L1

Title: Topics on regularity theory for fully non-linear integro-differential equations

Abstract: We will focus on local and non-local Monge Ampere type equations, equations with deforming kernels and convex envelopes of functions with optimal special conditions. We discuss global solutions and their regularity properties.

Title: Quasilinear Conservative Collisional Transport in Kinetic Mean Field models

Abstract: We shall focus the on the interplay of nonlinear analysis  and numerical approximations to mean field models in particle physics where kinetic transport flows in momentum are strongly nonlinearly  modified by macroscopic quantities in classical or spectral density spaces. Two noteworthy models arise: the classical Fokker-Plank Landau dynamics as a low magnetized plasma regimes in the modeling of perturbative non-local high order terms. The other one corresponds to perturbation under strongly magnetized dynamics for fast electrons  in momentum space  give raise to a coupled system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics on spectral energy waves  (quasi-particles) in a quantum process of their resonant interaction. Both models are rather different, yet there are derived form the Liouville-Maxwell system under different scaling. Analytical tools and some numerical  simulations show a presence of  strong hot tail anisotropy  formation taking the stationary states away from Classical equilibrium solutions stabilization for the iteration in a three dimensional cylindrical model. The semi-discrete schemes preserves the total system mass, momentum and energy, which are enforced by the numerical scheme. Error estimates can be obtained as well.

Work in collaboration with Clark Pennie and Kun Huang

Regularization of linear least squares problems is necessary in a variety of contexts. However, the optimal regularization parameter is usually unknown a priori and is often to be determined in an ad hoc manner, which may involve solving the problem for multiple regularization parameters. In this talk, we will discuss three randomized algorithms, building on the sketch-and-precondition framework in randomized numerical linear algebra (RNLA), to efficiently solve this set of problems. In particular, we consider preconditioners for a set of Tikhonov regularization problems to be solved iteratively. The first algorithm is a Cholesky-based algorithm employing a single sketch for multiple parameters; the second algorithm is SVD-based and improves the computational complexity by requiring a single decomposition of the sketch for multiple parameters. Finally, we introduce an algorithm capable of exploiting low-rank structure (specifically, low statistical dimension), requiring a single sketch and a single decomposition to compute multiple preconditioners with low-rank structure. This algorithm avoids the Gram matrix, resulting in improved stability as compared to related work.

Time-optimal control can be used to improve driving efficiency for autonomous
vehicles and it enables us explore vehicle and driver behaviour in extreme
situations. Due to the computational cost and limited scope of classical
optimal control methods we have seen new interest in applying reinforcement
learning algorithms to autonomous driving tasks.
In this talk we present methods for translating time-optimal vehicle control
problems into reinforcement learning environments. For this translation we
construct a sequence of environments, starting from the closest representation
of our optimisation problem, and gradually improve the environments reward
signal and feature quality. The trained agents we obtain are able to generalise
across different race tracks and obtain near optimal solutions, which can then
be used to speed up the solution of classical time-optimal control problems.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain or a parameter in the equation. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving an optimization problem constrained by an augmented system of equations that characterize the location of the branch points. The flexibility and robustness of the method also allow us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency. We will apply this technique on systems arising from biology, fluid dynamics, and engineering, such as the FitzHugh-Nagumo model, Navier-Stokes, and hyperelasticity equations.

This talk is about the mathematics behind an artistic project focusing on the vibrations of soap films.

Latent Dirichlet Allocation (LDA) is capable of analyzing thousands of documents in a short time and highlighting important elements, recurrences and anomalies. It is generally used in linguistics to study natural language: its word analysis reveals the theme(s) of a document, each theme being identified by a specific vocabulary or, more precisely, by a particular statistical distribution of word frequency.
In the climatologists' use of LDA, the document is a daily weather map and the word is a pixel of the map. The theme with its corpus of words can become a cyclone or an anticyclone and, more generally, a 'pattern'  that the scientists term motif. Artificial intelligence – a sort of incredibly fast robot meteorologist – looks for correlations both between different places on the same map, and between successive maps over time. In a sense, it 'notices' that a particular location is often correlated with another location, recurrently throughout the database, and this set of correlated locations constitutes a specific pattern.
The algorithm performs statistical analyses at two distinct levels: at the word or pixel level of the map, LDA defines a motif, by assigning a certain weight to each pixel, and thus defines the shape and position of the motif;  LDA breaks down a daily weather map into all these motifs, each of which is assigned a certain weight.
In concrete terms, the basic data are the daily weather maps between 1948 and nowadays over the North Atlantic basin and Europe. LDA identifies a dozen or so spatially defined motifs, many of which are familiar meteorological patterns such as the Azores High, the Genoa Low or even the Scandinavian Blocking. A small combination of those motifs can then be used to describe all the maps. These motifs and the statistical analyses associated with them allow researchers to study weather phenomena such as extreme events, as well as longer-term climate trends, and possibly to understand their mechanisms in order to better predict them in the future.

The preprint of the study is available as:
 Lucas Fery, Berengere Dubrulle, Berengere Podvin, Flavio Pons, Davide Faranda. Learning a weather dictionary of atmospheric patterns using Latent Dirichlet Allocation. 2021. ⟨hal-03258523⟩ https://hal-enpc.archives-ouvertes.fr/X-DEP-MECA/hal-03258523v1
 

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two projects related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells. 

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.