Stanford Applied Math Seminar

Wednesdays at 4:15 PM in 384H.

Spring 2016 Schedule

April 6
Graeme Milton, University of Utah
Extending the Theory of Composites to Other Areas of Science

Science is littered with examples where progress in one area has led to progress in other sometimes completely different areas thanks to similar underlying mathematical structures. Here we show that many of the linear wave-equations of physics can be reformulated in a way that is similar to the static or quasistatic equations of composites. The reformulation of Schr{\''o}dingers equation, for example, leads to new minimization variational principles. Desymmetrizing it leads to FFT methods for solving the multielectron Schrodinger equation which only requires FFT transforms in 2 spatial variables. Unlike density functional theory it is applicable even to excited states.

The theory of composites carries over to the response of inhomogeneous bodies, and integral representations and bounds for the dynamic response can be used in an inverse way to say something about what is inside the body from boundary measurements. Even the concept of functions can be generalized, based on the theory of composites. The new functions, superfunctions, have applications to accelerating FFT methods. A superfunction can be thought as a collection of subspaces, and there are associated rules for multiplication, division, addition, and subtraction of such superfunctions. This talk is based on a book I am editing, by the same title, and the work is coauthored with Maxence Cassier, Ornella Mattei, Mordehai Milgrom, and Aaron Welters.

April 13
Guillaume Bal, Columbia University
Boundary control in transport and diffusion equations

Consider a prescribed solution to a diffusion equation in a small domain embedded in a larger one. Can one (approximately) control such a solution from the boundary of the larger domain? The answer is positive and this form of Runge approximation is a corollary of the unique continuation property (UCP) that holds for such equations. Now consider a (phase space, kinetic) transport equation, which models a large class of scattering phenomena, and whose vanishing mean free path limit is the above diffusion model. This talk will present positive as well as negative results on the control of transport solutions from the boundary. In particular, we will show that internal transport solutions can indeed be controlled from the boundary of a larger domain under sufficient convexity conditions. Such results are not based on a UCP. In fact, UCP does not hold for any positive mean free path even though it does apply in the (diffusion) limit of vanishing mean free path. These controls find applications in inverse problems that model a large class of coupled-physics medical imaging modalities. The stability of the reconstructions is enhanced when the answer to the control problem is positive.

April 20
Messoud Efendiyev Mathematical modelling of biofilms: analysis and simulation
Coarse graining collective motion

In this talk, I will present new class of reaction -diffusion-transport equations which describe spatial spreading mechanisms of biomass. The feature of these equations is that they are comprising two kind of degeneracy : porous medium and fast diffusion. These classes of doubly-degenerate parabolic systems arising in the modelling of antibiotic disinfections of biofilms as well as biofilm growth in a porous media. Well-posedness, long-time dynamics of solutions in terms of global attractors and their Kolmogorov`s entropy as well as open questions will be discussed.

April 27
Doug James, Stanford

May 4
TBA, University of TBA

May 11
Joel Tropp, Caltech
Universality laws for randomized dimension reduction

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability.

This talk describes a phase transition in the behavior of the dimension reduction map as the embedding dimension increases. The location of this phase transition is universal for a large class of datasets and random dimension reduction maps. Furthermore, the stability properties of randomized dimension reduction are also universal. These results have many applications in numerical analysis, signal processing, and statistics.

Joint work with Samet Oymak.

May 18, 2:45pm in 384I
Torbjörn Lundh, Stanford
Vascular inspired mathematics

Most likely inspired by Hippocrates’ empirical findings, Aristotle stated that ”It belongs to the physician to know that circular wounds heal more slowly, but it belongs to the geometer to know the reasoned fact.” Posterior Analytics I,13 (79a15-16).

We will discuss at this ancient challenge, but also an even older: How to provide a well defined pressure by a compression bandage? Finally we will have a look at three more modern vascular questions: How to construct a geometrically optimized by-pass and a removable stent? And how to predict the position and shape of a guide wire in a vascular tree?

May 18
Benjamin Moll, Princeton
PDE Models in Macroeconomics

The purpose of my talk is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. My talk will mostly focus on macroeconomic models with heterogeneous agents, e.g. heterogeneous consumers, workers and firms. These share a common mathematical structure which can be summarized by a system of coupled nonlinear partial differential equations (PDEs): (i) a Hamilton–Jacobi–Bellman (HJB) equation describing the optimal control problem of a single atomistic individual and (ii) an equation describing the evolution of the distribution of a vector of individual state variables in the population (such as a Fokker–Planck equation, Fisher–KPP equation or Boltzmann equation). While plenty is known about the properties of each type of equation individually, our understanding of the coupled system is much more limited. Lasry & Lions have termed such a system a ‘mean field game’ and obtained some theoretical characterizations for special cases, but many open questions remain. In my talk I will present important examples of these systems of PDEs that arise naturally in macroeconomics, discuss what is known about their properties, and highlight some directions for future research. Background paper 1 amd Background paper 2

May 25, 3:30pm in 384H
Le Chen, University of Kansas
Intermittency fronts for the stochastic heat equation and beyond

Intermittency is a universal phenomenon that happens provided that the system has a multiplicative noise. Tall peaks and low valleys are typical pictures of an intermittent solution. The canonical model to study intermittency is the stochastic heat equation (SHE) subject to a multiplicative space-time white noise. When initial data are localized in space, these tall peaks will propagate in space at certain speed. I will show some simulations and present some results on the propagation rate of this model, the so-called intermittency fronts. The key tool in this study is a moment formula. If time is permitted, I will also present some related results on other SPDE's.

May 25, 4:40pm in 384H
Laurent Mertz, Universite Paris VI
Stochastic Variational Inequalities and Applications to Engineering Mechanics

In this presentation, we discuss cycle properties of a stochastic variational inequality arising in random mechanics and their applications to the risk analysis of failure of a simple mechanical structure.

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