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IMA Tutorial/Workshop:

New Paradigms in Computation

March 28-30, 2005

The primary goal of this workshop is to facilitate the use of the best computational techniques in important industrial applications. Key developers of three of the most significant recent or emerging paradigms of computation - fast multipole methods, level set methods, and multiscale computation -will provide tutorial introductions to these classes of methods. Presentations will be particularly geared to scientists using or interested in using these approaches in industry. In addition the workshop will include research reports, poster presentations, and problem posing by industrial researchers, and offer ample time for both formal and informal discussion, related to the use of these new methods of computation. If you wish to describe a problem in the problem posing session, please contact Arnd Scheel at deputy@ima.umn.edu.

Organizer

Robert V. Kohn
Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
kohn@courant.nyu.edu
http://www.math.nyu.edu/faculty/kohn/

Tutorial Lectures:

Overview of Multiscale Methods
Problems with Multiple Time Scales

Weinan E
Department of Mathematics and Program in Applied and Computational Mathematics
Princeton University
weinan@princeton.edu
http://www.math.princeton.edu/~weinan/

Fast Multipole Methods and their Applications

Leslie F. Greengard
Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
greengard@cims.nyu.edu
http://www.math.nyu.edu/faculty/greengar/

Advances in Advancing Interfaces: Level Set Methods, Fast Marching Methods, and Beyond

James A. Sethian
Department of Mathematics
University of California-Berkeley
sethian@math.berkeley.edu
http://math.berkeley.edu/~sethian/

Biographies and Lecture Abstracts

Weinan E (Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University) http://www.math.princeton.edu/~weinan/

Biography: Weinan E received his PhD from the University of California at Los Angeles in 1989. He was visiting member at the Courant Institute from 1989 to 1991. He joined the IAS in Princeton as a long term member in 1992 and went on to take a faculty position at the Courant Institute at New York University in 1994. He is Professor of Mathematics at Princeton University since 1999. His awards include the Alfred P. Sloan Foundation Fellowship, a Presidential Faculty Fellowship, the Feng Kang Prize in Scientific Computing and the Collatz Prize awarded by the International Council of Industrial and Applied Mathematics. He serves on the editorial board of various journals including the Journal of American Mathematical Society, Acta Mathematica Sinica, Journal of Computational Mathematics, Communications of Contemporary Mathematics, and Journal of Statistical Physics.

Overview of Multiscale Methods

Abstract: We will begin by reviewing the basic issues and concepts in multiscale modeling, including the various models of multi-physics, serial and concurrent coupling strategies, and the essential features of the kind of multiscale problems that we would like to deal with. We then discuss some representative examples of successful multiscale methods, including the Car-Parrinello method and the quasi-continuum method. Finally we discuss several general methodologies for multiscale, multi-physics modeling, such as the domain decomposition methods, adaptive model refinement and heterogeneous multiscale methods. These different methodologies are illustrated on one example, the contact line problem. Throughout this presentation, we will emphasize the interplay between physical models and numerical methods, which is the most important theme in modern multiscale modeling.

Problems with Multiple Time Scales

Abstract: We will discuss the mathematical background and numerical techniques for three types of problems with multiple time scales: stiff ODEs, Markov chains with disparate rates and rare events.

Leslie F. Greengard (Department of Mathematics, Courant Institute of Mathematical Sciences, New York University)

Biography: Leslie F. Greengard was born in London, England, and grew up in New York, Boston, and New Haven. He received his B.A. in mathematics from Wesleyan University in 1979, his Ph.D. in computer science from Yale University in 1987, and his M.D. from Yale University in 1987. From 1987 89 he was a National Science Foundation Postdoctoral Fellow at Yale University in the Department of Computer Science. He is presently a professor of mathematics at the Courant Institute of New York University, where he has been a faculty member since 1989. In 2001, he was awarded the Leroy P. Steele Prize by the AMS Council. Much of his work has been in the development of analysis-based fast algorithms such as the Fast Multipole Method for gravitation and electromagnetics and the Fast Gauss Transform for diffusion.

Fast Multipole Methods and their Applications

Abstract: In these lectures, we will describe the analytic and computational foundations of fast multipole methods (FMMs), as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a system of N particles from O(N2) to O(N) or O(N log N) operations. FMMs are equally useful, however, in solving partial differential equations by first recasting them as integral equations. We will present examples from electromagnetics, elasticity, and fluid mechanics.

For lecture notes, please see R. K. Beatson and L. Greengard, A short course on fast multipole methods, in Wavelets, Multilevel Methods and EllipticPDEs, M. Ainsworth, J. Levesley, W. Light, and M. Marletta, eds., Oxford University Press, 1997, pp. 1.37

A preprint is available at www.math.nyu.edu/faculty/greengar

Robert V. Kohn (Department of Mathematics, Courant Institute of Mathematical Sciences)New York University) http://www.math.nyu.edu/faculty/kohn/

Biography: Robert V. Kohn received his A.B. from Harvard University in 1974, his M.Sc. from the University of Warwick in 1975, and a Ph.D. From Princeton in 1979. He spent two years as an NSF Postdoc at New York University's Courant Institute of Mathematical Sciences, before he joined the faculty. He has been a Professor of Mathematics at the Courant Institute since 1981. His honors include SIAM's Ralph Kleinman Prize, an Ordway Visiting Professorship at the University of Minnesota, and a Sloan Research Fellowship. His research interests include mathematical aspects of materials science, nonlinear partial differential equations, nonconvex variational problems, and mathematical finance. In addition, he is among the leaders of the Courant Institute's professional masters program in mathematical finance.

James A. Sethian (Department of Mathematics, University of California-Berkeley) http://math.berkeley.edu/~sethian/

Biography: James A. Sethian was born on May 10, 1954, in Washington, DC. He received a B.A. in mathematics from Princeton University in 1976 and a Ph.D. in Applied Mathematics from the University of California, Berkeley, in 1982. After a National Science Foundation Postdoctoral Fellowship at the Courant Institute of Mathematical Sciences, he joined the faculty at UC Berkeley, where he is now Professor of Mathematics as well as Head of the Mathematics Department at the Lawrence Berkeley National Laboratory. He has been a plenary speaker at the International Congress of Industrial and Applied Mathematicians, and has been an invited speaker at the International Congress of Mathematicians. He has received SIAM's I. E. Block Community Lecture Prize, and in 2004 was awarded the Norbert Wiener Prize in Applied Mathematics by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics SIAM). He is an Associate Editor of SIAM Review, the Journal of Mathematical Imaging and Vision, and the Journal on Interfaces and Free Boundaries.

Advances in Advancing Interfaces: Level Set Methods, Fast Marching Methods, and Beyond

Abstract: Propagating interfaces occur in a variety of settings, including semiconductor manufacturing in chip production, the fluid mechanics of ink jet plotters, segmentation in cardiac medical imaging, computer-aided-design, optimal navigation in robotic assembly, and geophysical wave propagation. Over the past 25 years, a collection of numerical techniques have come together, including Level Set Methods and Fast Marching Methods for computing such problems in interface phenomena in which topological change, geometry-driven physics, and three-dimensional complexities play important roles. These algorithms, based on the interplay between schemes for hyperbolic conservation laws and their connection to the underlying theory of curve and surface evolution, offer a unified approach to computing a host of interface problems.

In this tutorial, the author will cover (i) the development of these methods, (ii) the fundamentals of Level Set Methods and Fast Marching Methods, including efficient, adaptive versions, and the coupling of these schemes to complex physics, and (iii) new approaches to tackling more demanding interface problems. The emphasis in this tutorial will be on a practical, "hands-on" view, and the methods and algorithms will be discussed in the context of on-going collaborative projects, including work on semiconductor processing, industrial ink jet design, and medical and bio-medical imaging.

Raju Mattikalli (The Boeing Company)

Industrial Problem: Observer Positioning:

Abstract: Consider the museum guard problem, i.e. finding positions for guards in a museum so that they can have the largest number of museum artifacts within sight. This problem was first defined by Victor Klee in 1973. There are many variations of the museum guard problem.

My interest is in an outdoor version of the museum guard problem. Consider the problem of positioning cameras to monitor mountainous terrain, say over a section of the Grand Canyon. The terrain can be assumed to have vertical surfaces, but no overhangs. The field of view of a camera can be represented as a cone with a fixed half angle and height. The objective is to define camera positions and orientations to achieve maximum coverage of the terrain surface.

Cameras are either fixed or mobile. Fixed (both over space and time) cameras can be mounted at any point zero to six feet high along a normal to the ground surface. The camera cone can have any orientation. Mobile cameras can be assumed to be mounted on constant speed airplanes capable of making turns with a radius no smaller than R. Mobile cameras can have time varying orientation, with a maximum angular rotation speed of T.

I will present 3 variations of the above problem.

LIST OF CONFIRMED PARTICIPANTS

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