Date: Friday, November 2, 2012
Time: 10:30 AM
Place: E & CS Grid Room (2120) 

Title: Toward Unified Approximation Theory and Efficient Solution Techniques
       over Complex Geometries

Xiangmin Jiao
Department of Applied Mathematics and Statistics
Stony Brook University

         Abstract: I will present our current work on establishing a rigorous
         and unified theoretical foundation for numerical approximation methods
         on complex geometries, and developing an integrated algorithmic
         framework for analyzing and solving some challenging modeling and
         design problems. Our approach tightly integrates the analysis and
         algorithms in approximation theory, numerical linear algebra, and
         computational geometry. Our point of departure is a matrix-based
         numerical calculus for the n-dimensional Taylor series, which leads to
         a new approximation theory based on rank-deficient, weighted least
         squares (RD-WLS) approximations. We prove that WLS can deliver the
         same approximation power as interpolation-based approximations under
         realistic assumptions, while allowing irregular meshes or point
         clouds. We address rank deficiency using robust, adaptive numerical
         techniques, with guaranteed consistency and stability. The RD-WLS
         formulation introduces two scaling matrices, which help achieving
         optimal accuracy and help enhancing the diagonal dominance of the
         resulting linear systems without compromising accuracy, so that the
         resulting linear system is more amenable to the optimal multigrid
         solvers  based on stationary iterative methods.

         Short Bio:

         Dr. Xiangmin Jiao is an associate professor in the Department of
         Applied Mathematics and Statistics and an adjunct associate professor
         in the Department of Computer Science of Stony Brook University. He
         received his Ph.D. in computer science in 2001 from University of
         Illinois at Urbana-Champaign (UIUC). After receiving his Ph.D., he was
         a research scientist at the Center for Simulation of Advanced Rockets
         (CSAR) at UIUC and then a visiting assistant professor at Georgia
         Institute of Technology, before joining Stony Brook in 2007. His
         current research interests focus on developing efficient and robust
         algorithms and implementations for numerical discretizations over
         complex geometries, dynamic surfaces, optimal multigrid solvers,
         applied computational and differential geometry, multiphysics
         coupling, and their applications in various engineering and physical
         sciences.