Review:"This well-written and expansive book is
ambitious in its scope in that it aims at sound and thorough pedagogy as
far as its subject matter is concerned, and it also aims at preparing the
reader for computational work: note the subtitle, viz., “A geometric
approach to modeling and analysis.” In that sense it is a particularly
timely book, seeing that we have computing power at our disposal like
never before. There are many good examples accompanying or even guiding
the text, as well as extensive problem sets for the properly serious
student. This book hits its targets square and should prove very valuable
to its readership, be they mathematicians, engineers, or physicists." Michael Berg, MAA
Reviews, November 2017.
Review:"The starting point of this
impressive textbook is the important fact that there are remarkable
situations where the variables that describe a dynamical system do
not lie in a vector space (i.e., a simple flat algebraic structure)
but rather lie in a geometrical setting allowing the differential
calculus, namely a differential manifold. ... In conclusion, this
book is extremely useful for each reader who wishes to develop a
modern knowledge of analytical mechanics." Mircea
Crâşmăreanu,
zbMATH 1381.70005,
2018.
Review:"This book presents a monograph on
foundational geometric principles of Lagrangian and Hamiltonian dynamics
and their application in studying important physical systems. ... The
emphasis in this book is on global descriptions of Lagrangian and Hamiltonian dynamics, where suitable
mathematical tools are available, via global analysis of dynamical
properties. This treatment is novel and unique and it is the most
important distinction and contribution of this text to the existing
literature. Throughout the book numerous examples of Lagrangian and
Hamiltonian systems are included, which are especially useful to
illustrate the concepts and the way in which the developed theory can be
applied in practical situations. ... The book under review succeeds in all
its objectives and sets the stage for a treatment of computational issues
associated with Lagrangian and Hamiltonian dynamics that evolve on a
configuration manifold. It is very clearly written and it will be
especially useful both for beginning researchers and for graduate students
in applied mathematics, physics, or engineering.
M. Eugenia Rosado María, AMS
Mathematical Reviews, 2018.
Accelerated
Merit Advancement to Professor, Step III
I have received a three-year accelerated merit advancement from Professor,
Step I, to Professor, Step III, which will take effect July 1, 2016.
Accelerations are typically awarded on the basis of unusual achievement
and exceptional promise of continued growth. During the three merit
reviews I have undergone since arriving at UCSD, I have received a
total of two accelerations, and one half-step bonus.
Kavli Fellow, National Academy of
Sciences I have been invited in participate in the German-American Kavli
Frontiers
of Science symposium, which is jointly organized by the Alexander von
Humboldt Foundation and the National Academy of Sciences.
The symposium bring together outstanding young scientists
to discuss exciting advances and opportunities in a broad range of
disciplines. A committee of NAS members selects the participants from
among young researchers who have already made recognized contributions to
science, including recipients of major fellowships and awards. [ Poster ]
In Memory of Jerrold E. Marsden (1942-2010)
On September 21, 2010, my dear advisor, mentor, role model, collaborator,
colleague, and friend, Jerry
Marsden, lost his battle with cancer. He will
be missed and remembered always.
[ ICIAM Memorial Events | SIAM News Obituary ]
Research Vignette on Streaming Video
Prior to joining UCSD, I was an assistant professor at Purdue
University, and the department of mathematics there commissioned a short
video vignette where I describe my research in a broadly accessible
fashion. This is available as a streaming video.
Academic
and Industrial, Science and Engineering Collaborations and Consulting
Opportunities Welcomed
I am constantly on the lookout for interesting and challenging
collaborations with scientists and engineers from academia or industry. I
am also available as a consultant for modeling, simulating, controlling,
and quantifying uncertainty for systems that evolve on complicated nonlinear
configuration spaces. Have a look at my YouTube channel for
examples
of applications, and if you would like to learn more about the role that
geometric numerical methods might play in your application, 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."
Graduate Positions Available in the Computational Geometric
Mechanics group
Research positions for graduate students 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 and engineering:
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.
Hosting of Fedor Lynen Research
Fellows of the Alexander von Humboldt Foundation
As an alumni of the German-American Frontiers of Science symposium, I am
able to host research stays for Feodor
Lynen Research Fellows
of the
Alexander von Humboldt
Foundation. This supports post-doctoral researchers
(up to four years after completing a doctorate) for 6 to 24 months, and
experienced researchers (up to 12 years after completing a doctorate) for
6 to 18 months, divisible into a maximum of three visits over three years.
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, 2002-2019, 2009.
[ PDF | arXiv:0705.3868 | Journal Link | 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,
15(3), 307-330, 2009.
[ PDF | arXiv:0712.4400 | Journal Link | 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, 3(1), 19-36,
2010.
[ PDF | arXiv:0704.3875 | Journal Link | BibTeX ]
Nonlinear Dynamics of the 3D Pendulum (with N.A.
Chaturvedi, T. Lee, and N.H.
McClamroch), Journal of Nonlinear Science, 21(1), 3-32, 2011.
[ PDF | arXiv:0707.1196 | Journal Link | BibTeX ]
Variational and Geometric Structures of Discrete Dirac
Mechanics (with T. Ohsawa), Foundations of
Computational Mathematics, 11(5), 529-562, 2011.
[ PDF | arXiv:0810.0740 | Journal Link | BibTeX ]
Dynamics of a 3D Elastic String Pendulum (with T. Lee and
N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3347-3352,
2009.
[ PDF | arXiv:0903.0332 | IEEE Xplore | BibTeX | Animation: Fig2 ]
Computational Dynamics of a 3D Elastic String Pendulum
Attached to a Rigid Body and an Inertially Fixed Reel Mechanism (with
T. Lee,
N.H. McClamroch), Nonlinear Dynamics, 64(1-2), 97-115, 2011.
[ PDF | arXiv:0909.2083 | Journal Link | BibTeX | Animation: Fig2(a), Fig3(a), Fig4(a) ]
Computational Geometric Optimal Control of Connected Rigid Bodies
in a Perfect Fluid
(with T. Lee, N.H. McClamroch), Proc. American Control Conf., 5985-5990,
2010.
[ PDF | arXiv:0909.3871 | BibTeX | Animation: Fig2(a), Fig3(a) ]
Discrete Hamilton-Jacobi Theory (with A.M. Bloch, T. Ohsawa),
SIAM Journal on Control and Optimization, 49(4), 1829-1856, 2011.
[ PDF | arXiv:0911.2258 | Journal Link | BibTeX ]
Discrete Hamiltonian Variational Integrators
(with J. Zhang),
IMA Journal of Numerical Analysis, 31(4), 1497-1532,
2011.
[ PDF | arXiv:1001.1408 | Journal Link | BibTeX ]
Geometric Tracking Control of a Quadrotor UAV on SE(3) (with
T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control,
5420-5425, 2010.
[ PDF | arXiv:1003.2005 | IEEE Xplore | BibTeX ]
Discrete Hamilton-Jacobi Theory and Discrete Optimal Control
(with T. Ohsawa, A.M. Bloch),
Proc. IEEE Conf. on Decision and Control, 5438-5443, 2010.
[ PDF | IEEE Xplore | BibTeX ]
Stokes-Dirac Structures through Reduction of
Infinite-Dimensional Dirac Structures
(with J. Vankerschaver, H. Yoshimura, J.E. Marsden),
Proc. IEEE Conf. on Decision and Control, 6265-6270, 2010.
[ PDF | arXiv:1010.2547 | IEEE Xplore | BibTeX ]
Discrete Dirac Structures and Implicit Discrete Lagrangian
and Hamiltonian Systems
(with T. Ohsawa),
XVIII International Fall Workshop on
Geometry and Physics, 91-102, AIP Conference Proceedings 1260, 2010.
[ PDF | Journal Link | BibTeX ]
Geometric Numerical Integration of Complex Dynamics of
Tethered Spacecraft
(with T. Lee, N. H. McClamroch),
Proc. American Control Conf., 1885-1891, 2011.
[ PDF | arXiv:1010.1724 | BibTeX ]
Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3)
(with T. Lee, N. H. McClamroch),
Asian Journal of Control, 15(3), 1-18, 2013,
[ PDF | Journal Link | BibTeX ]
Prolongation-Collocation Variational Integrators
(with T. Shingel),
IMA Journal of Numerical Analysis, 32(3), 1194-1216,
2012.
[ PDF | arXiv:1101.1995 | Journal Link | BibTeX ]
On the Geometry of Multi-Dirac Structures and Gerstenhaber
Algebras
(with J. Vankerschaver, H. Yoshimura),
Journal of Geometry and Physics, 61(8), 1415-1425, 2011.
2011.
[ PDF | arXiv:1102.2835 | Journal Link | BibTeX ]
General Techniques for Constructing Variational Integrators
(with T. Shingel),
Frontiers of Mathematics in China (Special issue on computational
mathematics, invited paper), 7(2), 273-303, 2012.
[ PDF | arXiv:1102.2685 | Journal Link | BibTeX ]
Stable Manifolds of Saddle Points for Pendulum Dynamics on
S^2 and SO(3) (with
T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control,
3915-3921, 2011.
[ PDF | arXiv:1103.2822 | BibTeX ]
Variational Integrators, invited article for Springer
Encyclopedia of Applied and Computational Mathematics, 1519-1525,
2015.
[ PDF | BibTeX ]
The Hamilton-Pontryagin Principle and Multi-Dirac Structures
for Classical Field Theories
(with J. Vankerschaver, H. Yoshimura),
Journal of Mathematical Physics, 53(7), 072903 (25 pages), 2012.
[ PDF | arXiv:1207.2814 | Journal Link | BibTeX ]
Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3)
(with
T. Lee, N.H. McClamroch), Proc. American Control Conf.,
4649-4654, 2012.
[ PDF | arXiv:1109.4457 | BibTeX ]
Hamilton--Jacobi Theory for Degenerate Lagrangian Systems
with Holonomic and Nonholonomic Constraints
(with T. Ohsawa, D. Sosa),
Journal of Mathematical Physics, 53(7), 072905 (29 pages), 2012.
[ PDF | arXiv:1109.6056 | Journal Link | BibTeX ]
Generating Functionals and Lagrangian PDEs
(with C. Liao, J. Vankerschaver),
Journal of Mathematical Physics, 54(8), 082901 (22 pages), 2013.
[ PDF | arXiv:1111.0280 | Journal Link | BibTeX ]
Dynamics and Control of a Chain Pendulum on a Cart
(with T. Lee and N. H. McClamroch),
Proc. IEEE Conf. on Decision and Control, 2502-2508, 2012.
[ PDF | arXiv:1211.4604 | BibTeX ]
Hamel's Formalism and Variational Integrators on a Sphere
(with A.M. Bloch, and D.V. Zenkov),
Proc. IEEE Conf. on Decision and Control, 7504-7510, 2012.
[ PDF | arXiv:1211.4607 | BibTeX ]
High-Fidelity Numerical Simulation of Complex Dynamics of
Tethered Spacecraft (with T. Lee, and N.H. McClamroch), Acta
Astronautica, 99, 215-230, 2014.
[ PDF | arXiv | Journal Link | BibTeX ]
Dirac Structures and Hamilton-Jacobi Theory for Lagrangian
Mechanics on Lie Algebroids (with D. Sosa),
Journal of Geometric Mechanics, 4(4), 421-442, 2012.
[ PDF | arXiv:1211.4561 | BibTeX ]
A novel formulation of point vortex dynamics on the sphere:
geometrical and numerical aspects
(with J. Vankerschaver), Journal of Nonlinear Science, 24(1), 1-37,
2014.
[ PDF | arXiv:1211.4560 | Journal Link | BibTeX ]
Spectral Variational
Integrators (with J. Hall), Numerische Mathematik, 130 (4),
681-740, 2015.
[ PDF | arXiv:1211.4534 | Journal Link | BibTeX ]
Symplectic Semiclassical Wave Packet Dynamics (with T.
Ohsawa), Journal of Physica A, 46(40), 405201 (28 pages), 2013.
[ PDF | arXiv:1302.1139 | Journal Link | BibTeX ]
Coordinate-free Formulation for Dynamics and Control of a
Chain Pendulum on a Cart (with T. Lee, and N.H. McClamroch),
under revision, 2013.
[ PDF | BibTeX ]
Lie Group Spectral Variational Integrators (with J. Hall),
Foundations of Computational Mathematics, 17(1), 199-257, 2017.
[ PDF | arXiv:1402.3327 | Journal Link | BibTeX ]
Geometric Representations of Whitney Forms and their
Generalization to Minkowski Spacetime (with J. Salamon, and J.
Moody), under revision,
2014.
[ PDF | arXiv:1402.7109 | BibTeX ]
Space-Time Finite-Element Exterior Calculus and Variational
Discretizations of Gauge Field Theories
(with J. Salamon, and J. Moody), Proc. Mathematical Theory of
Networks and Systems, 743-747, 2014.
[ PDF | BibTeX ]
A Novel Variational Formulation for Thermoelastic Problems
(with Z. Ebrahimzadeh, and M. Mahzoon),
Communications in Nonlinear Science and Numerical Simulation, 22
(1-3), 263-268, 2015.
[ PDF | Journal Link | BibTeX ]
Global Formulations of Lagrangian and Hamiltonian Mechanics
on Two-Spheres (with T. Lee, and N.H.
McClamroch), Proc. IEEE Conf. Decision and Control, 6010-6015, 2015.
[ PDF | arXiv:1503.02620 | IEEE Xplore | BibTeX ]
Global Formulations of Lagrangian and Hamiltonian Dynamics
on Embedded Manifolds (with T. Lee, and N.H.
McClamroch), Proc. IMA Conf. on Mathematics of Robotics, 2015.
[ PDF | BibTeX ]
Geometric formulations of Furuta pendulum control problems
(with T. Lee, and N.H. McClamroch), Mathematics in Engineering, Science
and Aerospace, 7(1), 69-81, 2016.
[ PDF | Journal Link | BibTeX ]
Spectral-Collocation Variational Integrators (with
Y. Li, and B. Wu),
Journal of Computational Physics, 332, 83-98, 2017.
[ PDF | Journal Link | BibTeX ]
Spectral variational integrators for semi-discrete
Hamiltonian wave equations (with
Y. Li, and B. Wu),
Journal of Computational and Applied Mathematics, 325, 56-73,
2017.
[ PDF | Journal Link | BibTeX ]
Lagrangian and Hamiltonian Taylor Variational Integrators
(with J.M. Schmitt, and T. Shingel), BIT Numerical Mathematics, 58(2),
457-488, 2018.
[ PDF | arXiv:1703.06599 | Journal Link | BibTeX ]
Variational integrators for interconnected Lagrange-Dirac
systems (with H. Parks),
Journal of Nonlinear Science, 27(5), 1399-1434, 2017.
[ PDF | arXiv:1603.01554 | Journal Link | BibTeX ]
Interpolation on Symmetric Spaces via the Generalized Polar
Decomposition (with E.S. Gawlik),
Foundations of Computational Mathematics, 18(3), 757-788, 2018.
[ PDF | arXiv:1605.06666 | Journal Link | BibTeX ]
Iterative Computation of the Frechet Derivative of the Polar
Decomposition
(with E.S. Gawlik),
SIAM Journal on Matrix Analysis and Applications, 38(4), 1354-1379,
2017.
[ PDF | arXiv:1608.04491 | Journal Link | BibTeX ]
Embedding-Based Interpolation on the Special Orthogonal Group
(with E.S. Gawlik), SIAM Journal on Scientific Computing, 40(2),
A721-A746, 2018.
[ PDF | arXiv:1608.05738 | Journal Link | BibTeX ]
Properties of Hamiltonian Variational Integrators
(with J.M. Schmitt), IMA Journal of Numerical Analysis, 38, 377-398,
2018.
[ PDF | Journal Link | BibTeX ]
Geometric Exponential Integrators (with X. Shen),
Journal of Computational Physics, submitted, 2018.
[ PDF | arXiv:1703.00929 | BibTeX ]
Constructing Equivalence-Preserving Dirac Variational
Integrators with Forces (with H. Parks),
IMA Journal of Numerical Analysis, submitted, 2017.
[ PDF | arXiv:1703.03045 | BibTeX ]
Lie Group Variational Integrators for Rigid Body Dynamics
using Quaternions (with X. Shen), Journal of Computational and
Applied Mathematics, submitted, 2017.
[ PDF | arXiv:1705.04404 | BibTeX ]
High-Order Retractions on Matrix Manifolds using Projected
Polynomials
(with E.S. Gawlik),
SIAM Journal on Matrix Analysis and Applications, 39(2), 801-828,
2018.
[ PDF | arXiv:1705.05554 | Journal Link | BibTeX ]
Adaptive Variational Integrators
(with J. Schmitt), Applied Numerical Mathematics, submitted, 2018.
[ PDF | arXiv:1709.01975 | BibTeX ]
Connecting Information Geometry and Geometric
Mechanics (with J. Zhang), Entropy (Special Issue on Information Geometry II), 19(10), 518 (31 pages), 2017.
[ PDF | Journal Link | BibTeX ]
Construction and comparison of multidimensional spectral
variational integrators and spectral collocation methods
(with Y. Li, and B. Wu), Applied Numerical Mathematics, accepted,
2018.
[ PDF | Journal Link | BibTeX ]
An Empirical Chaos Expansion Method for Uncertainty
Quantification
(with G. Wilkins), under revision, 2018.
[ PDF | arXiv:1709.08668 | BibTeX ]
Variational Discretizations of Gauge Field Theories using
Group-equivariant Interpolation, Foundations of Computational
Mathematics, submitted, 2018.
[ PDF | BibTeX ]
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 ]
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, Mathematical Sciences, National Science Foundation,
October 2014.
Panel member, Mathematical Sciences, National Science Foundation, Oct
2013.
Panel member, Mathematical Sciences, National Science Foundation, Nov
2011.
Panel member, Mathematical Sciences Graduate Research Fellowship,
National Science Foundation, Feb 2010.
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. in Applied and Interdisciplinary Mathematics, University of Michigan, Ann
Arbor.
Biographical Sketch
Melvin Leok is a tenured 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, the
SIAM Journal on Control and Optimization, the LMS Journal of Computation
and Mathematics, the Journal of Geometric Mechanics, and the Journal of
Computational Dynamics.
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).