Applied Math Colloquium - Fall 2020
Colloquia are held on Fridays at 11:30 a.m. via Webex, unless otherwise noted. Webex information will be posted at a later date.
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September 11
Cecilia Freire (Drexel)
Rates of Convergence to Statistical Equilibrium: A General Approach and Applications
Randomness is an intrinsic part of many physical systems. For example, it might appear due to uncertainty in the initial data, or in the derivation of the mathematical model, or also in observational measurements. In this talk, we focus on the study of convergence/mixing rates for stochastic/random dynamical systems towards statistical equilibrium. Our approach uses the weak Harris theorem combined with a generalized coupling technique to obtain such rates for infinite-dimensional stochastic systems in a suitable Wasserstein distance. We emphasize the importance of obtaining these results via algorithms that are well-defined in infinite dimensions. This allows to obtain convergence rates that are robust with respect to finite-dimensional approximations, thus beating the curse of dimensionality. In particular, we show two scenarios where this approach is applied in the context of stochastic fluid flows. First, to show that Markov kernels constructed from a suitable numerical discretization of the 2D stochastic Navier-Stokes equations converge towards the invariant measure of the continuous system. Second, to approximate the posterior measure obtained via a Bayesian approach to inverse PDE problems, particularly when applied to advection-diffusion type PDEs. Here we present an alternative proof of mixing rates for the exact preconditioned Hamiltonian Monte Carlo algorithm in infinite dimensions, a result that was still an open problem until quite recently. This talk is based on joint works with Nathan Glatt-Holtz (Tulane U).
September 18
Dwight Barkley (Warwick)
Mechanisms and Universality in the Subcritical Route to Turbulence
Recent years have witnessed a profound change in our understanding of the route to turbulence in wall-bounded shear flows. In stark contrast to the classical Hopf-Landau picture where turbulence arises through an increase in the temporal complexity of fluid motion, the route to turbulence in subcritical shear flows occurs via spatio-temporal intermittency and falls in the class of non-equilibrium statistical phase transitions known as directed percolation. In this talk I will focus on two aspects of the transition problem. The first is physical mechanisms underlying spatio-temporal intermittency in shear flows. The second is the universality in the subcritical route to turbulence.
September 25
Dominic Vella (Oxford)
Buffering by Buckling: New wrinkles on Gauss’ Pizza Theorem
The deformations of thin elastic objects are familiar from everyday life, from a piece of paper or clothes crumpling, to vibrations of bridges and other structures. A guiding principle of such deformations is that the material chooses to bend, rather than change its length, whenever possible. I will describe some of the consequences of this desire to avoid stretching and Gauss’ Remarkable Theorem, which include everyday examples like the geometrical rigidification of pizza slices by curving the crust. However, I will show that these consequences can be subverted by buckling instabilities such as wrinkling. Finally, I will talk about how fast these deformations happen.
October 2
Jonathan Weare (Courant Institute)
Long Timescale Behavior from Trajectory Data
The transition operator of a Markov process encodes all evolutionary statistics of the process. For example, the most slowly decorrelating functions of the process are encoded as certain eigenvectors of the transition operator. In cases of practical interest, however, the transition operator is not known in closed form. Instead our only access to the transition operator is through random trajectories of the Markov process. These observations have motivated a huge body of research focused on dynamical spectral estimation, which approximates eigenvectors of the transition operator and their corresponding eigenvalues using random trajectory data. Despite intense methodological development, the mathematical understanding of the error properties of dynamical spectral estimation remains incomplete. In this talk I will discuss our error analysis of a leading algorithm for dynamical spectral estimation, called the variational approach for conformation dynamics (VAC). We derive explicit mathematical relationships revealing the dependence of VAC error on key design parameters. In a second part of the talk I will describe algorithms that we are currently developing to compute predictions of long timescale phenomenon (i.e. rare events) using only relatively short trajectory data.
October 9
Kui Ren (Columbia)
Computational Inversion with the Quadratic Wasserstein Metrics
The quadratic Wasserstein distance has recently been proposed as an alternative to the classical $L^2$ distance for measuring data mismatch in computational inverse problems. Extensive computational evidences showing the advantages of using the Wasserstein distance has been reported. The objective of this talk is to provide some simple observations that might help us explain the numerically-observed differences between results based on the quadratic Wasserstein distance and those based on the $L^2$ distance for general linear and nonlinear inverse problems.
October 16
Richard Tsai (U Texas Austin)
Implicit Boundary Integral Methods and Applications
I will talk about the implicit boundary integral methods. It is a general framework that can be applied to solve a variety of problems that involve non-parametrically represented surfaces. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop systematically and in a unified fashion numerical schemes for treating a wide range of problems that involve both differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. We will apply this framework to solve some surface PDE problems, boundary integral equations, and optimal control problems.
October 23
Mason Porter (UCLA)
Topological Data Analysis of Spatial Systems
From the venation patterns of leaves to spider webs, roads in cities, and social networks, the structure of many systems are influenced significantly by space. Accordingly, the analysis of the effects of space on structure and function is an active area in the study of networks and other complex systems. In this talk, I'll give an introduction to spatial networks, topological data analysis (TDA), and the application of TDA to spatial complex systems. I'll discuss examples from voting in elections, city street networks, and spiders spinning webs under the influence of various drugs.
October 30
Ann Almgren (Lawrence Berkeley National Laboratory)
Low Mach Number Modeling
Low Mach number equation sets approximate the equations of motion of a compressible fluid by filtering out the sound waves, which allows the system to evolve on the advective rather than the acoustic time scale. Depending on the degree of approximation, low Mach number models retain some subset of possible compressible effects. In this talk I will discuss low Mach number models for reacting and stratified flows arising in combustion, astrophysics and atmospheric modeling.
November 6
No Colloquium
November 13
Vlad Vicol (Courant Institute NYU)
Shock Formation and Vorticity Creation for Compressible Euler
In this talk, I will discuss a long term project, joint with Tristan Buckmaster and Steve Shkoller, concerning the formation of singularities (shocks) for the compressible Euler equations with the ideal gas law. We provide a constructive proof of stable shock formation from smooth initial datum, of finite energy, and with no vacuum regions. Via modulated self-similar variables, the blow-up time and location can be explicitly computed, and at the blow-up time, the solutions can be shown to have precisely Holder 1/3 regularity. Additionally, for the nonisentropic problem sounds waves interact with entropy waves to produce vorticity at the shock.
November 20
Lou Kondic (NJIT)
Modeling Liquid Crystal Films on Nanoscale
This talk will focus on recently developed models and computational techniques for thin films, with focus on nematic liquid crystal films. Models and computations are formulated within the framework of the long wave approach, augmented by the inclusion of nematic-solid interaction forces via disjoining pressure model. Particular aspects that will be discussed involve inclusion of liquid-crystalline nature in the model in a tractable manner. The resulting asymptotic model allows for discussing dewetting type of instabilities of nematic films and in particular the influence of nematic properties on the nature of dewetting. The second part of the talk will build upon the first one by considering explicitly anisotropic nature of the nematic films and the influence which such anisotropy has on dewetting. The analytical techniques are supplemented by large scale GPU based simulations that allow for computing in large domains and for discussion of various instability mechanisms in a fully nonlinear setting.
November 27
No Colloquium - Thanksgiving Recess
December 4
Thomas Anderson (Michigan)
Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation
We outline a hybrid frequency-time method that promises efficient (O(1) sampling cost) and high-order dispersionless long-time transient solutions to the time-dependent obstacle scattering problem of the 2D/3D acoustic wave equation (and which readily generalizes to other wave contexts), and demonstrate its effectiveness in comparison to other popular time-stepping methods (hybrid convolution quadrature and pure time-stepping). It becomes useful to study temporal decay of wave solutions (including in ``trapping'' scenarios), a classical question treated by the well-known Lax Phillips scattering theory. We develop (computationally-amenable) ``domain-ofdependence'' bounds on solutions to wave scattering problems and establish rapid decay estimates using only (existing) Helmholtz resolvent estimates on the real frequency axis, for geometries that have previously posed as barriers to proving rapid decay.
Updated: November 30, 2020