活动地点：

活动类型：学术报告

主讲人：Prof. Stan Osher

活动时间：

活动内容：

Morning:

Using geometry and iterated refinement for inverse problems. (1) Total Variation Image Restoration

Stan Osher, joint with Jinjun Xu, Wotao Yin, Martin Burger and Donald Goldfarb

Abstract: Total Variation based regularization and image restoration was developed by Rudin-Osher-Fatemi in the late 80's. Recently, Yves Meyer characterized textures as elements of the dual of BV and did some extremely interesting analysis on the original ROF model. This led to practical algorithms to decompose images into structure plus texture.
Very promising results involving processing image gradients simultaneously with images were obtained by Lysaker-Osher-Tai, based on earlier work on processing surfaces by Tasdizen-Whitaker-Burchard-Osher. This has now led
to a new way of refining and enhancing the solutions to a wide class of inverse problems. I will discuss all this and present image restoration results which appear to be state-of-the-art.

Afternoon:

Computing Multivalued Physical Observables for Wave and Schrodinger Equations

Stanley Osher, joint with H. Liu, Y. Tsai and S. Jin

Abstract:
Motivated by obtaining Eulerian ray tracing algorithms, we develop a level set method for the computation of multivalued physical observables (density, velocity, etc.) for any symmetric hyperbolic system and for Schrodinger's equation. The main idea is to evolve the density near the n dimensional bicharacteristic manifold of the Hamilton-Jacobi equation, identified as the common zeros of n level set functions in phase space. The main advantages over the standard kinetic equation using the Liouville equation with a Dirac measure initial data are: (1) Our initial data is smooth-a singular integral involving the Dirac delta function is evaluated only in a postprocessing step, thus avoiding numerical difficulties and (2) a local level set method reduces the computational effort to an optimal number of
operations. These advantages enable us to compute all physical observables for multidimensional systems.