February 04, 2011, 1100 Webb

We will begin by solving a classic open problem in numerical analysis. We will introduce a simple, fast, linear technique, for computing interpolating polynomials (in one and higher dimensions) that avoids the Runge phenomenon and converges rapidly on scattered data. Using this as a fundamental building block, we re-think the classical ideas behind finite-difference methods, and show that it is possible to construct very high-order finite-difference schemes on irregular grids with complex geometries. We will conclude with numerical experiments that compare our finite-difference code with state-of-the-art low- and high-order finite-element codes. We will also show examples of our code solving exterior problems without any artificial truncation of the domain

Shiv obtained his Ph.D. in numerical analysis at Yale University in 1994. After spending a year as a Visiting Instructor at the Mathematics department of North Carolina State University, Raleigh, he joined UCSB as an Assistant Professor. His early work included the development of first fast rank-revealing QR algorithm, and the first numerically stable inverse iteration algorithm for computing the eigendecomposition of symmetric matrices. Then, in joint work with Prof. Ali Sayed of UCLA, he developed the first numerically stable and fast algorithms for Toeplitz and other matrices with displacement structure. He also worked jointly with Professors Gu (UC Berkeley), Sayed and Golub (Stanford), in providing fast algorithms for finding globally optimal solutions to linear systems with bounded errors in the data. More recently, in joint work with Prof. Gu, Dewilde (TU Delft) and others, he has developed an entire family of fast numerical algorithms for matrices with HSS (Hierarchically Semi-Separable) structure. In collaboration with Prof. Mhaskar (Cal. State. Los Angeles), he developed the first numerically stable algorithm for constructing linear interpolatory polynomials that can work in arbitrary dimensions with arbitrary scattered data.