future modeling and simulation experts. DATAHYKING will focus on:
Developing reliable and efficient simulation methods;
Designing robust consensus-based optimisation, also for machine learning;
Developing multifidelity methods for uncertainty quantification and data assimilation;
Applications in traffic flow, finance and granular flow, also in collaboration with industry.
The 5th edition of the conference Asymptotic Behavior of systems of PDEs arising in physics and biology (ABPDE 5) will take place in Lille on June 5-9. People interested in attending and possibly giving a presentation can register here.
This short note presents an extension of the hybrid, model-adaptation method introduced in [T. Laidin, arXiv 2202.03696, 2022] for linear collisional kinetic equations in a diffusive scaling to the nonlinear mean-field Vlasov-Poisson-BGK model. The aim of the approach is to reduce the computational cost by taking advantage of the lower dimensionality of the asymptotic model while reducing the overall error. It relies on two criteria motivated by a perturbative approach to obtain a dynamic domain adaptation. The performance of the method and the conservation of mass are illustrated through numerical examples.
I have defended my Habilitation thesis! The manuscript is available here, go check it out.
This habilitation thesis covers a large part of the research I have carried out since the end of my PhD training. This work has been essentially aimed at modeling, analyzing and simulating systems composed of a large number of interacting particles or agents. These systems can be described through the framework of collisional kinetic equations, such as the seminal Boltzmann’s equation, the granular gases equations, or kinetic systems of collective behavior. Such models describes complex and sometimes vital situations in a modern, evolving world, such as pollution, traffic and disease spreading, to name but a few. To understand them mathematically, to analyze them, and to be able to solve them accurately and efficiently with numerical methods is an important modern problem. With my scientific collaborators, we have worked on three main topics that are presented in this text. The first one concerns the development of new spectral methods to calculate efficiently and with the highest possible accuracy the so-called collision operators which intervene in these models, to analyze them and to implement them efficiently. The second one concerns the understanding of the granular gas equation, a model which is still largely open and misunderstood, and in particular its hydrodynamic limits of the compressible type. The third, finally, concerns the study of asymptotic behaviors in long time and small parameters of these equations, and the development of numerical methods preserving these behaviors, the so-called AP methods.
We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on pollinators density. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interactions graph and the quantity of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.
Spectral methods, thanks to their high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the collisional kinetic equations of Boltzmann type, such as the Boltzmann-Nordheim equation. This equation, modeled on the seminal Boltzmann equation, describes using a statistical physics formalism the time evolution of a gas composed of bosons or fermions. Using the spectral-Galerkin algorithm introduced in [F. Filbet, J. Hu, and S. Jin, ESAIM: Math. Model. Numer. Anal., 2011], together with some novel parallelization techniques, we investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, we are able to observe numerically both Bose-Einstein condensation and Fermi-Dirac relaxation.
The 4th edition of the conference Asymptotic Behavior of systems of PDEs arising in physics and biology (ABPDE4) will take place in Lille on November 16-19. People interested in attending and possibly giving a presentation can register here
We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and BGK equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.
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