IAM Distinguished Colloquium Series

Monday, 17 Oct 2011, 3:00-4:00 pm
Prof. Joel A. Tropp, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Low-Rank Matrix Decompositions

Computer scientists have long known that randomness can be used to improve the performance of algorithms. A familiar application is the process of dimension reduction, in which a random map transports data from a high-dimensional space to a lower-dimensional space while approximately preserving some geometric properties. By operating with the compact representation of the data, it is theoretically possible to produce approximate solutions to certain large problems very efficiently. Recently, it has been observed that dimension reduction has powerful applications in numerical linear algebra and numerical analysis. This talk provides a high-level introduction to randomized methods for computing standard matrix approximations, and it summarizes a new analysis that offers (nearly) optimal bounds on the performance of these methods. In practice, the techniques are so effective that they compete with – or even outperform – classical algorithms. Since matrix approximations play a ubiquitous role in areas ranging from information processing to scientific computing, it seems certain that randomized algorithms will eventually supplant the standard methods in some application domains. This is joint work with Gunnar Martinsson and Nathan Halko (the paper is available here).

Joel A. Tropp is Assistant Professor of Applied & Computational Mathematics at the California Institute of Technology. He earned the Ph.D. degree in Computational Applied Mathematics from the University of Texas at Austin in 2004. Dr. Tropp’s work lies at the interface of applied mathematics, electrical engineering, computer science, and statistics. The bulk of this research concerns the theoretical and computational aspects of sparse approximation, compressive sampling, and randomized linear algebra. He has also worked extensively on the properties of structured random matrices. Dr. Tropp has received several major awards for young researchers, including the 2007 ONR Young Investigator Award and the 2008 Presidential Early Career Award for Scientists and Engineers. He is also winner of the 6th Vasil A. Popov prize and the 2011 Monroe H. Martin prize.

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Monday, 14 Nov 2011, 3:00-4:00 pm
Dr. William L. Oberkampf, Consulting Engineer, Austin, Texas
Perspectives on Verification, Validation, and Uncertainty Quantification FULL TALK

Verification and validation (V&V) are the primary means to assess mathematical model and numerical accuracy in computational simulations. Code verification deals with the assessment of the reliability of the software coding and the numerical algorithms used, while solution verification deals with numerical error estimation of the computational solution to the mathematical model. Validation assesses the accuracy of the mathematical model as compared to an appropriate fiducial reference. In the natural sciences, this reference is commonly experimental measurements of the system of interest. Uncertainty quantification attempts to characterize the uncertainties represented by, and due to, the mathematical model, the numerical solution error, and the experimental data. Important research questions in uncertainty quantification deal with (a) model updating and calibration, as opposed to predictive uncertainty estimation, (b) estimation of model form uncertainty for cases where experimental data are available, and (c) extrapolation of estimated model form uncertainty to conditions for which no experimental data are available. This talk will briefly discuss all of these issues within the framework of how computational simulations are used in a decision-making environment.

William L. Oberkampf received his PhD in 1970 from the University of Notre Dame in Aerospace Engineering. He has 41 years of experience in research and development in fluid dynamics, heat transfer, flight dynamics, and solid mechanics. He served on the faculty of the Mechanical Engineering Department at the University of Texas at Austin from 1970 to 1979. From 1979 until 2007 he worked in both staff and management positions at Sandia National Laboratories in Albuquerque, New Mexico. During his career he has been deeply involved in both computational simulation and experimental activities. During the last 20 years he has been focused on verification, validation, uncertainty quantification, and risk analyses in modeling and simulation. He retired from Sandia as Distinguished Member of the Technical Staff and is a Fellow of the American Institute of Aeronautics and Astronautics. He has over 160 journal articles, book chapters, conference papers, and technical reports, and has taught 35 short courses in the field of verification and validation. He recently co-authored, with Christopher Roy, the book "Verification and Validation in Scientific Computing" published by Cambridge University Press.

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Monday, 05 Mar 2012, 3:00-4:00 pm
Prof. Andy Woods, Department of Earth Sciences, University of Cambridge, United Kingdom
Modelling Carbon Sequestration Processes

In this talk we will present a series of models of the motion of CO₂ following injection into the subsurface, including models of the migration of a buoyant plume as it experiences capillary trapping, drainage and dissolution in a heterogeneous rock. The models will focus on identifying leakage pathways and the fraction of the CO₂ which may leak from the system. We will also present models of the dynamics of near surface leakage including possible lake eruptions driven by CO₂ seeps.

Professor Andy Woods has research interests in a wide range of fluid mechanical problems, ranging from the dynamics of explosive volcanic systems and natural ventilation to oil recovery, geothermal energy and carbon sequestration. He has been a Green Scholar at Scripps Institution of Oceanography and won several prestigious awards including the Marcello Carapezza Prize and the Wagner Medal. He is currently the B.P. Professor of Applied Mathematics and Geophysics at the University of Cambridge, as well as the Director of the B.P. Institute for Multiphase Flow.

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Monday, 12 Mar 2012, 3:00-4:00 pm
Prof. Andrew Bernoff, Department of Mathematics, Harvey Mudd College, Claremont, California
Langmuir Layers: Exploring A (Nearly) Two-Dimensional Fluid Experiment

A Langmuir Layer is a molecularly thin layer of a polymer, lipid or liquid crystal on the surface of another fluid. In this (nearly) two-dimensional layer, we can observe bubbles of a fluid phase that even when stretched or highly contorted always appear to return to a circular shape. The force driving these evolutions is line tension, a two-dimensional analog of surface tension. We report on a combined experimental, theoretical, and numerical study of Langmuir layers and show how we can deduce the strength of the line tension in the system by comparing theory and experiment. As time permits we will also describe other phenomena observed in Langmuir systems, including collapse of gas phase bubbles, co-existence of three or more fluid phases, and formation of dogbone and labyrinth patterns due to dipolar repulsion in the layer. This work is the result of collaboration with Prof. Elizabeth Mann, an experimental physicist at Kent State University, Prof. J. Adin Mann, Jr., a chemical engineer at Case Western Reserve University, and Prof. James Alexander, a mathematician also at Case Western Reserve University and is supported by the National Science Foundation.

Andrew Bernoff is a Professor of Mathematics at Harvey Mudd College. His research specializes in bridging the gaps between mathematics, physics, biology and engineering with a particular emphasis on using dynamical systems methods to understand experiments and natural phenomena. Prof. Bernoff was an undergraduate at MIT where he received BS degrees in mathematics and physics. While an undergraduate, he founded the MIT Integration Bee. In 1978 he was awarded a Marshall Scholarship to pursue a PhD at the University of Cambridge in England. His PhD studies were on the application of dynamical systems methods in fluid mechanics in the Department of Applied Mathematics and Theoretical Physics (DAMTP). Prof. Bernoff has spent time on the faculty at Northwestern, Duke and the University of California at Berkeley before settling in at Harvey Mudd College, where he is the Diana and Kenneth Jonsson Professor of Mathematics and Chair of the Mathematics Department. He is passionate about mentoring undergraduate research, coaching the Harvey Mudd College Putnam Team, and supporting Harvey Mudd College’s Clinic Program, a year-long practicum in which teams of undergraduates work for industrial sponsors on real-world problems and applications. His NSF-supported research program centers on understanding the behavior of fluids at small scales and modeling the swarming of organisms, in particular locusts, and is built on collaborations at multiple colleges and universities.

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Monday, 19 Mar 2012, 3:00-4:00 pm
Prof. John M. Guckenheimer, Mathematics Department, Cornell University, Ithaca, New York
Complex Oscillations

Biological and chemical systems display bursting and mixed mode oscillations. This lecture will survey recent advances in the theory of dynamical systems with multiple time scales that has dramatically improved our understanding of these complex temporal behaviors. A natural classification of different types of bursting and mixed modes is developed and used as a foundation for numerical methods that analyze multiple time scale models. These methods are applied to mixed mode oscillations of chemical reactions that were intensively studied thirty years ago without producing models that faithfully reproduced experimental observations. Additional examples are drawn from neuroscience.

John Guckenheimer obtained his Ph.D. from University of California at Berkeley in 1970 and is currently the Abram R. Bullis Professor in Mathematics at Cornell University. His research is a blend of theoretical investigation, development of computer methods and studies of nonlinear systems that arise in diverse fields of science and engineering. Two of the primary themes have been bifurcation theory and the effects of multiple time scales in shaping dynamical behavior. Application areas in which he has worked include population biology, fluid dynamics, neurosciences, animal locomotion and control of nonlinear systems. His work on algorithm development includes contributions to methods for computing bifurcations, periodic orbits and invariant manifolds of vector fields and for the analysis of fractal dimensions of attractors.

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Monday, 26 Mar 2012, 3:00-4:00 pm
Prof. Richard Montgomery, Department of Mathematics, University of California at Santa Cruz, California
From Brake to Syzygy in the Three-Body Problem

A brake orbit for the Newtonian three-body problem is a solution for which all three velocities are zero at some instant: the brake instant. If we follow such an orbit there will be a later instant at which the three bodies become colinear: the instant of syzygy. In this manner we can define a flow-induced "Poincare map" from brake initial conditions to syzygy configurations. Appropriately viewed, this brake-to-syzygy map is a map between planar domains. Understanding its image destroyed certain myths that the speaker had regarding action-minimizing orbits. The map fits in towards a possible global understanding of the planar three-body problem which we will explain. Key is a viewpoint on the planar three-body problem in which the configuration of all three bodies is represented as a single point in 3-space (its "shape") and in which Newton's equations become a mechanical system on this 3-space. Some movies of Paul Klee-like periodic brake orbits inspired by this work will be shown.

Richard Montgomery got undergraduate degrees in mathematics and physics from Sonoma State in Northern California in 1980. He got his PhD under Jerry Marsden at Berkeley in 1986 and after that had a Moore Instructorship at MIT for two years, then two years of postdoc in Berkeley. His research fields are geometric mechanics, celestial mechanics, control theory, and differential geometry. He is perhaps best known for his rediscovery, with Alain Chenciner, of Cris Moore's figure eight solution to the three-body problem, which led to a slew of new 'choreography' solutions. He also established the existence of the first-known abnormal minimizer in subRiemannian geometry (in control lingo this is an abnormal extremal for a problem linear in controls, with control quadratic cost function), and is known for investigations using gauge-theoretic ideas of how a falling cat lands on its feet. He has written one book on subRiemannian geometry. In addition to mathematics and mechanics, he is a minor slowly fading legend in the kayaking world of California for first descents done in the early 1980s. He has two daughters, is married, and lives and works in Santa Cruz, CA.

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