# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Structure Determination through Eigenvectors of Sparse Operators

In many applications, the main goal is to obtain a global low dimensional representation of the data, given some local noisy geometric constraints. In this talk we will show how the problems listed below can be efficiently solved by constructing suitable operators on their data and computing a few eigenvectors of sparse matrices corresponding to the data operators.

##### Simulations of 5-D Plasma Turbulence in Fusion Energy Devices

This talk will start with a brief status report on magnetic fusion energy research. One of the key challenges in fusion has been the occurrence of fine-scale turbulent fluctuations, which cause plasma to leak out of a magnetic trap, so we would like to be able to predict and reduce this turbulence. A major advance in this field has been the recent development of codes for comprehensive 5-D gyrokinetic simulations of microturbulence in the core region of fusion devices. These simulations have been made feasible by significant advances not only in raw computer power, but also in asymptotic simplification of the problem formulation, and in algorithmic development. Remaining challenges and some opportunities for contributions from applied and computational mathematics will be described.

##### Group representation patterns in digital signal processing

In the lecture we will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. We will begin the lecture by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then we will explain how to generalize the construction to obtain a larger collection of functions that we call "The oscillator dictionary." Functions in the oscillator dictionary admit many interesting properties, in particular, we will explain several of these properties which arise in the context of problems of current interest in areas such as communication and radar. Joint work with Nir Sochen (Tel Aviv).

##### Spectral-element and adjoint methods in computational seismology

We provide an introduction to the use of spectral-element and adjoint methods in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss how these equations may be solved numerically based upon the spectral-element method (SEM) to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by global and regional seismographic networks. We also discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the *inverse *problem.

##### Symmetric functions of a large number of variables

##### Multiscale Methods for Hydrodynamics of Polymer Chains in Solution

The hydrodynamics of complex fluids, such as polymer solutions and colloidal suspensions, has attracted great interest due to recent advances in fabrication of micro- and nano-fluidic devices. I will first review recent advances in mesoscopic numerical methods for simulating the interaction between complex fluid flow and suspended macro molecules and structures. Computational issues at play include coarse-graining to bridge the large gap in timescales and length scales, coupling between disparate methods such as molecular dynamics and Navier-Stokes solvers, the inclusion of thermal fluctuations.

##### Emissions Market Models

The main goal of the talk is to introduce a new cap-and-trade scheme design for the control and the reduction of atmospheric pollution. The tools developed for the purpose of the study are intended to help policy makers and regulators understand the pros and cons of the emissions markets at a quantitative level.

We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits by the regulator in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from those firms anticipating that they will not need them, creating a financial market for pollution credits.

##### Computational Astrophysics and the Dynamics of Accretion Disks

The ever increasing performance of computer hardware and improvements to the accuracy of numerical algorithms are revolutionizing scientific research in many disciplines, but perhaps none more so than astrophysics. I will begin by describing why computation is crucial for the solution of a variety of problems at the forefront of research in astronomy and astrophysics, with particular emphasis on understanding accretion flows onto black holes. I will outline the challenge of developing, testing, and implementing numerical algorithms for the investigation of these problems. Finally, I will present results that demonstrate how computation can help us understand the basic physics of magnetized accretion disks.

##### Systems Engineering for Water Management

It is estimated that we harvest and utilize about 65% of the readily available fresh water resources of the world. In general, perhaps because water is perceived as an abundantly available resource, we use water rather poorly. Typically less than half the water taken from the environment serves the objective for which it was intended. The UNESCO World Water reports 2003 and 2005 identify in no uncertain terms a water crisis.

In this lecture we provide an overview of a 10 year collaborative research and development effort, between the University of Melbourne and a local company Rubicon Systems Australia, and more recently with National ICT Australia.

##### Stable Internet Routing Without Global Coordination

Global Internet connectivity results from a competitive cooperation of tens of thousands of independently-administered networks (called Autonomous Systems), each with their own preferences for how traffic should flow. The responsibility for reconciling these preferences falls to interdomain routing, realized today by the Border Gateway Protocol (BGP). However, BGP allows ASes to express conflicting local policies that can lead to global routing instability. This talk proposes a set of guidelines for an AS to follow in setting its routing policies, without requiring coordination with other ASes. Our approach exploits the Internet's hierarchical structure and the commercial relationships between ASes to impose a partial order on the set of routes to each destination.

##### Unusual Classical Ground States of Matter

A classical ground-state configuration of a system of interacting particles is one that minimizes the system potential energy. In the laboratory, such states are produced by slowly cooling a liquid to a temperature of absolute zero, and usually the ground states are crystal structures. However, our theoretical understanding of ground states is far from complete. For example, it is difficult to prove what are the ground states for realistic interactions. I discuss recent theoretical/computational methods that we have formulated to identify unusual crystal ground states as well as disordered ground state [1,2,3,4]. Although the latter possibility is counterintuitive, there is no fundamental reason why classical ground states cannot be aperiodic or disordered.

##### Trouble with a chain of stochastic oscillators

I will discuss some recent (but modest) results showing the existence and slow mixing of a stationary chain of Hamiltonian oscillators subject to a heat bath. Such systems are used as simple models of heat conduction or energy transfer. Though the unlimite goal might be seen to under stand the "fourier" like law in this setting, I will be less ambitious. I will show that under some hypotheses, the chain posses a unique stationary state. Surprisingly, even these simple results require some delicate stochastic averaging. Furthermore, it is the existence of a stationary measure (not the uniquness) which is difficult. This is joint work with Martin Hairer.

##### A generalization of compact operators and its application to the existence of local minima without convexity

We will introduce a certain property for a continuous (non-linear) operator that allows for the existence of local minima for functionals when the derivative complies with such a condition, without the need to check either weak lower semicontinuity or convexity. It turns out that this property is a generalization of the standard compactness for a continuous, non-linear operator. We illustrate the relevance of this condition by applying it to several problems in one space dimension.

##### Compressive Optical Imaging

Recent work in the emerging field of compressed sensing indicates that, when feasible, judicious selection of the type of image transformation induced by imaging systems may dramatically improve our ability to perform reconstruction, even when the number of measurements is small relative to the size and resolution of the final image. The basic idea of compressed sensing is that when an image is very sparse (i.e. zero-valued at most locations) or highly compressible in some basis, relatively few incoherent observations suffice to reconstruct the most significant non-zero basis coefficients. These theoretical results have profound implications for the design of new imaging systems, particularly when physical and economical limitations require that these systems be as small, mechanically robust, and inexpensive as possible.

##### Parallel-in-time algorithms and long-time integration

We investigate some issues related to the integration of Hamiltonian systems when using integrators that are parallel in time (the so-called class of parareal integrators, introduced by JL Lions, Y. Maday and G. Turinici in C. R. Acad. Sci., Paris, Sér. I, Math. 332, No.7, 661-668 (2001)). We show that, when appropriately adjusted, this original class of integrators enjoy excellent properties of conservation over long times. We present some elements of numerical analysis that explain the numerical observations. We also present a possible symmetrized version of such algorithms, with similar, agreeable properties. This is joint work with Yvon Maday (University Paris 6) and Frederic Legoll (Ecole des Ponts).

##### The Empirical Mode Decomposition: the method, its progress, and open questions

The Empirical Mode Decomposition (EMD) was an empirical one-dimensional data decomposition method invented by Dr. Norden Huang about ten years ago and has been used with great success in many fields of science and engineering. In this talk, I will introduce, from the perspective of a physical scientist, the thinking behind and the algorithm of EMD; and its most recent developments, especially the Ensemble EMD (EEMD), a noise-assisted data analysis method, and the multi-dimensional EMD based on EEMD. I will also outline some open questions that we currently do not have answers, or even clues to the answers, such as how to optimize EMD algorithm, what is the mathematical nature of EMD. To a significant degree, this is a talk intended for obtaining helps from mathematicians.

##### On the interplay between coding theory and compressed sensing

Compressed sensing (CS) is a signal processing technique that allows for accurate, polynomial time recovery of sparse data-vectors based on a small number of linear measurements. In its most basic form, robust CS can be viewed as a specialized error-control coding scheme in which the data alphabet does not necessarily have the structure of a finite field and where the notion of a “parity-check” is replaced by a more general functionality. It is therefore possible to combine and extend classical CS and coding-theoretic paradigms in terms of introducing new minimum distance, reconstructions complexity, and quantization precision constraints.

##### Stochastic finite element approximations of elliptic problems of higher stochastic order

In this talk we will address numerical methods for two stochastic elliptic models, where the coefficients are perturbed by colored noise or white noise. An overview of the development of numerical methods will be given for the first model. We will focus on a stochastic Galerkin finite element method for the second model. Such a model is unbiased in that the expectation of the solution solves the same equation with statistically averaged coefficients. The developed numerical algorithms are based on finite element discretization in the physical space, and Wiener chaos expansion in the probability space. Since in many practically important examples solutions of the stochastic elliptic SPDEs have infinite variance, we investigate the convergence of our algorithms in appropriately weighted Wiener chaos spaces.

##### Surface Correspondence via Discrete Uniformization

Many applied-science fields like medical imaging, computer graphics and biology use meshes to model surfaces. It is a challenging problem to determine whether, how and to what extent such surfaces correspond to each other, e.g. to see whether they are differently parametrized views of one object, or whether they indicate movement of part of an object with respect to its other parts. In this talk we will show how the Uniformization theory can be used to establish correspondences between simply-connected surfaces. We will present an algorithm for automatically finding corresponding points between two discrete surfaces (meshes).

##### Recent results on critical nonlinear Schrödinger equations

Dr. Li will review some recent progress on critical nonlinear Schrödinger equations. This talk will focus on the scattering conjecture and the solitary wave conjecture for both the mass-critical case and the energy-critical case. If time permits, Dr. Li will also discuss some related results for other types of dispersive equations.