Date/Time : 19 Jan 2022 16:00
Venue : via Zoom
Speaker : Frédéric Rousset, Université Paris-Sud

Join Zoom Meeting
https://nus-sg.zoom.us/j/89641856972?pwd=dmVFYXo1bXQ5UGxXZCtHYzdWbTh3UT09

Meeting ID: 896 4185 6972
Passcode: 203052

Abstract: We shall study the well-posedness of singular kinetic equations of Vlasov type arising in plasma physics (quasineutral limit) or fluid mechanics (variational solutons of the incompressible Euler equation). Singular means that the force field is given by the gradient of a macroscopic quantity (the density for example) without smoothing. A stability condition on the profile of the initial data is necessary for well-posedness in Sobolev spaces. (joint work with D. Han-Kwan (Ecole Polytechnique))

Add to calendar :

Date/Time : 11 Nov 2021 16:00
Venue : via Zoom
Speaker : Ruiwen Shu, University of Oxford

Join Zoom Meeting
https://nus-sg.zoom.us/j/82284835444?pwd=ZDFaR1RDeDFWQU1iQmFidHYwbUhCQT09

Meeting ID: 822 8483 5444
Passcode: 965013

Abstract:  Erdős and Turán proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erdős-Turán inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erdős-Turán inequality is obtained based on a thorough understanding of these equilibrium distributions.

Add to calendar :

Date/Time : 01 Nov 2021 10:00
Venue : via Zoom
Speaker : Yingzhou Li, Fudan University

Join Zoom Meeting
https://nus-sg.zoom.us/j/86776918257?pwd=cHJ3eDZNclUweWhBaitPZFRieGtNQT09

Meeting ID: 867 7691 8257
Passcode: 061477

Abstract: Coordinate descent methods are considered for eigenvalue problems based on a reformulation of the leading eigenvalue problem as a nonconvex optimization problem. The convergence of several deterministic coordinate methods is analyzed and compared. We also analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes. Under generic assumptions, we prove that the algorithm iterate will almost surely escape strict saddle points of the objective function. As a result, the algorithm is guaranteed to converge to local minima if all saddle points are strict. Numerical examples of applications to quantum many-body problems demonstrate the efficiency and provide benchmarks of the proposed coordinate descent methods.

Add to calendar :

Date/Time : 18 Oct 2021 10:00
Venue : via Zoom
Speaker : Zhennan Zhou, Peking University

Join Zoom Meeting
https://nus-sg.zoom.us/j/85127734314?pwd=amUyZjZEdTQycEhETWNESlFDMEdoUT09

Meeting ID: 851 2773 4314
Passcode: 348196

Abstract:  In this talk, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. In the first part of the talk, we present a conservative and positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models. Secondly, we are concerned with a kinetic model for neuron networks. Individual neurons are characterized by their voltage and conductance, the dynamics of the voltage is influenced by the conductance and when the voltage is reaching a threshold, it is immediately reset to a lower value. By exploring a series of toy models, we aim to identify the cause of the emergence of time-periodic solutions in such Fokker-Planck equations.

Add to calendar :

Date/Time : 15 Oct 2021 16:00
Venue : via Zoom
Speaker : Dietmar Gallistl, Universität Jena

Join Zoom Meeting
https://nus-sg.zoom.us/j/86885266448?pwd=TzJKSVhmZzdITEN3TFh2b0RMMTEzZz09

Meeting ID: 868 8526 6448
Passcode: 006862

Abstract:  This contribution proposes a compatible finite element discretization for the approximation of the inf-sup constant for the divergence. The new approximation replaces the $H^{-1}$ norm of a gradient by a discrete $H^{-1}$ norm which behaves monotonically under mesh-refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. The scheme enables an approximation with arbitrary polynomial degrees. It can be viewed as a Rayleigh-Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. In particular, the computed approximations are guaranteed upper bounds for the inf-sup constant. The novel error estimates prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are presented.

Add to calendar :

Date/Time : 07 Oct 2021 16:00
Venue : via Zoom
Speaker : Xavier Fernández-Real, EPFL

Join Zoom Meeting
https://nus-sg.zoom.us/j/86013696968?pwd=VHJFQkVhVmg5N0Urdk0yV3A2MEJQQT09

Meeting ID: 860 1369 6968
Passcode: 328086

Abstract:  The fractional or thin one-phase problem is the study of minimizers of the functional
J_s(u) = [u]_{H^s(B_1)}^2 + |{u > 0} \cap B_1|.
The aim of this talk is to study homogeneous stable solutions to the fractional one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions n ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons’ cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open.

The goal of this talk is twofold and is based on a recent work with X. Ros-Oton. On the one hand, we present our first main contribution: the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of “large solutions” for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions n ≤ 5 is one-dimensional, independently of the parameter s∈(0,1).

Add to calendar :

Date/Time : 27 Sep 2021 10:00
Venue : via Zoom
Speaker : Timo Sprekeler, National University of Singapore

Join Zoom Meeting
https://nus-sg.zoom.com.cn/j/86200274068?pwd=eHlLS0JHMWprU0VjVTN6WWtOVE95QT09

Meeting ID: 862 0027 4068
Passcode: 578977

Abstract:  This talk presents recent progress in the analysis and the finite element approximation of periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. In the first part of the talk, we discuss optimal rates for the convergence of $(u_{\varepsilon})$ to the solution of the homogenized problem in $L^{\infty}$ and $W^{1,p}$ norms, and we present a $W^{2,p}$ corrector estimate. These results are used in the second part of the talk to construct and analyze a finite element method for the approximation of such problems. This talk is based on work with Hung Tran (University of Wisconsin-Madison), and work with Yves Capdeboscq (Université de Paris, Sorbonne Université) and Endre Süli (University of Oxford).

Add to calendar :