Stanford Applied Math Seminar

Wednesdays at 4:30 PM in 384H.

Spring 2019 Schedule

April 3
Jan Wehr, University of Arizona
Aggregation and deaggregation of robot swarms

A system of stochastic differential delay equations models the motion of a light-sensitive robot, exploring an inhomogeneous landscape. It can be approximated by a stochastic differential equation without delay and, in this form, studied using a multiscale expansion. The resulting dynamics depends on the value of the delay parameter, which can be positive or negative---I will explain what this means and how it is realized experimentally. The results for a single robot are applied to predict an aggregation-deaggregation transition in many-robot systems, in a very good agreement with experiments. A nontechnical summary can be found at

April 10
Konstantin Khanin, University of Toronto
On Geometrical Renormalization for KPZ

We discuss two renormalization approaches to the problem of KPZ universality. The first one is based on information regarding optimal actions and is related to the Airy sheet process. The second approach is defined in pure geometrical terms. We shall discuss some results and conjectures related to the second renormalization scheme.

April 12, Friday 2:30pm in 383N
Sepideh Mirrahimi, Universite Paul Sabatier, Toulouse
Singular limits for models of selection and mutations with heavy tails

In this work, we perform an asymptotic analysis of a nonlocal reaction-diffusion equation, with a fractional laplacian as the diffusion term and with a nonlocal reaction term. Such equation models the evolutionary dynamics of a phenotypically structured population. We perform a rescaling considering large time and small effect of mutations, but still with algebraic law. We prove that asymptotically the phenotypic distribution density concentrates as a Dirac mass which evolves in time. This work extends an approach based on Hamilton-Jacobi equations with constraint, that has been developed to study models from evolutionary biology, to the case of fat-tailed mutation kernels. However, unlike previous works within this approach, the WKB transformation of the solution does not converge to a viscosity solution of a Hamilton-Jacobi equation but to a viscosity supersolution of such equation which is minimal in a certain class of supersolutions.

April 17, 12:30pm in 384I
Wuchen Li, UCLA
Wasserstein Information Geometric Learning

Optimal transport (Wasserstein metric) nowadays play important roles in data science. In this talk, we brief review its development and applications in machine learning. In particular, we will focus its induced differential structure. We will introduce the Wasserstein natural gradient in parametric models. The metric tensor in probability density space is pulled back to the one on parameter space. We derive the Wasserstein gradient flows and proximal operator in parameter space. We demonstrate that the Wasserstein natural gradient works efficiently in several statistical machine learning problems, including Boltzmann machine, generative adversary models (GANs) and variational Bayesian statistics.

April 17,
Daniel Tartakovsky, Stanford University
Method of Distributions for Hyperbolic Conservation Laws with Random Inputs

Parametric uncertainty, considered broadly to include uncertainty in system parameters and driving forces (source terms and initial and boundary conditions), is ubiquitous in mathematical modeling. The method of distributions, which comprises PDF and CDF methods, quantifies parametric uncertainty by deriving deterministic equations for either probability density function (PDF) or cumulative distribution function (CDF) of model outputs. Since it does not rely on finite-term approximations (e.g., a truncated Karhunen-Loeve transformation) of random parameter fields, the method of distributions does not suffer from the ``curse of dimensionality''. On the contrary, it is exact for a class of nonlinear hyperbolic equations whose coefficients lack spatio-temporal correlation, i.e., exhibit an infinite number of random dimensions.

April 24, 3:15pm in 384-I
Georgia Papadakis, Stanford University
Tailoring optical and thermal properties with nanophotonics

Understanding and tailoring light-matter interactions lies at the heart of many modern solid-state technologies, ranging from imaging, to solar photovoltaics, data storage and quantum computations. This inspires research in artificial composite materials with properties acquired by design, often termed metamaterials. In this talk, I will demonstrate how metamaterial principles along with simple photonic concepts, such as surface plasmon polaritons, phonons and excitons, guided waves and optical bandgaps, can be utilized and combined to introduce novel material functionalities that surpass natural bounds. These include optical magnetism, unattainable with natural materials, ultra-lightweight mirrors with van der Waals heterostructures, and actively tunable dielectric response. Controlling the flow of light via its interaction with structured matter has further implications in photonic thermal management, with applications in nanoscale cooling, thermal circuitry, and waste heat recovery. In the second part of this talk, I will discuss how the framework of fluctuational electrodynamics helps us understand and control the flow of heat. Our recent results show that, by borrowing ideas from optoelectronics, we can design nanoscale thermal switches, the thermal counterpart to the electronic field-effect transistor. Thermal management is also crucial in energy recycling, in particular thermophotovoltaics, an emerging technology where conversion of heat to electricity is mediated by emission of electromagnetic radiation. I will discuss fundamental limits (e.g. blackbody limit), current limitations (e.g. intrinsic semiconductor processes such as non-radiative recombination), and how these are mitigated using near-field radiative heat transfer.materials

April 24,
Wei Cai, Southern Methodist University
Stochastic Algorithms for Electromagnetic Problems of Nano-Structures

In this talk, we will present some stochastic algorithms and numerical results for solving electromagnetic problems in nano-particles and random meta-materials. Firstly, we will present a path integral Monte Carlo method for computing magnetic polarizability tensors of nano-particles of complex geometries for material sciences applications. The method relies on a Feynman-Kac formula involving reflecting Brown motions (RBMs) and accurate computation of the local time of the RBMs using a random walk-on-spheres technique. Secondly, in order to optimize functional properties of 3-D random meta-materials (MMs), we will present a stochastic representation scheme for random MMs with volume exclusion constrains and given correlations, a fast volume integral equation electromagnetic solver for the scattering of a large number of meta-atoms of typical geometric shapes (cubes, spheres, and ellipses) in layered media, and a procedure to optimize the optical properties of the MMs. A new fast multipole method for 3-D Helmholtz equation for layered media will be presented based on new multipole expansion (ME) and multipole to local translation (M2L) operators for layered media Green's functions.

May 1,
Yao Yao, Georgia Tech
Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations

In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gomez-Serrano, Jaemin Park and Jia Shi. >

May 8,
Svetlana Jitomirskaya, University of California, Irvine
Cantor spectrum of a model of graphene in magnetic field

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero and Hausdorff dimension bounded by 1/2. The latter statement is proved by technique which also leads to resolution of (a half of) the Thouless conjecture for the Harper's operator. Based on joint works with S. Becker, R. Han, and also I. Krasovsky

May 15,
Gautam Iyer, Carnegie Mellon University
Dissipation Enhancement by Mixing

We quantitatively study the interaction between diffusion and mixing in both the continuous, and discrete time setting. In discrete time, we consider a mixing dynamical system interposed with diffusion. In continuous time, we consider the advection diffusion equation where the flow of the advecting vector field is assumed to be sufficiently mixing. We explicitly estimate the dissipation time based on the mixing rate. Moreover, in the discrete time setting, we show that the $L^2$ energy decays double exponentially in time, and this double exponential rate is achieved for by a large class of toral automorphisms. We will also briefly discuss how dissipation enhancement can be used to control the growth of certain non-linear equations.

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