University of Alberta Edmonton Canada

Multiscale Computation and Analysis for Schrödinger-Poisson System in Heterogeneous Materials Liqun Cao Chinese Academy of Sciences, China [email protected] In this talk, we discuss the multiscale computation for Schrödinger-Poisson system arising from the electronic properties of semiconductors such as quantum wells, wires and dots, and CMOS transistor. We first introduce the Schrödinger equations with the effective mass approximation which has been particularly successful in the case of heterostructures. Combining Allaire’s work and our result, we give an interpretation why the effective mass approximation is very successful for calculating the band structures of semiconductor nanostructures in the vicinity of Γ point, from the viewpoint of mathematics. Second, we analyze the relationship between microscopic Maxwell’s equations and macroscopic Maxwell’s equations. Furthermore, we discuss mathematical modeling of electromagnetics at nano-scale and offer an explanation as to why Schrödinger-Poisson equations and Maxwell-Schrödinger equations have been widely used to semiconductor nanostructures. Third, the multiscale method for Schrödinger-Poisson system in heterogeneous materials is presented. Numerical simulations are carried out to validate the theoretical results. Finally, some unsolved problems are advanced. Framelet-Based Algorithm for Medical Imaging Applications Raymond H. Chan The Chinese University of Hong Kong, Hong Kong [email protected] Framelets have been used successfully in various problems in image processing, including inpainting, impulse noise removal, super-resolution image restoration, etc. In this talk, we will introduce two new applications of the tight-frame algorithm in medical imaging. The first one is segmentation which is the process of identifying object outlines within images. We apply the framelet-based approach to identify tube-like structures such as blood vessels in magnetic resonance angiography images. Our method iteratively refines a region that encloses the possible boundary or surface of the vessels. Results on real 2D/3D images demonstrate that the method is very efficient and outperforms other existing methods. The second application is image reconstruction for parallel magnetic resonance imaging (MRI).