Video Research Vignette Available
As part of Purdue's annual presidential review of the College of Science,
the department of mathematics has commissioned a short video vignette
where I describe my research in a broadly accessible fashion. This is
available as a streaming video.
Selected as one of Purdue's two nominees for the
Packard Fellowship
I have been selected in an internal competition as one of Purdue
University's two nominees in 2009 for the Packard
Fellowship for Science and Engineering. Every year, the
Packard Foundation invites the presidents of 50 universities to nominate two professors each
from their institutions. An advisory panel of distinguished scientists and
engineers then selects 20 Fellows to receive individual awards of
$875,000, payable over five consecutive years.
Visiting Control and Dynamical Systems at Caltech for
the Spring Quarter
I will be on leave from Purdue University from April 2009 to mid June
2009, and will be in residence at my alma mater, the California Institute of
Technology, during their spring quarter as a Visiting
Assistant Professor of Control and
Dynamical Systems.
Graduate Positions Available in the Computational Geometric
Mechanics group
Research positions for graduate students, and are available in the broad
area of geometric numerical
methods in geometric mechanics and control. A description of my current
research may be found in my research
statement, bibliography, collected publications, or
the abstracts below.
These positions are funded by the following research grants from the National Science Foundation in applied and computational mathematics:
Please contact me by email if you are interested in any
of these positions. These are full-time positions, and students who are
interested in internship opportunities should read my policy on internships.
Academic
and Industrial, Science and Engineering Collaborations
Welcomed
I am constantly on the lookout for interesting and challenging
collaborations with scientists and engineers from academia or industry.
If you have an interesting application for which geometric
numerical methods might be relevant, please feel free to contact me.
Much of my theoretical work is motivated by specific applications.
To paraphrase William Morris, "Have nothing in your [mathematics] that
you do not know to be useful, or believe to be beautiful."
Computational Geometric Mechanics: Applying
discrete
differential geometry and discrete Lagrangian and Hamiltonian mechanics to
the construction of geometric structure-preserving numerical algorithms.
Computational Geometric Control Theory:
Construction of
real-time digital feedback control of mechanical systems using techniques
from geometric control theory and computational geometric mechanics.
Numerical Analysis: Derivation of accurate and
efficient
numerical schemes with good long-time geometric stability properties
by combining geometric integration with adaptive, spectral, and
multiscale techniques.
Discrete Poincaré Lemma (with M. Desbrun,
and J.E. Marsden), Appl. Numer. Math. 53 (2-4), 231-248,
2005.
[ PDF | Journal Link | BibTeX ]
A Lie Group Variational Integrator for the Attitude Dynamics of a
Rigid Body with Applications to the 3D Pendulum (with T. Lee, and
N.H. McClamroch), Proc. IEEE Conf. on Control Applications, 962-967,
2005.
[ PDF | IEEE Xplore | BibTeX | Animation: Chaotic Motion ]
Controlled Lagrangians and Stabilization of the Discrete
Cart-Pendulum System (with A.M. Bloch, J.E. Marsden, and D.V.
Zenkov), Proc. IEEE Conf. on Decision and Control, 6579-6584, 2005.
[ PDF | IEEE Xplore | BibTeX ]
Discrete Routh Reduction (with S.M. Jalnapurkar, J.E.
Marsden, and M. West), J. Phys. A: Math. Gen. 39, 5521-5544 (Geometric Integration
special issue, invited paper), 2006.
[ PDF | arXiv:math.NA/0508330 | Journal Link | BibTeX ]
Polyhedral Potential and Lie Group Variational Integrator
Computations for the Full Two Body Problem (with
E. Fahnestock, T. Lee, N.H. McClamroch, D.J. Scheeres), Proc. AIAA/AAS
Astrodynamics Specialist Conf., AIAA-2006-6289, 2006.
[ PDF | arXiv:math.NA/0608695 | Journal Link | BibTeX ]
Optimal Control of a Rigid Body using Geometrically Exact
Computations on SE(3) (with T. Lee, and N.H. McClamroch),
Proc. IEEE Conf. on Decision and Control, 2710-2715, 2006.
[ PDF | arXiv:math.OC/0602588 | IEEE Xplore | BibTeX | Animation: Fig1(a), Fig2(a), Fig3(a), Fig4(a) ]
Deterministic Global Attitude Estimation
(with T. Lee, A.K. Sanyal, and N.H. McClamroch),
Proc. IEEE Conf. on Decision and Control, 3174-3179, 2006.
[ PDF | arXiv:math.OC/0602589 | IEEE Xplore | BibTeX ]
Controlled Lagrangians and Potential Shaping for
Stabilization of Discrete Mechanical Systems
(with A.M. Bloch, J.E. Marsden, and D.V. Zenkov),
Proc. IEEE Conf. on Decision and Control, 3333-3338, 2006.
[ PDF | arXiv:math.OC/0602590 | IEEE Xplore | BibTeX ]
A Discrete Variational Integrator for Optimal Control Problems in
SO(3) (with A.M. Bloch, I.I. Hussein, and A.K. Sanyal), Proc. IEEE
Conf. on Decision and Control, 6636-6641, 2006.
[ PDF | arXiv:math.OC/0509536 | IEEE Xplore | BibTeX ]
Global Attitude Estimation using Single Direction
Measurements (with T. Lee, N.H. McClamroch, and A.K. Sanyal),
Proc. American Control Conf., 3659-3664, 2007.
[ PDF | arXiv:math.OC/0609481 | IEEE Xplore | BibTeX ]
Lie Group Variational Integrators for the Full Body
Problem, (with T. Lee, and N.H. McClamroch), Comput. Methods
Appl. Mech. Engrg. 196(29-30), 2907-2924, 2007.
[ PDF | arXiv:math.NA/0508365 | Journal Link | BibTeX | Animation: Fig4 ]
Lie Group Variational Integrators for the Full Body Problem in
Orbital Mechanics (with T. Lee, and N.H. McClamroch), Celestial
Mechanics and Dynamical Astronomy 98(2), 121-144, 2007.
[ PDF | Journal Link | BibTeX | Animation: Fig2 ]
Propagation of Uncertainty Sets for Rigid Body Attitude Flows
(with N.A. Chaturvedi, T. Lee, N.H. McClamroch, and A.K. Sanyal), Proc.
IEEE Conf. on Decision and Control, 2689-2694, 2007.
[ PDF | arXiv:math.DS/0702737 | IEEE Xplore | BibTeX ]
A Combinatorial Optimal Control Problem for Spacecraft
Formation Reconfiguration
(with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and
Control, 5370-5375, 2007.
[ PDF | arXiv:math.OC/0702738 | IEEE Xplore | BibTeX | Animation: Fig2 ]
Global Optimal Attitude Estimation using Uncertainty
Ellipsoids (with A.K. Sanyal, T. Lee, and N.H. McClamroch), Systems
and Control Letters, 57(3), 236-245, 2008.
[ PDF | arXiv:math.OC/0606083 | Journal Link | BibTeX ]
Time Optimal Attitude Control for a Rigid Body (with T. Lee,
and N.H. McClamroch), Proc. American Control Conf., 5210-5215, 2008.
[ PDF | arXiv:0709.2514 | IEEE Xplore | BibTeX | Animation: Fig1(a), Fig2(a) ]
Matching and stabilization of discrete mechanical systems
(with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc.
Appl. Math. Mech., 7, 1030603-1030604, 2007.
[ PDF | Journal Link | BibTeX ]
Optimal Attitude Control of a Rigid Body using Geometrically
Exact Computations on SO(3) (with T. Lee, and N.H. McClamroch),
Journal of Dynamical and Control Systems, 14(4), 465-487, 2008.
[ PDF | arXiv:math.OC/0601424 | Journal Link | BibTeX | Animation: Fig2(a), Fig3(a), Fig4(a), Fig5(a) ]
Global Symplectic Uncertainty Propagation on SO(3) (with T.
Lee, and N.H. McClamroch), Proc. IEEE Conf. on
Decision and Control, 61-66, 2008.
[ PDF | arXiv:0803.1515 | IEEE Xplore | BibTeX ]
Discrete Control Systems (with T. Lee, and N.H. McClamroch),
invited article for Springer Encyclopedia of Complexity and System
Science, accepted, 2008.
[ PDF | arXiv:0705.3868 | BibTeX ]
Geometric Structure-Preserving Optimal Control of the Rigid
Body (with A.M. Bloch, I.I.
Hussein, and A.K. Sanyal), Journal of Dynamical and Control Systems,
accepted, 2008.
[ PDF | arXiv:0712.4400 | BibTeX ]
Computational Geometric Optimal Control of Rigid Bodies (with
T. Lee, and N.H. McClamroch), Brockett Legacy Special Issue,
Communications in Information and Systems, 8(4), 445-472, 2008.
[ PDF | arXiv:0805.0639 | Journal Link | BibTeX ]
Dynamics of Connected Rigid Bodies in a Perfect Fluid (with
T. Lee, and N.H.
McClamroch), Proc. American Control Conf., 408-413, 2009.
[ PDF | arXiv:0809.1488 | IEEE Xplore | BibTeX | Animation: Fig1 ]
Controlled Lagrangians and Stabilization of Discrete Mechanical
Systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov),
Discrete and Continuous Dynamical Systems - Series S, accepted, 2009.
[ PDF | arXiv:0704.3875 | BibTeX ]
Nonlinear Dynamics of the 3D Pendulum (with N.A.
Chaturvedi, T. Lee, and N.H.
McClamroch), Journal of Nonlinear Science, submitted, 2008.
[ PDF | arXiv:0707.1196 | BibTeX ]
Discrete Dirac Structures and Variational Discrete Dirac
Mechanics (with T. Ohsawa), Foundations of Computational Mathematics,
submitted, 2008.
[ PDF | arXiv:0810.0740 | BibTeX ]
Dynamics of a 3D Elastic String Pendulum (with T. Lee and
N.H. McClamroch), Proc. IEEE Conf. on Decision and Control,
accepted, 2009.
[ PDF | arXiv:0903.0332 | BibTeX | Animation: Fig2 ]
A Discrete Theory of Connections on Principal Bundles (with
J.E. Marsden, and A.D. Weinstein), preprint, 2004.
[ PDF | arXiv:math.DG/0508338 | BibTeX ]
Discrete Exterior Calculus (with M. Desbrun, A.N. Hirani, and
J.E. Marsden), preprint, 2003.
[ PDF | arXiv:math.DG/0508341 | BibTeX ]
Estimating the Attractor Dimension of the Equatorial Weather
System, Acta Phys. Pol. A 85, S27-S35, 1994.
[ PDF | Journal Link | BibTeX ]
Papers in Preparation
Computational Dynamics of a 3D Elastic String Pendulum
Attached to a Rigid Body and a Reel Mechanism (with T. Lee,
N.H. McClamroch).
Dirac-Hamilton-Jacobi Theory for Implicit Lagrangian and
Hamiltonian Systems (with T. Ohsawa, D. Sosa).
Dirac Structures in Lagrangian Mechanics on Lie Groupoids
(with D. Sosa).
Discrete Hamilton-Jacobi Theory (with A. Bloch, T. Ohsawa).
Discrete Hamiltonian Variational Integrators (with J.
Zhang).
Discrete Augmented Variational Principles in Mechanics and
Control (with
T. Ohsawa).
Optimal Control of Connected Rigid Bodies in a Perfect Fluid
(with T. Lee, N.H. McClamroch).
Referee, Advances in Difference Equations,
Aerospace Science and Technology,
Applied Numerical Mathematics, Celestial
Mechanics and Dynamical Astronomy, Communications in
Contemporary Mathematics, Communications in Numerical
Methods in Engineering, Computational Science &
Discovery, Discrete and Continuous Dynamical
Systems, ESAIM: Control, Optimisation and Calculus
of Variations, Foundations of Computational Mathematics,
IEEE Transactions on Automatic Control, IET Control
Theory & Applications, IMA Journal of
Numerical Analysis, Journal of Computational
Physics, Journal of Mathematical Physics, Journal of
Nonlinear Science, Journal of Physics A, Journal of
Symplectic Geometry, Mathematical and Computer Modelling,
Nonlinearity, Numerical Algorithms,
Numerische Mathematik, Physica D,
Physics Letters A,
Proceedings of the Royal Society A, SIAM Applied Dynamical
Systems, SIAM Multiscale Modeling and Simulation,
SIAM Journal on Numerical Analysis, SIAM Journal on
Scientific Computing, SIGMA
(Symmetry, Integrability and Geometry: Methods and
Applications), Soft Computing and Automation Journal,
Transport in Porous Media.
Reviewer, IEEE Conference on Decision and Control 2005,
2006, 2007, 2008, IEEE Multi-conference on
Systems and Control 2007, American Control Conference 2008,
Mathematical Reviews, Springer Books, Swiss
National Science Foundation.
Panel member, Applied Mathematics, National Science Foundation, Mar
2009.
Panel member, Computational Mathematics, National Science Foundation,
Mar 2008.
Erdös, P.; Graham, R. L.; Montgomery, P.;
Rothschild, B. L.; Spencer, J.; Straus, E. G.;
Euclidean Ramsey theorems. I., J. Combinatorial Theory Ser. A, 14
(1973), 341-363.
Golubitsky, M. A.; Rothschild,
B. L.; Primitive subalgebras of exceptional Lie algebras.
Bull. Amer. Math. Soc. 77 1971 983-986.
Golubitsky, M. A.; Marsden, J.
E.; The Morse lemma in infinite dimensions via singularity
theory. SIAM J. Math. Anal. 14 (1983), no. 6, 1037-1044.
Desbrun, M.; Leok, M.; Marsden, J. E.;
Discrete Poincaré
Lemma, Appl. Numer. Math. 53(2-4), 231-248, 2005.
Amit
Sanyal,
Mechanical Engineering, University of Hawaii at Manoa.
Alan Weinstein,
Mathematics, University of California, Berkeley.
Matthew West,
Aeronautics and Astronautics, Stanford University.
Dmitry Zenkov,
Mathematics, North Carolina State University.
Students
Taeyoung Lee,
Ph.D. in Aerospace Engineering (Distinguished Achievement Award,
Ivor K. Mclvor Award, Rackham Predoctoral Fellow, and Rackham
International Student Fellow), University of Michigan,
Ann Arbor. (co-advised with N. Harris McClamroch)
Masako
Kishida, M.S.E. in Aerospace Engineering, M.S. in Applied
and Interdisciplinary Mathematics, University of Michigan, Ann
Arbor. (co-advised with Dennis S. Bernstein)
Melvin Leok is a tenured associate professor of mathematics at the
University of California, San Diego, where his research is supported in
part by grants from the National Science Foundation in applied and computational
mathematics, including a Faculty Early Career Development (CAREER) award.
He serves on the editorial boards of the Journal of Nonlinear Science, and
the Journal of Geometric Mechanics.
Prior to joining UCSD, he was a tenure-track assistant professor of
mathematics at Purdue University, a visiting assistant professor of
control and dynamical systems at the California Institute of Technology,
and a T.H. Hildebrandt research assistant professor of mathematics at the University of Michigan,
Ann Arbor. At Purdue, he was a nominee for the Packard Fellowship for
Science and Engineering, and at Michigan, he received a Horace H. Rackham
Faculty Fellowship and Grant, and a Margaret and Herman Sokol Spring/Summer
Research Grant.
He received his B.S. with honors and M.S. in Mathematics in 2000,
and his Ph.D. in Control and Dynamical Systems with a minor in Applied and
Computational Mathematics under the direction of Jerrold Marsden in 2004,
all from the California Institute of Technology.
His primary research interests are in computational geometric mechanics,
computational geometric control theory, discrete geometry, and
structure-preserving numerical schemes, and particularly how these
subjects relate to systems with symmetry and multiscale systems.
He was the recipient of the SciCADE New Talent Prize in 2007 for his
work on Lie Group and Homogeneous Variational Integrators, and
the SIAM Student Paper Prize, and the Leslie Fox
Prize (second prize) in Numerical Analysis, both in 2003, for his work on
Foundations of Computational Geometric Mechanics. While a
doctoral student at Caltech, he held a Poincaré Fellowship
(2000-2004), a Josephine de Kármán Fellowship (2003-2004),
an International Fellowship from the Agency for Science, Technology, and
Research (2002-2004), a Tau Beta Pi Fellowship (2000-2001), and a Tan
Kah Kee Foundation Postgraduate Scholarship (2000).
As a Caltech
undergraduate, he received the Loke Cheng-Kim Foundation Scholarship
(1996-2000), the Carnation Scholarship (1998-2000), the Herbert J. Ryser
Scholarship (1999), the E.T. Bell Undergraduate Mathematics Research Prize
(1999), and the Jack E. Froehlich Memorial Award (1999).
Visiting Control and Dynamical Systems at Caltech for
the Spring Quarter
I have received a National Science Foundation Faculty Early Career
Development (CAREER) Award in support of my work on
Computational Geometric Mechanics:
Foundations, Computation, and Applications. This is being
funded by the Applied Mathematics Program of the Division
of Mathematical Sciences. [ Purdue University News Service Release | Inside Purdue, Pages 8-9 | Notices of the AMS, Pages 269-270 ]
Research positions for graduate students, and are available in the broad
area of geometric numerical
methods in geometric mechanics and control. A description of my current
research may be found in my research
statement, bibliography, collected publications, or
the abstracts below.
These positions are funded by the following research grants from the National Science Foundation in applied and computational mathematics:
