October 18, 2012

Gunther Uhlmann, University of Washington

Multiwave imaging and photoacoustic tomography

Abstract: Multi-wave imaging methods, also called hybrid methods, attempt to combine the high resolution of one imaging method with the high contrast capabilities of another through a physical principle. One important medical imaging application is breast cancer detection. Ultrasound provides a high (sub-millimeter) resolution, but suffers from low contrast. On the other hand, many tumors absorb much more energy of electromagnetic waves (in some specific energy bands) than healthy cells. Photoacoustic tomography (PAT) consists of sending relatively harmless optical radiation into tissues that causes heating which results in the generation of propagating ultrasound waves (the photo-acoustic effect). Such ultrasonic waves are readily measurable. The inverse problem then consists of reconstructing the optical properties of the tissue. In Thermoacoustic tomography (TAT) low frequency microwaves, with wavelengths on the order of 1m, are sent into the medium. The rationale for using the latter frequencies is that they are less absorbed than optical frequencies. Transient Elastography (TE) images the propagation of shear waves using ultrasound. We will discuss these imaging techniques with emphasis on PAT.

October 25, 2012

Ronald Coifman, Yale University

Information Integration/Organization and numerical harmonic analysis

Abstract: We provide an overview of recent developments in methodologies for empirical organization of data. We present a geometric/analytic mathematical framework for learning, which revolves around building a network or a graph whose nodes are observations. In our framework, connections between observations are constantly reconfigured in order to achieve learning for specific tasks. In particular we will provide a synthesis of a range of ideas from mathematics and machine learning, which address the transition from a local similarity model to a global configuration. This is analogous to Newtonian Calculus, which from of a local linear model of variability, calculates a global solution to a differential, or partial differential equation. We apply these fundamentals to jointly organize the rows and columns of a matrix, viewed either as the matrix of a linear operator, or as a Database. Here the rows are viewed as functions on the columns and the columns as functions of the rows, a dual geometry is built to optimize prediction and processing . We relate these methods to ideas from classical Harmonic Analysis and indicate tools to measure success of information extraction. In particular we introduce methodologies that resemble “signal processing” on data matrices, enabling functional regression, prediction, denoising, compression fast numerics, and so on. We illustrate these ideas to organize and map out in an automatic and purely data driven fashion on music databases of audio segments, text documents, psychological questionnaires, medical profiles, physical data, financial data.

November 8, 2012

Chris Bretherton, University of Washington

Clouds, Aerosols and Climate: An Applied Problem in Need of Applied Mathematicians

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January 10, 2013

Percy Deift, Courant Institute of Mathematical Sciences

Toeplitz Determinants with Fisher-Hartwig Singularities

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February 14, 2013

Rachel Kuske, University of British Columbia

Noise and the Piecewise World: Sliding, Grazing and Zigzags in Random Environments

Abstract: Modeling systems with discontinous dynamics has received increased attention in biological, engineering, and environmental applications. Furthermore, across nature and engineering, delay is an inherent feature in systems with feedback and control. While there have been recent advances for analyzing the complex deterministic behavior of such systems, there are many open questions around understanding and predicting noise-driven and noise-sensitive phenomena in the piecewise continuous context. Stochastic effects can often change the picture dramatically, particularly if multiple time scales are present. This talk covers new ideas for exploring the interplay of nonlinearities, delays, randomness, and piecewise smooth dynamics, and explaining the often elusive and surprising phenomena that are observed. Progress depends on the exchange of mathematical techniques and phenomenological intution from seemingly unrelated canonical models of biophysics, mechanics, and chemical dynamics. The approaches and dynamics are illustrated in the application areas of relay control, balance, and blood diseases.

March 7, 2013

Anna Gilbert, University of Michigan

A Survey of Sparse Approximation

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April 4, 2013

Max Gunzburger, Florida State University

A Nonlocal Vector Calculus and Nonlocal Methods for Diffusion and Mechanics

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May 16, 2013

Liam Paninski, Columbia University

Challenges and Opportunities in Statistical Neuroscience

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May 30, 2013

Stanley Osher, University of California, Los Angeles

The Magic of L1 Type Regularization

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October 6, 2011:

James Murray, Princeton University/University of Washington

Vignettes from a mathematician’s odyssey in biology: from pilot ejection injuries to the benefits of cannibalism

Abstract: I shall describe a series of problems for which the models are both practical and the mathematics simple. Pilot ejection can cause extreme stresses on the spine which frequently result in major injuries. A basic model of the physical process shows how such injuries could occur. Many diseases require a careful assessment of medication level: liver disease is a prime example. I shall describe the medical dilemma and show how the model provided a scientific method for determining the appropriate medication level. It is now used in England. Breathalysers are notoriously inaccurate. I shall describe two models which show that it is not possible to design a better breathalyzer. Prostate specific antigen (PSA) blood tests are considered indicators of possible prostate cancer but are notoriously misunderstood and misinterpreted. Our model explains a known anomaly and highlights inadequacies in current government guidelines. Cannibalism is widespread in nature and generally of benefit to the species. I shall describe a model for salamander cannibalism and then describe how important cannibalism has been for humans. Finally I shall show how certain spatial patterns in development are influenced by geometry and scale and, time permitting, how the woods on Bainbridge Island can exhibit jungle like aspects.

October 27, 2011:

Eliot Fried, University of Washington

Some features and challenges of the Navier-Stokes-alpha-beta equation

Abstract: The Navier-Stokes-alpha-beta equations regularize the Navier-Stokes equations by the addition of dispersive and dissipative terms. The dispersive term is proportional to the divergence of the corotational rate of the symmetric part of the velocity gradient. The dissipative term is proportional to the bi-Laplacian of the velocity. The coefficients of these terms involve factors alpha and beta, respectively, both having dimensions of length. Calculating the energy spectrum for an assembly of stretched spiral vortices reveals an inertial range where Kolmogorov's -5/3 law holds and shows that choosing beta less than alpha yields a better approximation of the inertial range of the Navier-Stokes equations. Direct numerical simulations of three-dimensional periodic turbulent flow confirm this and also show that vorticity structures behave more realistically when beta is less than alpha. However, the simulations indicate that optimal choices of alpha and beta are resolution dependent. This suggests the possibility of developing multigrid methods that capitalize on resolution dependence by using the Navier-Stokes-alpha-beta equations at coarse grid levels, with different choices of alpha and beta at each level, to accelerate convergence to solutions of the Navier-Stokes equation at the finest grid level. Results obtained from a two-dimensional spectral multigrid algorithm of this type show promise.

November 10, 2011:

Felix Otto, University of Bonn

Pattern formation and Partial Differential Equations

Abstract: In three specific examples, we shall demonstrate how the theory of partial differential equations (PDEs) relates to pattern formation in nature: Spinodal decomposition and the Cahn-Hilliard equation, Rayleigh-B\'enard convection and the Boussinesq approximation, rough crystal growth and the Kuramoto-Sivashinsky equation. These examples from different applications have in common that only a few physical mechanisms, which are modeled by simple-looking evolutionary PDEs, lead to complex patterns. These mechanisms will be explained, numerical simulation shall serve as a visual experiment. Numerical simulations also reveal that generic solutions of these deterministic equations have stationary or self-similar statistics that are independent of the system size and of the details of the initial data. We show how PDE methods, i. e. a priori estimates, can be used to understand some aspects of this universal behavior. In case of the Cahn-Hilliard equation, the method makes use of its gradient flow structure and a property of the energy landscape. In case of the Boussinesq equation, a ``driven gradient flow'', the background field method is used. In case of the Kuramoto-Sivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers' equation.

January 5, 2012

Marsha Berger, Courant Institute of Mathematical Sciences

Cell-cut methods for flows in complicated geometry

The Cartesian grid embedded boundary approach has attracted much interest in the last decade due to the ease of grid generation for complicated geometries. This approach uses rectangular Cartesian meshes over most of the domain, with irregular, or cut cells at the where the mesh intersects a solid body. It is in routine use in design projects at NASA Ames to automate the solution of steady inviscid compressible flow. Extending the method however to compute time-dependent flow or viscous flow is much more complicated. In this talk we first briefly describe our approach to embedded boundary computations, and illustrate what distinguishes it from level set or other immersed boundary approaches. We present examples showing the current state for computing steady invsicid flow at NASA Ames. The second half of the talk will concentrate on the algorithmic issues that arise when trying to extend the method to compute time-dependent and viscous flows. We present our preliminary work in two space dimensions illustrating a simpli- fied but second-order accurate approach to solving the so-called small-cell problem faced by explicit difference schemes. We also show our recent work in computing high Reynolds number flow without using the anisotropic re- finement which is possible with a body-fitted grid.

January 19, 2012

Walter Strauss, Brown University

Steady Rotational Water Waves

Precise study of water waves began with the derivation of the basic mathematical equations of fluids by the great Euler in 1752. In the two and a half centuries since then, the theory of fluids has played a central role in the development of mathematics. Water waves are fluids with a free surface. I will discuss periodic waves that travel at a constant speed. Using local and global bifurcation theory, we now know how to prove that there exist very many such waves. They may have either small or large amplitudes. I will outline the existence proof, joint with Adrian Constantin, and then exhibit some computations, joint with Joy Ko, of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs either at the crest, on the bed directly below the crest, or in the interior of the fluid. The vorticity also affects the pressure beneath the fluid and the behavior of the fluid particles. This work is a perfect example of the synergy between theory and computation.

January 26, 2012

Gigliola Staffilani, Massachusetts Institute of Technology

Dispersive equations and their role beyond PDE

Arguably the star in the family of dispersive equations is the Schrodinger equation. Among many mathematicians and physicists it is regarded as fundamental, in particular to understand complex phenomena in quantum mechanics. But not many people may know that this equation, when defined in a periodic setting for example, has a very reach and more abstract structure that touches several fields of mathematics, among which analytic number theory, symplectic geometry, dynamical systems and probability. In this talk I will illustrate in the simplest possible way how all these different aspects of a unique equation have a life of their own while interacting with each other to assemble a beautiful and subtle picture.

April 12, 2012

Lai-Sang Young, Courant Institute

Dynamics of neuronal networks modeling visual cortex

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I will discuss joint work with Aaditya Rangan in which we model a small patch of layer 2/3 of the primary visual cortex (V1) as a large network of point neurons. Network architecture is chosen to reflect a few coarse structures of V1, and network parameters are constrained by biological data from laboratories. We sought to explain macroscopic observations from dynamics on the neuronal level, and found parameter regions for which our network exhibits simultaneously a number of empirically observed V1 phenomena including orientation tuning, localized receptive fields and surround suppression. After an overview, I will focus on one phenomenon, such as spontaneous pattern formation in V1 in the absence of visual stimuli, and discuss the dynamical mechanisms behind it as well as when Fokker-Planck type descriptions are valid and when they are not.

April 26, 2012

Larry Abbott, Columbia University

Harnessing Neural Network Dynamics

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May 17, 2012

Michael Shelley, Courant Institute

Active Fluids

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The mechanics of fluids and structures can sometimes be extremely useful in explaining biological phenomena. While bird flight and fish swimming are well-known examples, I will discuss two other cases where the role of mechanics seems not so obvious but turns out to be central and even surprising. The first concerns understanding experimental observations of a simple undulating organism -- the nematode C. elegans -- negotiating a fluid-filled space full of obstacles. The second case focuses on the pronuclear complex in C. elegans embryo and how it achieves proper position and orientation within the cell so that early development can successfully proceed.