Stanford Applied Math Seminar

Wednesdays at 4:30 PM in 384H.

Winter 2017 Schedule

January 17
Li-Cheng Tsai, Columbia University
KPZ equation limit for the six vertex model

The Six Vertex (6V) model, initially introduced as a model for ice, is an integrable model for tiling in two dimensions. In this talk we consider symmetric and stochastic 6V models, and show that, under certain scaling into the ferroelectric/disordered phase critical point, fluctuations described by the Kardar--Parisi--Zhang (KPZ) equation arises. Our approach utilizes the one- and two-point Markov duality enjoyed by the stochastic 6V model. One-point duality gives the (so-called) microscopic Hopf--Cole transform, and thereby exposes the connection to the KPZ equation. On the other hand, two-point duality provides exact, analyzable formulas that help to establish certain self-averaging, which is the key step in the proof given the aforementioned transform. Joint work with Ivan Corwin, Promit Ghosal, and Hao Shen

January 24
Houman Owhadi, Caltech
A game theoretic approach to numerical approximation and algorithm design

This talk will cover interplays between Game Theory, Numerical Approximation and Gaussian Process Regression. We will illustrate this interface between statistical inference and numerical analysis through problems related to numerical homogenization, operator adapted wavelets, fast solvers, and computation with dense kernel matrices. This talk will cover joint work with F. Schäfer, C. Scovel, T. Sullivan and L. Zhang.

January 31
Yuwei Fan Stanford University
On the moment model reduction for kinetic equation

Model reduction of kinetic equation turns a high dimensional problem to a low dimensional quasi-linear system, which not only provides further understanding of the problem, but also essentially improves the efficiency of the numerical simulation. As a quasi-linear system with Cauchy data, the well-posedness of the model deduced is required to be hyperbolic. In the existed models, some of them are hyperbolic, and some of them may be regularized to be hyperbolic, while there are seldom progress on the else models. In this talk, I will start from Grad's moment method, and point out that all Grad's moment system are not hyperbolic even around the thermodynamic equilbrium, and then proposed a globally hyperbolic regularization for Grad's moment system to obtain the globally hyperbolic moment system. By exploring the essential of the regularization, we extend the regularization to a generic framework to moment model reduction for kinetic equation. The fascinating point is, with only routine calculation, symmetric hyperbolic models can always be deduced with any ansatz for generic kinetic equation by the framework we proposed. By this framework, existing models are re-presented and brand new models are discovered.

February 7
Jonathan Luk, Stanford
Multi-dimensional shock formation for hyperbolic systems

Shock formation for hyperbolic systems in one spatial dimension has been well-understood since the 70s. More recently, there has been much progress in understanding shock formation in more than one dimensions in the special case of quasilinear scalar wave equations. The problem for general hyperbolic systems, especially for those featuring more than one characteristic speeds, still remains widely open.
In this talk, I will discuss some of the recent progress and present a joint result with Speck (MIT) in which we exhibit a solution regime of stable shock formation for the full 3D compressible Euler equations with non-trivial vorticity and variable entropy. This is the first multi-dimensional shock formation result featuring multiple characteristic speeds. I will emphasis and discuss the geometric methods used in the proof.

February 23, Friday, 3pm in 384I
David Loaiza, Aperio
Inventing the Future of Finance, Alternative Data and Aperio

Times are changing. Traditionally, Corporate access and financial modeling were paramount to the investing process. Today, we are surrounded by alternative data from credit cards, smart devices and sensors. That means that there are radical ways to gain deeper understanding of a company’s revenue performance and trends leveraging big data. Aperio models terabytes of raw data, delivering powerful insights to our Long/Short Equity teams.
Dr. David Loaiza is a Managing Director and the Chief Data Scientist for Aperio. He is responsible for Point72’s big data research team. Prior to joining Point72, David was the Chief Data Scientist and Global Head for Compliance Analytics at JP Morgan Chase. He led teams overseeing Anti-Money Laundering, Trade Surveillance, Know Your Customer risk scoring, Sanction List and Fair Lending analytics.
Dr. Loaiza was also a White House Fellow working as a special assistant to the White House Office of Management and Budget. Before the White House, he was the technical advisor and a technical team leader for the U.S. Delegations monitoring the denuclearization activities in North Korea. He has fifteen years of experience leading research programs at Los Alamos National Laboratory. Dr. Loaiza has a Ph.D. in nuclear physics and an MBA.

February 28,
Chrysoula Tsogka, Stanford
Multifrequency interferometric imaging with intensity-only measurements

We consider here the problem of coherent imaging using intensity-only measurements. The main challenge in intensity-only imaging is recovering phase information that is not directly available in the data, but is essential for coherent image reconstruction. Imaging without phases arises in many applications such as crystallography, ptychography and optics where images are formed from the spectral intensities. The earliest and most widely used methods for imaging with intensity-only measurements are alternating projection algorithms. The basic idea is to project the iterates on the intensity data sequentially in both the real and the Fourier spaces. Although these algorithms are very efficient for reconstructing the missing phases in the data, and performance is often good in practice, they do not always converge to the true, missing phases. This is especially so if strong constrains or prior information about the object to be imaged, such as spatial support and non-negativity, are not reliably available. Rather than using phase retrieval methods, we propose a different approach in which well-designed illumination strategies exploit the spatial and frequency diversity inherent in the problem. These illumination strategies allow for the recovery of interferometric data that contain relative phase information which is all that is needed to reconstruct a so-called holographic image. There is no need for phase reconstruction in this approach. Moreover, we show that this methodology leads to holographic images that suffer no loss of resolution compared with those that use full phase information. This is so when the background media through which the probing signals propagate are assumed known. We also consider propagation media with fluctuations in the index of refraction which are unknown and therefore modeled as a random process. In such media the recovered relative phases fluctuate and this introduces noise and instability in the image formation process. Using adequately designed filters (masks), the uncertainty in the recovered phases is reduced and statistically stable imaging results are obtained. The robustness of our approach will be explored with numerical simulations carried out in an optical (digital) microscopy imaging regime.

March 7,
Charles-Albert Lehalle, Capital Fund Management, Paris
From Empirical Behaviour of High Frequency Traders to Optimal High Frequency Interactions with a Limit Orderbook

As a motivation, I will start by a quick summary of [1], that gives empirical evidences on the behaviour of HFT around news and gives some statistics about their activity in Europe. After a short discussion on the way they provide less liquidity on expected news than usual, even once corrected from volatility variations, I will take evidences from [3] and [4] to show that HFT do use the imbalance of orderbooks to take decisions at the finest level. Since imbalance of orderbook is known to play a role in orderbook dynamics [5], I will then present the stochastic control framework of [2] dedicated to take decisions at this level. Then we will see how to obtain results in this framework and conclude by comparing them to previous empirical evidences. This will suggest HFT are using control frameworks that are probably close to ours.

March 14,
Domokos Szasz, Budapest University of Technology and Economics
Fourier law from Hamiltonian dynamics

It is a classical and fundamental quest of mathematical statistical physics to rigorously understand the heat transfer equation of Laplace from Hamiltonian dynamics. In 2008 the physicists Gaspard and Gilbert came up with a billiard model and suggested a two step approach for it: 1. for the energies of the particles derive – in the rare interaction limit – a Markov jump process (dynamical part); 2. take the hydrodynamic limit of the obtained jump process; this is expected to lead to the Laplace equation (stochastic part). Since their model was still unsuitable for mathematics, we first introduced its tractable variant: the disk-piston model. For it we can show that its rare interaction limit is, indeed, a Markov jump process. The talk is based on joint works with P. Balint, Th. Gilbert, P. Nandori and I.P. Toth

March 16, Friday, 4pm in 380X
Mitchell Luskin, University of Minnesota
Mathematical Modeling and Numerical Analysis for Incommensurate 2D Materials

Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the electronic and optical properties of 2D materials. One of the main issues encountered in the mathematical and computational modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. Basic concepts like the electronic density of states and the Kubo-Greenwood formulas for transport properties will be formulated and analyzed in the incommensurate setting. New computational approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

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