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MA 273A Advanced Techniques in Computational Mathematics
- Geometric Numerical Integration
TTh 2:00pm-3:20pm, APM 2402.
Final Presentation Schedule
The final presentations will be held on Thursday, December 8, between
3:00pm and 6:00pm, in APM 2402. The schedule is as follows:
||Geometric Integration Techniques in Molecular and Spin Dynamic
||Finite Element Exterior Calculus for Lagrangian Field Theories
||Symplectic-Momentum and Energy-Momentum methods applied to the
||Symplectic splitting methods for molecular dynamics and backward error
||Investigation of the XY model for persistent homology analysis of
||Variational integrators applied to the double spherical pendulum
||Symplectic Integrators for Optimal Control Theory
||Shooting-based variational integrators and spectral
||Shortest Paths via Geodesic Flow
- This course is relevant to engineers, scientists, and mathematicians with an interest in long-time simulations of mechanical systems, including applications to robotic motion planning, astrodynamics, rigid-body, molecular and stellar dynamics. The application areas addressed will be tailored to the interests of the course participants.
- Many differential equations of interest in the physical sciences and engineering exhibit geometric properties that are preserved by the dynamics. Recently, there has been a trend towards the construction of numerical schemes that preserve as many of these geometric invariants as possible.
- Such methods are of particular interest when simulating mechanical systems that arise from Lagrangian or Hamiltonian mechanics, wherein the preservation of physical invariants such as the energy, momentum, and symplectic form can be important when simulating long-time dynamics of such systems.
- In applications arising from astrodynamics and robotics, the dynamics evolve on nonlinear manifolds such as Lie groups, and in particular the rotation group, and the special Euclidean group. Numerical schemes that respect the underlying nonlinear manifold structure will also be discussed.
- This course will begin with an overview of classical numerical integration schemes, and their analysis, followed by a more in depth discussion of the various geometric properties that are of importance in many practical applications, followed by a survey of the various geometric integration schemes that have been developed in recent years. Issues pertaining to the analysis and implementation of such schemes will also be addressed.
- A strong undergraduate background in linear algebra, and differential equations; Some familiarity with numerical methods, classical mechanics and differential geometry would be helpful, but not essential; Programming experience in any language, e.g., C/C++, FORTRAN, MATLAB.
- Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics
Leimkuhler, Reich, Cambridge University Press, 2005. ISBN: 0521772907
[ Electronic Version ]
This book is currently unavailable from the publisher, but an electronic copy will be made available to students in this course.
You should consider these additional references as a potential source of advanced topics in geometric numerical integration, which would be suitable for further study, as part of your course project.
- Geometric Numerical Integration, Springer Series in Computational Mathematics
Hairer, Lubich, Wanner, 2nd Edition, Springer-Verlag, 2006. ISBN: 3540306633
[ Electronic Version ]
- Geometric numerical
integration illustrated by the Störmer-Verlet method, Hairer,
Lubich, Wanner, Acta Numerica, 399-450, 2003.
- Discrete mechanics and variational
integrators, Marsden, West, Acta Numerica, 357-514, 2001.
- Asynchronous Variational
Integrators, Lew, Marsden, Ortiz, West, Arch. Rational Mech. Anal.,
167(2), 85-146, 2003.
- Splitting methods,
McLachlan, Quispel, Acta Numerica, 341-434, 2002.
- Linear algebra algorithms as
dynamical systems, Chu, Acta Numerica, 1-86, 2008.
- Lie-group methods, Iserles,
Munthe-Kaas, Norsett, Zanna, Acta Numerica, 1-148, 2005.
Applications, Marsden, Ratiu, Third Edition, Electronic Version.
- Introduction to Mechanics
and Symmetry, Marsden, Ratiu, Second Edition, Springer-Verlag, 2002.
9 (An Introduction to Lie Groups), Mechanics and Symmetry, Second
- Your grade in the course is based on your project and your 20-minute
project presentation during the last week of classes.
- The topic of your project should be decided in consultation with the
instructor before the end of the first month of class.