The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Tianling Jin.
We introduce the stochastic Mather measure for discounted approximation of degenerate viscous Hamilton-Jacobi equations by using a nonlinear adjoint method. We apply it to the characterization of the limit of discounted approximation. This is a joint work with Hung Vinh Tran (Univ. of Chicago).
In the 1990s, it was discovered that Burgers' equation \(\rho_t + \rho \rho_x = 0\) interacts nicely with stochastic initial data. This observation is due to several authors, beginning with special cases and alternative notions of solution, culminating with the 1998 work of Jean Bertoin. This paper showed that if \(\rho(x,0)\) is a Levy process without positive jumps, then for fixed \(t \gt 0\) the solution \(\rho(x,t)\) has the same property, and gave a description for the evolution of the Laplace exponent. Extending results of this type to more general scalar conservation laws \(\rho_t = H(\rho)_x\) was a challenge considered by Menon and Srinivasan in 2010, who conjectured a closure property for Feller processes with an evolution described (equivalently) by either a kinetic equation or a Lax pair. In this talk we discuss a special case where this can be verified.
This talk is focused on Hamilton-Jacobi equations posed on networks. The difficulty lies in the discontinuity of the Hamiltonian with respect to the space variable. Existence relies on the notion of flux-limited (viscosity) solutions. Uniqueness relies on the construction of an appropriate vertex test function. Some applications and extensions will be also presented at the end of the talk. The main results are obtained in collaboration with Régis Monneau.
The asymptotic stability of coherent states , like kinks in one dimension poses a great challenge. This is due to the long range nature of the dispersive equation. This talk will focus on one such problem. We study the 1D Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. We prove global existence and (in L-infinity) scattering as well as a certain kind of strong smoothness for the solution at time-like infinity; it is based on several new classes of normal-form transformations. The analysis also shows the limited smoothness of the solution, in the presence of the resonances.
For the water waves system we have shown the formation in finite time of splash and splat singularities. A splash singularity is when the interface remain smooth but self-intersects at a point and a splat singularity is when it self-intersects along an arc. In this talk I will discuss new results on stationary splash singularities for water waves and in the case of a parabolic system a splash can also develop but not a splat singularity.
Lecture 1: Solving the 3-D Puzzle of Rotation Assignment in Single Particle Cryo-Electron Microscopy
Monday, June 1, 2015, 4 - 5 PM, Ryerson 251.
Single particle cryo-electron microscopy (EM) recently joined X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy as a high-resolution structural method for biological macromolecules. In single particle cryo-EM, the 3-D structure needs to be determined from many noisy 2-D projection images of individual, ideally identical frozen-hydrated macromolecules whose orientations and positions are random and unknown (i.e., random X-ray transform). Maximum likelihood estimation has emerged as a popular and powerful approach for estimating the unknown pose parameters. Iterative algorithms such as expectation-maximization are guaranteed to converge to a local maximum of the likelihood function, but not necessarily to the global one. Finding and certifying the global maximum is challenging because the parameter space is exponentially large in the number of images. In this talk we introduce an approach for maximizing the likelihood based on semidefinite programming and the Fourier transform over the rotation group SO(3). The resulting algorithm does not require any initial 3D model, and provides a certificate whenever it finds the global maximum. Numerical experiments demonstrate that the global solution is obtained with high probability whenever the signal-to-noise ratio is above a certain threshold. The approach is quite general and can be applied to handle other groups of transformations that arise in other applications in signal processing, image analysis, computer vision and computer graphics.
Lecture 2: Vector diffusion maps and the graph connection Laplacian
Tuesday, June 2, 2015, 4:30 - 5:30 PM, Eckhart 202
Motivated by the problem of denoising 2D cryo-EM images, in this talk we consider vector diffusion maps (VDM), a mathematical framework for organizing and analyzing high dimensional data sets, 2D images and 3D shapes. VDM is based on the discrete graph connection Laplacian which is a generalization of the graph Laplacian from spectral graph theory. The connection Laplacian provides an extension of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps.
Lecture 3: Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem
Wednesday, June 3, 2015, 4 - 5 PM, Eckhart 202
A key assumption in classical algorithms for cryo-EM is that the sample consists of (rotated versions of) identical molecules. However, in many datasets this assumption does not hold. Many molecules of interest exist in more than one conformational state. These structural variations are of great interest to biologists, as they provide insight into the functioning of the molecule. In this talk we tackle the problem of classifying the images to their different conformations. Our approach is based on estimating the covariance matrix of the underlying three-dimensional structures. What makes this problem challenging is that only 2D projections of the molecules are observed, but not the molecules themselves. This problem can be viewed as an instance of noisy low-rank matrix completion/sensing, and we derive an estimator for the covariance matrix as a solution to a certain linear system. The linear operator to be inverted, which we term the projection covariance transform, is an important object in covariance estimation for problems in computerized tomography involving structural variation.