NSF CAREER Award DMS-0747659
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.
Postdoctoral and
Graduate Positions Available in the Computational Geometric
Mechanics @ Purdue group
Research positions for postdoctoral scholars
and 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, 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.
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 ]
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 | 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 ]
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 ]
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 ]
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 | 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 ]
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 ]
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 ]
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 ]
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 | 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 | 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 ]
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 | 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 ]
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, accepted, 2007.
[ PDF | arXiv:math.OC/0601424 | Animation: Fig2(a), Fig3(a), Fig4(a), Fig5(a) ]
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 ]
Time Optimal Attitude Control for a Rigid Body (with T. Lee,
and N.H. McClamroch), Proc. American Control Conf., accepted, 2008.
[ PDF | arXiv:0709.2514 ]
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,
submitted, 2007.
[ PDF | arXiv:0712.4400 ]
Global Symplectic Uncertainty Propagation on SO(3) (with T.
Lee and N.H. McClamroch), Proc. IEEE Conf. on
Decision and Control, submitted, 2008.
[ PDF | arXiv:0803.1515 ]
Computational Geometric Optimal Control of Rigid Bodies (with T. Lee, and N.H. McClamroch), Brockett Legacy Special Issue, Communications in Information and Systems, submitted, 2008.
[ PDF | arXiv:0805.0639 ]
Lagrangian Mechanics and Variational Integrators on
Two-Spheres (with T. Lee and N.H. McClamroch), under revision,
2007.
[ PDF | arXiv:0707.0022 ]
Nonlinear Dynamics of the 3D Pendulum (with N.A.
Chaturvedi, T. Lee and N.H.
McClamroch), under revision, 2007.
[ PDF | arXiv:0707.1196 ]
Controlled Lagrangians and Stabilization of Discrete Mechanical
Systems I (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov),
under revision, 2008.
[ PDF | arXiv:0704.3875 ]
Geometric and Symplectic Integration minisymposium, SciCADE,
Saint-Malo, France, Jul 2007.
Effective Computational Methods for Highly Oscillatory Problems,
Newton Institute, University of Cambridge, UK, Jul 2007.
Minicourse, First International Summer School on Geometry, Mechanics,
and Control, Cantabria, Spain, Jun 2007.
Geometric Methods in Dynamical Systems minisymposium, SIAM DS07,
Snowbird, UT, May 2007.
Applied Mathematics Seminar, Imperial College, London, UK, May 2007.
Highly Oscillatory Problems Seminar, Newton Institute, University of
Cambridge, UK, May 2007.
Center for Applied Mathematics Colloquium, University of Notre Dame,
Notre Dame, IN, Apr 2007.
Mathematics Colloquium, University of Iowa, Iowa City, IA, Mar 2007.
CDS/CIMMS Lunchtime Seminar, Caltech, Pasadena, CA, Dec 2006.
Applied Mathematics Colloquium, University of Maryland, Baltimore
County, Nov 2006.
Mathematical Sciences Colloquium, Rensselaer Polytechnic Institute,
Troy, NY, Oct 2006.
Applications of the Geometric Phase in Classical Mechanics
minisymposium, SIAM Annual Meeting, Boston, MA, Jul 2006.
Geometry and Dynamical Systems
with Applications Seminar, Arizona State
University, Tempe, AZ, Apr 2006.
Dynamics Seminar, University of Colorado, Boulder, CO,
Feb 2006.
Mathematics Colloquium, Colorado School of Mines, CO, Feb 2006.
Applied and Interdisciplinary Mathematics Seminar, University of
Michigan, Ann Arbor, MI, Feb 2006.
Mathematics Colloquium, University of South Carolina, Columbia, SC,
Feb 2006.
Mathematics Colloquium, Texas A&M University, College Station, TX,
Feb 2006.
Mathematics Colloquium, Colorado State University, Fort Collins, CO,
Feb 2006.
Mathematics Special Colloquium, Purdue University, West Lafayette, IN,
Feb 2006.
Joint Numerical Analysis and Differential Equations Seminar, North
Carolina State University, NC, Jan
2006.
Contemporary Dynamical Systems special session, AMS National Meeting,
San Antonio, TX, Jan 2006.
Applied Mathematics Seminar, Mathematics, University of Waterloo,
Canada, Dec 2005.
Control Seminar, College of Engineering, University of Michigan, Ann
Arbor, MI, Oct 2005.
Differential Geometry and Analysis Seminar, University of Toledo,
Toledo, OH. Sep 2005.
Computer and Computational Sciences, CCS-2, LANL, Los
Alamos, NM. Aug 2005.
Geometric Dynamics and its Applications minisymposium, SIAM DS05,
Snowbird, UT, May 2005.
Special Seminar, Mathematics, University of California, Berkeley, CA,
Apr 2005.
Mathematical Physics Seminar, University of Minnesota, Twin Cities,
MN, Mar 2005.
Joint Numerical Analysis and Differential Equations Seminar, North
Carolina
State University, NC, Jan
2005.
Mathematics Colloquium, University of California, San Diego, CA, Dec
2004.
Flight Dynamics and Control Seminar, University of Michigan, Ann
Arbor, MI, Oct 2004.
Geometry Seminar (two sessions), University of Michigan, Ann Arbor,
MI, Oct
2004.
Applied Mathematics Seminar, University of California, San Diego, CA,
Aug 2004.
Temasek Laboratories, National University of Singapore, Jul 2004.
Mathematics Seminar (four sessions), National University of Singapore,
Jul 2004.
Geometric Dynamics and Applications minisymposium, AIMS Fifth
International Conference on Dynamical Systems and Differential Equations,
Pomona, CA, June 2004.
Applied Mathematics and Numerical Analysis Seminar, University of
Minnesota, Twin Cities, MN, Jan 2004.
Referee, Aerospace Science and Technology,
Applied Numerical Mathematics, Celestial
Mechanics and Dynamical Astronomy, Communications in
Contemporary Mathematics, Communications in Numerical
Methods in Engineering, Discrete and Continuous Dynamical
Systems, Foundations of Computational Mathematics,
IEEE Transactions on Automatic Control, IMA Journal of
Numerical Analysis, Journal of Computational
Physics, Journal of Mathematical Physics, Journal of
Nonlinear Science, Journal of Physics A,
Nonlinearity, Numerical Algorithms, Physica D,
Physics Letters A,
Proceedings of the Royal Society A, SIAM Applied Dynamical Systems,
SIAM Multiscale Modeling and Simulation, 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, Mathematical Reviews.
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,
Doctoral candidate (Rackham Predoctoral Fellow and Rackham International
Student Fellow),
Aerospace Engineering, University of Michigan,
Ann Arbor. (co-advised with N. Harris McClamroch)
Masako
Kishida, Master's student, Aerospace Engineering, and Applied
and Interdisciplinary Mathematics Program, University of Michigan, Ann
Arbor. (co-advised with Dennis S. Bernstein)
Melvin Leok is a tenure-track assistant professor of mathematics at Purdue
University, West Lafayette, 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.
Prior to joining Purdue, he was a T.H. Hildebrandt
research assistant professor of mathematics at the University of Michigan,
Ann Arbor, where 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).
Research positions for postdoctoral scholars
and 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, or the abstracts below.
These positions are funded by the following research grants from the National Science Foundation in applied and computational mathematics:
