Abstract : The aim of this minisymposium is to gather researchers involved in the mathematical analysis of non-linear and non-local partial differential equations (PDEs), with emphasis on those modelling aggregation and/or diffusion phenomena. These PDEs are relevant in applications to physics, biology, population dynamics, data science, etc. The spectrum of possible mathematical approaches involves techniques from functional analysis, optimal transport theory, variational methods, etc. It is at the core of our minisymposium to touch upon recent advances in the study of aggregation-diffusion PDEs obtained, e.g., using generalised gradient flows, incompressible limits, particle approximations, numerical methods, symmetrisation and rearrangements, and Fourier analysis.
Organizer(s) : Antonio Esposito, David Gómez-Castro
[04031] Quantified overdamped limit for Vlasov-Fokker-Planck equations with singular interaction forces
Format : Talk at Waseda University
Author(s) :
Young-Pil Choi (Yonsei University)
Abstract : In this talk, I will discuss a quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean-Vlasov equation. Our strategy only requires weak
integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov-Poisson-Fokker-Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.
[04055] A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model
Format : Talk at Waseda University
Author(s) :
Noemi David (University of Lyon)
Tomasz Dębiec (University of Warsaw)
Mainak Mandal (Technische Universität Dresden)
Markus Schmidtchen (Technische Universität Dresden)
Abstract : In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).
[03802] Nonlocal particle approximations of the porous medium equation
Format : Online Talk on Zoom
Author(s) :
Katy Craig (University of California, Santa Barbara)
Olga Turanova (Michigan State University)
Karthik Elamvazhuthi (University of California, Riverside)
Matt Haberland (Cal Poly, San Luis Obispo)
Abstract : Given a desired target distribution and an initial guess of its samples, what is the best way to evolve the locations of the samples so that they accurately represent the desired distribution? A classical solution to this problem is to evolve the samples according to Langevin dynamics, a stochastic particle method for the Fokker-Planck equation. In today’s talk, I will contrast this with a nonlocal, deterministic particle method inspired by the porous medium equation. Using the Wasserstein gradient flow structure of the equations and Serfaty’s scheme of Gamma-convergence of gradient flows, I will show that, as the number of samples increases and the interaction scale goes to zero, the interacting particle system indeed converges to a solution of the porous medium equation. I will close by discussing practical implications of this result to both sampling and the training dynamics two-layer neural networks. This is based on joint work with Karthik Elamvazhuthi, Matt Haberland, and Olga Turanova.
[05056] A convergent discretization of the porous medium equation with fractional pressure
Format : Talk at Waseda University
Author(s) :
Félix del Teso (Universidad Autónoma de MadridU)
Abstract : We carefully construct and prove convergence of what is to our knowledge the first numerical discretization of the porous medium equation with fractional pressure,
\begin{equation}\tag{FPE}
\frac{\partial u}{\partial t}-\nabla\cdot\left(u^{m-1}\nabla(-\Delta)^{-\sigma}u\right)=0,
\end{equation}
for $\sigma\in(0,1)$. The model was introduced by Caffarelli and Vázques in 2011, and is currently one of two main nonlocal extensions of the local porous medium equation. It has finite speed of propagation and comes from a nonlocal Fick's law, but as opposed to the other extension, it does not satisfy the comparison principle. Without comparison, the analysis is difficult. Uniqueness is only known in 1d, where one can exploit that the ``cumulative density'' $v(x,t)=\int_{-\infty}^yu(y,t)dy$ satisfies
\begin{equation}
\frac{\partial v}{\partial t}+|\partial_xv|^{m-1}(-\Delta)^s v=0,\quad s=1-\sigma,
\end{equation}
which is a nonlocal quasilinear parabolic equation in nondivergence form that can be analyzed through viscosity solution methods.
Our numerical method then loosely speaking consists in discretizing this ``integrated'' equation with a difference quadrature scheme and then compute the solution $u$ of (FPE) via numerical differentiation. Using upwinding in non-traditional way, we obtain a new type of monotone schemes that allows for convergence analysis via the Barles-Perthame-Souganidis half-relaxed limit method. Combining this result with tightness arguments, we then prove convergence of the approximations of the original problem in the Rubinstein-Kantorovich/Wasserstein-1 distance uniformly in time.
Our results cover both absolutely continuous and Dirac or point mass initial data, and in the latter case, machinery for discontinuous viscosity solutions are needed in the analysis.
00036 (2/3) : 1D @G602 [Chair: David Gómez-Castro]
[05097] Uniform spectral gap in nonlocal-to-local approximations of diffusion
Format : Talk at Waseda University
Author(s) :
José Cañizo (Universidad de Granada)
Abstract : We consider the nonlocal diffusion equation $\partial_t u = \frac{1}{\epsilon^2} (J_\epsilon * u - u) + \nabla \cdot (x u)$, where $J_\epsilon(x) = \epsilon^{-d} J(x/\epsilon)$, posed for $t \geq 0$ and $x \in \mathbb{R}^d$. This equation approximates the standard Fokker-Planck equation as $\epsilon \to 0$, and serves as a good test ground for several techniques used to study the long-time behaviour of nonlocal PDE. In particular, entropy techniques are not easy to apply here, and lead to functional inequalities which are not well understood. Using probabilistic techniques we show that the above equation has a uniformly positive spectral gap as $\epsilon \to 0$, and we link this to quantitative versions of the central limit theorem. This problem has links to numerical analysis and to some models in mathematical biology, which will also be discussed.
[03996] Concentration phenomena arising in Aggregation Fast-Diffusion equations
Format : Talk at Waseda University
Author(s) :
Alejandro Fernández-Jiménez (University of Oxford)
Abstract : We will discuss about the asymptotic behaviour of the family of Aggregation-Diffusion Equations
$$
\partial_t\rho=\Delta\rho^m+div(\rho\nabla (V+W\ast\rho )),
$$
for $0 < m < 1$. We rely on compactness arguments, and the gradient flow structure of the problem to obtain convergence of the solutions when $t\to\infty$. Then, we pass to the mass equation and we use viscosity solutions to characterise the limit and discuss the existence of Dirac deltas.
The talk presents joint work with Prof. J.A. Carrillo and Prof. D. Gómez-Castro.
[05394] Sharp uniform-in-time propagation of chaos on the torus
Format : Online Talk on Zoom
Author(s) :
Rishabh Sunil Gvalani (Max Planck Institute for Mathematics in the Sciences)
Matías Delgadino (The University of Texas at Austin)
Abstract : We prove uniform-in-time propagation of chaos for weakly interacting diffusions with gradient drift on the torus. Our results are sharp both in terms of the rate (i.e. $N^{-\frac 12}$) and validity (they hold in the full subcritical range of temperatures). The proof relies on directly controlling the path-space relative entropy between the $N$ particle system and $N$ i.i.d copies of the synchronously-coupled limiting McKean SDE. This is joint work with Matías Delgadino.
[04109] Limiting gradient flow structure of deep linear neural networks
Format : Online Talk on Zoom
Author(s) :
Xavier Fernandez-Real (EPFL)
Abstract : We present recent results with L. Chizat, M. Colombo, A. Figalli, on the infinite-width limit of deep linear neural networks initialized with random parameters. We show that, when the number of neurons diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear neural network.
00036 (3/3) : 1E @G602 [Chair: José Antonio Carrillo]
Abstract : We investigate the existence of weak type solutions for a class of aggregation–diffusion PDEs with nonlinear mobility obtained as deterministic large particle limit of a suitable nonlocal versions of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. At the same time, we expose a rigorous gradient flow structure for this class of equations in terms of an Energy-Dissipation balance, which we obtain via the asymptotic convergence of functionals. The well-posedness is also investigated for aggregation/diffusion equation modeling the evolution of opinion formation on an evolving graph.
[03893] Non-local PDEs on graphs
Format : Online Talk on Zoom
Author(s) :
André Schlichting (University of Münster)
Abstract : This talk reviews some recent results on nonlocal PDEs describing the evolution of a density on discrete graph structures. These structures can arise from applications in the data science field, or they can also be obtained by a numerical discretization of a continuum problem. We also show how those equations are linked to their continuous counterpart in suitable local limits.
