John Von Neumann Lecture: The Effect of Local Features on Global Expansions
Tuesday, July 8
2:30 PM - 3:30 PM
Room: Town & Country
Global expansions, such as Fourier expansions, form the basis of many approximations which are successfully used in a wide variety of applications. Fourier expansions are used in applications ranging from medical CT scans, which are based on the Fourier expansion coefficients of an image, to spectral methods which are widely used to simulate complicated flows.
While methods based on global approximations converge exponentially when the underlying function and all its derivatives are smooth, they lose accuracy in the presence of discontinuities. The term "Gibbs phenomenon" refers to the fact that the presence of a local discontinuity degrades the global convergence of such methods, although this fact was first pointed out by Albert Michelson. Over the years, a variety of techniques have been developed to alleviate or overcome the Gibbs phenomenon. These techniques include filtering (in the physical space as well as in the transform plane) and reprojections (re-expansing the numerical solution in a different basis). In this talk, we will present the history of the Gibbs phenomenon, and review the recent literature concerning overcoming of this phenomenon, including a discussion of methods for edge detection, which allow us to determine the location of the discontinuities.
David I. Gottlieb