Stanford Applied Math Seminar

Wednesdays at 4:15 PM in 380C.

Winter 2014 Schedule

January 15
Vitali Vougalter, University of Cape Town
Existence and nonlinear stability of stationary states for the semi-relativistic Schroedinger-Poisson system

We study the stationary states of the semi-relativistic Schroedinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize our global well-posedness result for the semi-relativistic Schroedinger-Poisson system to spaces with higher regularity

January 22,
Inviscid limits for the stochastic Navier-Stokes equation

We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow. We prove that the limiting inviscid invariant measures are supported on bounded vorticity solutions of the 2D Euler equations. This is joint work with N. Glatt-Holtz and V. Sverak.

January 29
Guo Luo, Caltech
Potentially Singular Solutions of the 3D Incompressible Euler Equations

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of the singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup. We also discuss a 1D model which can be viewed as a local approximation to the Euler equations near the solid boundary of the cylinder. The finite-time blowup of this 1D model is proved under certain convexity conditions for the velocity field.

February 5
Frederik J Simons, Princeton University
Seismic Tomography with Sparsity Constraints, at the Global and the Exploration Scale

Seismic tomography leads to often gigantic systems of linear equations, and as data volumes increase, every effort has to be made to reduce the inversion problem to manageable size. At the same time as achieving a reduction of computational complexity, new sparsity-seeking methods should be able to give additional insight into the nature of the problem itself, and the character of the solution. Practically speaking, this means that if we are solving systems that relate data (functionals of seismograms) to model parameters (the physical properties of the Earth) through the use of sensitivity kernels (partial derivatives of the measurements with respect to the unknown Earth parameters), we have opportunities for dimensional reduction on the data, the model, and the kernel side. In this presentation I discuss strategies to make inroads on all three sides of the seismic tomographic inverse problem, using one-, two- and three-dimensional spherical wavelets, and $\ell_2$-$\ell_1$ combination misfit norms. Examples come from global tomography on the basis of finite-frequency travel time measurements, and from exploration tomography using spectral-element adjoint waveform inversion.

February 12
Matt Lorig, Princeton University
Pricing Variance Swaps on Time-Changed Markov Processes

Often, different financial derivatives are subject to the same sources of risk. In such cases, the prices of these derivatives ought to be related. In this talk, we will explore the relationship between a variance swap, which is a financial derivative whose payoff depends on the entire path of a stock S over a fixed time interval [0,T], and a European contract, which is a financial derivative whose payoff depends only on the value of S at the terminal time T. We will prove that, when a stock is modeled as a time-changed Markov process, the variance swap has the same value as a European contract whose payoff function is the solution of an integro-differential equation (which we will solve). The significance of this result is that the path-dependent variance swap contract can be priced relative to liquidly traded (and efficiently priced) path-independent European options in a semi-nonparametric fashion.

February 19
Guillaume Bal, Columbia University
High-contrast high-resolution coupled physics imaging modalities

Abstract: Several recent coupled-physics medical imaging modalities aim to combine a high-contrast, low-resolution, modality with a high-resolution, low-contrast, modality and ideally offer high-contrast, high-resolution, reconstructions. Such modalities often involve the reconstruction of constitutive parameters in partial differential equations (PDE) from knowledge of internal functionals of the parameters and PDE solutions. This talk presents results of uniqueness, stability and explicit reconstructions for several hybrid inverse problems. In particular, we provide explicit characterizations of what can (and cannot) be reconstructed and offer optimal (elliptic) stability estimates for a large class of coupled-physics imaging modalities including Magnetic Resonance Elastography, Transient Elastography, Photo-Acoustic Tomography and Ultrasound Modulation Tomography. Numerical simulations confirm the high-resolution, high-contrast, potential of these novel modalities.

February 26
Bedros Afeyan, Polymath Research
Nonlinear Optics of Plasmas: Mathematical Opportunities and Challenges

High intensity laser-matter interactions give rise to new nonlinear kinetic states of self-organization in plasmas. These nonlinear coherent structures pose challenges and present opportunities for applied mathematical models that combine non-equilibrium statistical mechanics, nonlinear dynamics, plasma physics, statistical optics and nonlinear ultrafast optics. Combining these disciplines to make sense of matter in extreme conditions will require new insights and a close collaboration between microscopic computational approaches, mesoscopic reduced models and macroscopic observable predictions. Attention will be given to the physics background and the numerical approaches popular in the field of high energy density plasmas. The outstanding, compelling applied mathematical questions will be highlighted throughout the presentation.

March 5
Liliana Borcea, University of Michigan
Electromagnetic wave propagation in random waveguides

I will present a study of long range propagation of electromagnetic waves in random waveguides with perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We analyze the wave field by decomposing it in evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering. I will explain how we can use such theory for imaging in random waveguides.

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