Past PDE CDT Lunchtime Seminar

2 December 2021
12:00
Ana Djurdjevac

Further Information: 

A Zoom link to the talk will be circulated to the mailing list on Wednesday, 1 December.  Please contact Benjamin Fehrman to be added.

Abstract

We will consider the viscous Burgers driven by a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. We will prove that any solution of the equation in question can be exponentially stabilised. Combining this result with an earlier result on local exact controllability we will show global exact controllability by a localised control. This is a joint work with A. Shirikyan.

  • PDE CDT Lunchtime Seminar
11 November 2021
16:00
Joshua Daniels-Holgate
Abstract

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

  • PDE CDT Lunchtime Seminar
28 October 2021
12:00
Abstract

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

  • PDE CDT Lunchtime Seminar
17 June 2021
12:00
Alexis Michelat

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

The integral of mean curvature squared is a conformal invariant that measures the distance from a given immersion to the standard embedding of a round sphere. Following work of Robert Bryant who showed that all Willmore spheres in the 3-sphere are conformally minimal, Robert Kusner proposed in the early 1980s to use the Willmore energy to obtain an “optimal” sphere eversion, called the min-max sphere eversion.

We will present a method due to Tristan Rivière that permits to tackle a wide variety of min-max problems, including ones about the Willmore energy. An important step to solve Kusner’s conjecture is to determine the Morse index of branched Willmore spheres, and we show that the Morse index of conformally minimal branched Willmore spheres is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.

  • PDE CDT Lunchtime Seminar
10 June 2021
17:00
Casey Rodriguez

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

  • PDE CDT Lunchtime Seminar
13 May 2021
12:00
Yufei Zhang

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional PDEs, such as those arising from stochastic control and games. In the first part, we show that in certain cases, DNNs can break the curse of dimensionality in representing high-dimensional value functions of stochastic control problems. We then exploit policy iteration to reduce the associated nonlinear PDEs into a sequence of linear PDEs, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions and their derivatives converge globally, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration. Numerical experiments on Zermelo's navigation problem and on consensus control of interacting particle systems are presented to demonstrate the effectiveness of the method. This is joint work with Kazufumi Ito, Christoph Reisinger and Wolfgang Stockinger.

  • PDE CDT Lunchtime Seminar
11 March 2021
12:00
Mathias Schäffner

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

  • PDE CDT Lunchtime Seminar
25 February 2021
12:00
Arianna Giunti

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.

  • PDE CDT Lunchtime Seminar
18 February 2021
17:00
Cole Graham

Further Information: 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

We explore one facet of an old problem: the approximation of hyperbolic conservation laws by viscous counterparts. While qualitative convergence results are well-known, quantitative rates for the inviscid limit are less common. In this talk, we consider the simplest case: a one-dimensional scalar strictly-convex conservation law started from "generic" smooth initial data. Using a matched asymptotic expansion, we quantitatively control the inviscid limit up to the time of first shock. We conclude that the inviscid limit has a universal character near the first shock. This is joint work with Sanchit Chaturvedi.

  • PDE CDT Lunchtime Seminar
4 February 2021
12:00
Dr Matias G. Delgadino
Abstract

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how the convexity properties of the thermodynamic energy affects the homogenization limit or the stability of the log-Sobolev inequality.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • PDE CDT Lunchtime Seminar

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