Lefschetz Center for Dynamical Systems Seminar
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: I will describe a new method for completing the boundaries of partially occluded objects. Like computations in primary visual cortex (and unlike all previous models of contour completion) our computation is Euclidean invariant. This invariance is achieved by representing the various states of the computation in a shiftable-twistable basis of 3D functions which resemble receptive fields in V1 and V2. Our method builds on Mumford's use of a Fokker-Planck equation to characterize the distribution of natural boundary shapes, Williams and Jacobs' notion of a stochastic completion field to model illusory contours in a probabilistic fashion, and Simoncelli et al.'s notion of a shiftable-steerable basis to perform Euclidean invariant computations in the plane.
This work is joint with Lance Williams in the computer science department at the University of New Mexico.
Brown Analysis Seminar
Applied Mathematics Colloquium
Abstract: Spectral methods are a powerful tool for the high-accuracy numerical solution of ODEs and PDEs in simple domains. Sometimes this technology seems complicated, but it is amazing what can be accomplished in a few inches of Matlab based on the simple idea of a Chebyshev collocation derivative. This talk will be an online tour of the forty short programs from my book soon to be published by SIAM, "Spectral methods in Matlab". Along the way we will solve, always to many digits of precision, problems including the Laplace, Poisson, biharmonic, wave, Mathieu, Orr-Sommerfeld, Airy, Schrodinger, Helmholtz, KdV, Allen-Cahn, and Kuramoto-Sivashinsky equations. The codes are available on the Web, and the longest is 35 lines long.
Scientific Computing Seminar
Abstract: Despite many advances in computational technique, solving Maxwell's equations in three dimensional exterior domains remains a challenging problem. This talk is about an integral equation based approach for scattering problems. The novelty of the method is that the incident field is synthesised using electromagnetic sources which lie strictly inside the scatterer. This is in contrast to the usual technique in which sources are distributed on the surface of the scatterer. One advantage of the interior source approach is that the integral equation has an analytic kernel with the consequence that the numerical solutions can converge exponentially (spectral convergence) as a function of the number of sources. In addition the difficulties of working with singular kernels are absent. The talk will explain the mathematical and physical bases of interior source methods and will present some recently developed theory to explain the rapid convergence. The results will be illustrated with specific scattering calculations.
PDE Seminar
Department of Mathematics Colloquium
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