Multiscale Methods in Geometric | Center for Computation & Technology

	Frontiers of Scientific Computing Lecture Series
	Multiscale Methods in Geometric
	Michael Holst
	Professor of Mathematics, University of California, San Diego
	Johnston 338 February 25, 2005 - 02:30 pm
Abstract: There is currently tremendous interest in flows arising in geometric analysis, due in part to Perelman's recent attack on the Poincare and geometrization conjectures using Hamilton's Ricci flow program. In this lecture, we consider elliptic problems arising as geometric constraints in Einstein flow, and develop a weak solution theory and estimates in Sobolev and Besov classes for the constraints which generalize previous results. We then derive a priori and a posteriori error estimates for Petrov-Galerkin approximations, and develop some nonlinear approximation algorithms based on adaptive multiscale methods. We then outline a class of projection methods for enforcing these types of elliptic constraints during numerical evolution, yielding discrete solutions which exactly satisfy the constraints at each point in time. We develop an approximation theory framework which shows that this class of methods retains the accuracy and stability properties of standard time integration methods which do not enforce constraints. We finish by illustrating some of the techniques with examples using the Finite Element ToolKit (FEtk). The first portion of this of this talk covers joint work with David Bernstein at LBL, and the second portion covers joint work with Lee Lindblom and several members of the TAPIR group at Caltech.

Speaker's Bio: Full Professor, Department of Mathematics, UC San Diego, 2003-Present. Visiting Associate in Physics, Department of Physics, Caltech, 2003-2005. Associate Professor, Department of Mathematics, UC San Diego, 2000-2003. Assistant Professor, Department of Mathematics, UC San Diego, 1998-2000. von Karman Instructor, Applied Mathematics, Caltech, 1995-1997. Prize Postdoctoral Fellowship, Applied Mathematics, Caltech, 1993-1995. Ph.D., University of Illinois at Urbana-Champaign, 1993. M.S., University of Illinois at Urbana-Champaign, 1990. B.S., Colorado State University, 1987. Professor Holst came to the UCSD Mathematics Department in Summer 1998. Prior to arriving at UCSD, he was an assistant professor at UC Irvine during 1997-1998, and from 1993-1997 he was a Prize Research Fellow and a von Karman Instructor of Applied Mathematics at the California Institute of Technology. Professor Holst was awarded an NSF CAREER Award in 1999 for his research in computational and applied mathematics. He was promoted to tenured Associate Professor of Mathematics in 2000, and to Full Professor of Mathematics in 2003. Holst went on leave to Caltech in 2003 to pursue his research in computational astrophysics, but returned to UCSD in 2004. He is currently PI, Co-PI, and/or on the steering committees for a number of interdisciplinary research projects and centers at UCSD, including: * The La Jolla Interfaces in Science Program (http://ljis.ucsd.edu); * The Center for Theoretical Biological Physics (http://ctbp.ucsd.edu); * The National Biomedical Computation Resource (http://nbcr.ucsd.edu); * The Bioinformatics Ph.D. Program (http://bioinformatics.ucsd.edu); * and The Computational and Applied Mathematics Research Group within the UCSD Mathematics Department (http://cam.ucsd.edu). Professor Holst's general research background and interests are in a broad area called computational and applied mathematics; his specific research areas are partial differential equations (PDE), numerical analysis, approximation theory, applied analysis, and mathematical physics. His research projects center around developing mathematical techniques (theoretical techniques in PDE and approximation theory) and mathematical algorithms (numerical methods) for using computers to solve certain types of mathematical problems called nonlinear PDE. These types of problems arise in nearly every area of science and engineering; this is just a reflection of the fact that physical systems that we try to manipulate (e.g., the flow of air over an airplane wing, or the chemical behavior of a drug molecule), or build (e.g., the wing itself, or a semiconductor), or simply study (such as the global climate, or the gravitational field around a black hole) are described mathematically by nonlinear PDE. In simple cases, these problems can be simplified so that purely mathematical techniques can be used to solve them, but in most cases they can only be solved using sophisticated mathematical algorithms designed for use with computers. Computational simulation of PDE is now critical to almost all of science and engineering; the mathematicians provide the mathematical tools and understanding so that scientists in physics, chemistry, biology, engineering, and other areas can confidently use the modern techniques of computational science in the pursuit of new understanding in their fields of study. To learn more about Professor Holst's particular research program, please see his webpage: http://cam.ucsd.edu/~mholst

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