arXiv subjects

Nader Masmoudi

Publications and source records attributed to Nader Masmoudi.

At least 37 records · Page 2Linked to original sources

arXiv2023

SEEDS: Exponential SDE Solvers for Fast High-Quality Sampling from Diffusion Models

A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is obtained by solving differential equations (DE) defined by the learnt model, a process which has shown to be prohibitively slow. Numerous efforts on speeding-up this process have consisted on crafting powerful ODE solvers. Despite being quick, such solvers do not usually reach the optimal quality achieved by available slow SDE solvers. Our goal is to propose SDE solvers that reach optimal quality without requiring several hundreds or thousands of NFEs to achieve that goal. We propose Stochastic Explicit Exponential Derivative-free Solvers (SEEDS), improving and generalizing Exponential Integrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exact solutions of diffusion SDEs, we craft SEEDS to analytically compute the linear part of such solutions. Inspired by the Exponential Time-Differencing method, SEEDS use a novel treatment of the stochastic components of solutions, enabling the analytical computation of their variance, and contains high-order terms allowing to reach optimal quality sampling $\sim3$-$5\times$ faster than previous SDE methods. We validate our approach on several image generation benchmarks, showing that SEEDS outperform or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.

cs.LG↗

arXiv2023

Stability Analysis of a Non-Separable Mean-Field Games for Pedestrian Flow in Large Corridors

We investigate the existence and stability of small perturbations of constant states of the generalized Hughes model for pedestrian flow in an infinitely large corridor. We show that constant flows are stable under a condition on the density. Our findings indicates that when the density is less than half of the maximum density $ρ_{m}/2$, which is the Lasry-Lions monotonicity condition, we can control the perturbation and prove positive stability results for the nonlinear Generalized Hughes model. However, due to wave propagation phenomena, we are unable to provide an answer for stability results when the density is higher. Our approach involves constructing an explicit solution for the linear problem in Fourier analysis and demonstrating, through a fixed-point argument, how to construct the solution for the full nonlinear mean-field games system.

math.AP↗

arXiv2023

Non-separable Mean Field Games for Pedestrian Flow: Generalized Hughes Model

In this paper, we present a new generalized Hughes model designed to intelligently depict pedestrian congestion dynamics, allowing pedestrian groups to either navigate through or circumvent high-density regions. First, we describe the microscopic settings of the model. The corresponding optimization problems are deterministic and can be formulated by a closed-loop model predictive control strategy. This microscopic setup leads in the mean-field limit to the Generalized Hughes model which is a class of non-separable mean field games system, i.e., Fokker-Planck equation and viscous Hamilton-Jacobi Bellman equation are coupled in a forward-backward structure. We give an overview on the mean field games in connection to our intelligent fluid model. Therefore, we show the existence of weak solutions to the Generalized Hughes model and analyze the vanishing viscosity limit of weak solutions. Finally, we illustrate the generalized Hughes model with various numerical experiments.

math.AP↗

arXiv2023

Diffusive limits of the steady state radiative heat transfer system: Curvature effects

This paper is devoted to the diffusive limit of the nonlinear radiative heat transfer system with curved boundary domain (\textit{two dimensional disk}). The solution constructed in \cite{ghattassi2022convergence} by the leading order interior solution and the boundary layer corrections fails here to approximate the solutions in $L^\infty$ sense for the diffusive limit. The present paper aims to construct a geometric correction to the boundary layer problem and obtain a valid approximate solution in $L^\infty$ sense. The main tools to overcome the convergence problem, are to use matched asymptotic expansion techniques, fixed-point theorems, linear and nonlinear stability analysis of the boundary layer problem. In particular, the spectral assumption on the leading order interior solution, which was proposed for the flat case in \cite{Bounadrylayer2019GHM2}, is shown to be still valid which guarantee the stability of the boundary layer expansion with geometric corrections. Moreover, the convergence result established in \cite[Lemma 10]{ghattassi2022convergence} remain applicable for the approximate solution with geometric corrections.

math.AP↗

arXiv2023

Tollmien-Schlichting waves in the subsonic regime

The Tollmien-Schlichting (T-S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo and Nguyen gave the first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T-S waves of temporal mode for certain boundary layer profiles with Mach number $m<\frac{1}{\sqrt 3}$. In this paper, we construct the T-S waves of both temporal mode and spatial mode to the linearized compressible Navier-Stokes system around the boundary layer flow in the whole subsonic regime $m<1$, including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi-incompressible and quasi-compressible systems, with a key ingredient being the solution of an Orr-Sommerfeld type equation using a new Airy-Airy-Rayleigh iteration instead of Rayleigh-Airy iteration introduced by Grenier, Guo and Nguyen. We believe the method developed in this work can be applied in solving other related problems for subsonic flows.

math.AP↗

arXiv2023

A dynamical approach to the study of instability near Couette flow

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier-Stokes equations with small viscosity $ν>0$, when the perturbations are in the critical spaces $H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size $ν^{\frac{1}{2}-δ_0}$ with any small $δ_0>0$, which implies that $ν^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo-spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues.

math.AP↗

arXiv2023

Time quasi-periodic vortex patches of Euler equation in the plane

We prove the existence of time quasi-periodic vortex patch solutions of the 2$d$-Euler equations in $\mathbb{R}^2$, close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -- as well as the degeneracy of a normal frequency -- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.

math.AP↗

arXiv2023

Space quasi-periodic steady Euler flows close to the inviscid Couette flow

We prove the existence of steady \emph{space quasi-periodic} stream functions, solutions for the Euler equation in vorticity-stream function formulation in the two dimensional channel ${\mathbb R}\times [-1,1]$. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin's cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.

math.AP↗

arXiv2022

Higher Regularity Theory for a Mixed-Type Parabolic Equation

In this paper, we study the higher regularity theory of a mixed-type parabolic problem. We extend the recent work of \cite{DMR} to construct solutions that have an arbitrary number of derivatives in Sobolev spaces. To achieve this, we introduce a counting argument based on a quantity called the "degree". In the second part of this paper, we apply this existence theory to the Prandtl system near the classical Falkner-Skan self-similar profiles in order to supplement the stability analysis of \cite{IM22} with a rigorous construction argument.

math.AP↗

arXiv2022

Diffusive limits of the steady state radiative heat transfer system: Boundary layers

In this paper, we study the diffusive limit of the steady state radiative heat transfer system for non-homogeneous Dirichlet boundary conditions in a bounded domain with flat boundaries. A composite approximate solution is constructed using asymptotic analysis taking into account of the boundary layers. The convergence to the approximate solution in the diffusive limit is proved using a Banach fixed point theorem. The major difficulty lies on the nonlinear coupling between elliptic and kinetic transport equations. To overcome this problem, a spectral assumption ensuring the linear stability of the boundary layers is proposed. Moreover, a combined $L^2$-$L^\infty$ estimate and the Banach fixed point theorem are used to obtain the convergence proof. This results extend our previous work \cite{ghattassi2020diffusive} for the well-prepared boundary data case to the ill-prepared case when boundary layer exists.

math.AP↗

arXiv2022

Asymptotic stability of two-dimensional Couette flow in a viscous fluid

In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity $ν>0$ in $\mathbb{T}\times\mathbb{R}$. It's generally known the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the size and the regularity of the initial perturbation that makes the nonlinear asymptotic stability holds. More precisely, we proved that if the initial perturbation is in some Gevrey-$\frac{1}{s}$ class with size $εν^β$ where $s\geq \frac{1-3β}{2-3β}$ and $β\in [0,\frac{1}{3}]$, then the nonlinear asymptotic stability holds.

math.AP↗

arXiv2022

Stability of the nonlinear Milne Problem for radiative heat transfer system

This paper focuses on the nonlinear Milne problem of the radiative heat transfer system on the half-space. The nonlinear model is described by a second order ODE for temperature coupled to transport equation for radiative intensity. The nonlinearity of the fourth power Stefan-Boltzmann law of black body radiation, bring additional difficulty in mathematical analysis, compared to the well-developed theory for Milne problem of linear transport equation. With the help of the monotonicity property of the second order ODE, we prove the existence of the nonlinear Milne problem on a finite interval using monotonic convergence theorems. Then the solution is extended to the half-space using a uniform weighted estimate and the compactness method. Moreover, the solutions are proved to converge to constants as $x\to \infty$. Therefore, the linear stability analysis is used to study the uniqueness of the nonlinear Milne problem. The existence and uniqueness for the linearized system is established under a spectral assumption on the solution of the nonlinear problem. The spectral assumption is shown to be satisfied when the boundary data is close to the well-prepared case by using a generalized Hardy's inequality. The uniqueness of the solution to the half-line nonlinear Milne problem is established in a neighborhood of solutions satisfying a spectral assumption and the energy estimate.The current work extends the study of Milne problem for linear transport equations and provides a comprehensive study on the nonlinear Milne problem of radiative heat transfer systems.

math.AP↗

arXiv2022

Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations

For any positive regularity parameter $β< \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^β$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.

math.AP↗

arXiv2022

Localisation of perturbations of a constant state in a traffic flow model

We consider, in the Aw-Rascle-Zhang traffic flow model, the problem of the asymptotic stability of constant flows. By using a perturbative approach, we show the stability in a larger space of perturbation than previous results. Furthermore, we are able to compute where the perturbation is mainly localised in space for a given time, based on the localisation of the perturbation initially. These new ideas can be applied to various other models of hyperbolic conservation laws with relaxations.

math.AP↗

arXiv2022

On singularity formation for the two dimensional unsteady Prandtl system around the axis

We consider the two dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a closed one dimensional equation. First, we give a precise description of singular solutions for this reduced problem. A stable blow-up pattern is found, in which the blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. Second, in the case where, for a general analytic solution, this trace of the derivative on the axis follows the stable blow-up pattern, we show persistence of analyticity around the axis up to the blow-up time, and establish a universal lower bound of $(T-t)^{7/4}$ for its radius of analyticity.

math.AP↗

arXiv2022

Asymptotic stability for two-dimensional Boussinesq systems around the Couette flow in a finite channel

In this paper, we study the asymptotic stability for the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity $ν$ and small thermal diffusion $μ$ in a finite channel. In particular, we prove that if the initial velocity and initial temperature $(v_{in},ρ_{in})$ satisfies $\|v_{in}-(y,0)\|_{H_{x,y}^2}\leq \e_0 \min\{ν,μ\}^{\f12}$ and $\|ρ_{in}-1\|_{H_x^{1}L_y^2}\leq \e_1 \min\{ν,μ\}^{\f{11}{12}}$ for some small $\e_0,\e_1$ independent of $ν, μ$, then for the solution of the two-dimensional Navier-Stokes Boussinesq system, the velocity remains within $O(\min\{ν,μ\}^{\f12})$ of the Couette flow, and approaches to Couette flow as $t\to\infty$; the temperature remains within $O(\min\{ν,μ\}^{\f{11}{12}})$ of the constant $1$, and approaches to $1$ as $t\to\infty$.

math.AP↗

arXiv2022

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

math.AP↗

arXiv2021

KAM theory for active scalar equations

In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $({\rm gSQG})_α$ in the patch form close to Rankine vortices. We show that invariant tori survive when the order $α$ of the singular operator belongs to a Cantor set contained in $(0,\frac12)$ with almost full Lebesgue measure. The proof is based on several techniques from KAM theory, pseudo-differential calculus together with Nash-Moser scheme in the spirit of the recent works \cite{Baldi-Berti2018,Berti-Bolle15}. One key novelty here is a refined Egorov type theorem established through a new approach based on the kernel dynamics together with some hidden Töpliz structures.

math.AP↗