# CAMP/Nonlinear PDE's Seminar

Wednesdays at 4 PM in Eckhart 202.

## Fall 1999

September 15
Alexandre Chorin, University of California, Berkeley
Optimal prediction.

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. I will present a simple derivation of the basic methodology, emphasizing nonlinear aspects. I will show that perturbation theory provides a useful device for dealing with quasi-linear problems, and provide nonlinear examples that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, I will explain the differences and similarities between optimal prediction, data acquisition, and duality methods for finding weak solutions. (Joint work with A. Kast, R. Kupferman and D. Levy).

September 22
Yuan-Nan Young, University of Chicago
Stratified Kolmogrov shear flow

Kolmogorov shear flow has been a investigated in the context of generation of 2D turbulent flow in both physics and geophysics. More recently, Kolmogorov shear flow has been revisited for its richness in various dynamical behavior. In this study we impose a stablizing temperature gradient onto the Kolmogorov shear flow and investigate the weakly nonlinear dynamics of this system. In addition, we also explore the limit of large Peclet number cases (small molecular diffusivity) and the nondiffusive limit. In the nondiffusive cases, the dynamics reduce to internal layer boundary dynamics and we will derive the amplitude equations for the internal boundary layers.

September 27 (Monday)
John Gibbon, Imperial College, London
Singularity formation in a special 3D Euler flow

The possibility is discussed in both analytical and numerical terms of a finite time singularity forming in the 3D incompressible Euler velocity field represented by $${\bf U}(x,y,z,t) = \left\{u(x,y,t),\,v(x,y,t),\; z\gamma(x,y,t) + W(x,y,t)\right\}$$ which is confined in a box of infinite extent in $z$ with periodic cross-section in $(x,\,y)$. Recent numerical simulations by Ohkitani and Gibbon provide evidence that a finite time singularity may develop in regions of negative $\gamma$. We discuss these results and the analytical issues surrounding them. The rapid amplification of regions of negative $\gamma$ accounts for the relatively large size of the corresponding full three-dimensional vortex that opens out just prior to its breakdown. A theorem of the same type as that of Beale, Kato and Majda is also proved where it is shown that $\int_{0}^{t}\gamma(\tau)\,d\tau$ controls any singularities that occur. This involves the modification of a two-dimensional logarithmic inequality of Kato.

October 6
Michael Vogelius, Rutgers University,
Determining internal objects of small area based on boundary measurements

In this talk I will discuss recent work concerning some asymptotic formulae, that, among other things, permit very efficient identification of small internal objects from boundary measurements. The boundary measurements could for instance be steady state electric data (voltages and currents) or they could be time harmonic electric- and magnetic data (corresponding to nonzero frequency, and the full Maxwell's equations).

October 12 (Tuesday)
Percy Deift, New York University
Asymptotics for random permutations

We show that the length of the longest increasing subsequence of a random permutation of length N behaves statistically as N becomes large like the largest eigenvalue of a random Hermitean matrix. This is joint work with Jinho Baik and Kurt Johansson.

October 19
Svetlana Jitomirskaya, University of California, Irvine
Quasiperiodic operators. Arithmetics and localization.

We will sketch a proof of (a correct version of) the Aubry-Andre conjecture on the metal-insulator transition for the almost-Mathieu operator. We will also discuss other recent results on nonperturbative localization for related models.

October 27
Harley Flanders, Jacksonville University
Automatic Differentiation With Applications to ODE, Integration, and More.

Automatic Differentiation (AD) is an algorithmic method of computing numerical values of derivatives, either high order derivatives of a function of one variable (the focus of this talk) or low order derivatives of functions of many variables. It is neither symbolic nor approximate, and is as accurate as the computer. AD is remarkably fast and accurate; its application to initial value problems for systems of ODE appears better than RK and other numerical methods. It also applies to implicit ODE's and to DAE's. The talk will cover how AD works, and how it may be applied. There will be a demo of the speaker's "ODE" software package

November 3
Vered Rom-Kedar, Weizmann Institute of Science
Instabilities in near integrable Hamiltonian systems (or: From the Tropics to the Poles in forty days).

The shaded energy-momentum diagram, a novel way to present the global phase space structure of low dimensional near-integrable Hamiltonian systems will be presented. Then, the appearance of a parabolic resonance will be explained, and it will be demonstrated that its appearance causes strong phase space instabilities in two degrees of freedom systems. Implications to higher dimensional systems will be presented. The subject will be presented in the physical context of a simple model for the motion of weather balloons. We use the parabolic-resonance instabilities to explain field experiments that show that the averaged poleward velocity of high altitude weather balloons may, on rare occasions, be much higher than the observed averaged poleward winds, while their eastward velocity is much slower than the observed averaged eastward winds. Collaborators: N. Paldor, Y. Dvorkin, A. Litvak-Hinnenzon.

November 10
Charles Fefferman, Princeton University
Conformal Invariants

A method of constructing scalars, differential operators, and pseudodifferential operators on a manifold, that have a transformation law under conformal mappings.

November 17
Alex Lipton-Lifschitz, Deutsche Bank
A unified approach to pricing path-dependent options

Creating and pricing options is one of the most important and challenging tasks of financial engineering. In this talk I explain why path-dependent options are needed in practice and show how to price them from a unified viewpoint based on PDE tools such as similarity reductions and probabilistic tools such as Levy's local time.

November 24
Vadim Kaloshin, Princeton University
Bad embedding properties of infinite dimensional fractal sets into finite dimensional spaces.

Consider a compact set $X \subset \Rm^n$ of Hausdorff dimension $D$. A classical result of Marstrand and Matilla says that for an almost every linear projection of $X$ into a subspace of dimension at least $D$ Hausdorff dimension is preserved, i.e. Hausdorff dimension of $X$ and its image are the same. Now consider a compact set $X$ in a Hilbert space $H$ and its image under linear projection into a finite dimensional space. It turns out that preservation of Hausdorff dimension under projections is {\it{no longer true}}. Namely, for any $D$ we construct a compact set of Hausdorff dimension $D$ in the real Hilbert space $l^2$ such that for {\it all} linear projections $\pi$ of $B$ into $\Rm^n$, no matter how large $n$ is, the Hausdorff dimension of $\pi(X)$ is less than $2$, no matter how large $D$ is. Regularity property (like Whitney's embedding theorem and H\"older exponent of the inverse map) of projections of fractal sets from an infinite-dimensional space to a finite-dimensional space will be discussed.

December 1
Donald Estep, Georgia Institute of Technology
Estimating the error of numerical solutions of systems of reaction-diffusion equations.
One of the pressing problems in the analysis of reaction-diffusion equations is obtaining accurate and reliable estimates of the error of numerical solutions. Recently, we made significant progress using a new approach that at the heart is computational rather than analytical. I will describe a framework for deriving and analyzing a posteriori error estimates, discuss practical details of the implementation of the theory, and illustrate the error estimation using a variety of well-known models. I will also briefly describe an application of the theory to the class of problems that admit invariant rectangles and discuss the preservation of invariant rectangles under discretization.

December 8
Boris Shraiman, Bell Labs
Geometry of turbulent fluctuations and multipoint statistics.

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