New additions to the Computational Geometric Mechanics group

Over the next few months, we will be welcoming two new postdoctoral scholars, a visiting scholar, and a visiting graduate student to the Computational Geometric Mechanics group at UCSD.

• Dr. Joris Vankerschaver, Ph.D. in Mathematics, Ghent University, 2007.
  
• Dr. Tomoki Ohsawa, Ph.D. in Applied and Interdisciplinary Mathematics, University of Michigan, 2010.
  
• Dr. Wencheng Li, an assistant professor of computational mathematics at Northwestern Polytechnical University in Xian, China, who will be visiting on a one year fellowship from the Chinese Scholarship Council.
  
• Cuicui Liao, a Ph.D. student in Mathematics at the Harbin Institute of Technology, who will be visiting on a one year scholarship from the Chinese Scholarship Council.
  

Preprint: Discrete Hamiltonian Variational Integrators

[ PDF | arXiv:1001.1408 [math.NA] ]

(with J. Zhang)

We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the symplectic partitioned Runge-Kutta methods. We also characterize the group invariance properties of discrete Hamiltonians which lead to a discrete Noether's theorem.

Preprint: Discrete Hamilton-Jacobi Theory

[ PDF | arXiv:0911.2258 [math.OC] ]

(with A.M. Bloch and T. Ohsawa)

We develop a discrete analogue of the Hamilton-Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton-Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time, and is shown to recover the Hamilton-Jacobi equation in the continuous-time limit. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi's solution to the Hamilton-Jacobi equation. We also prove a discrete analogue of the geometric Hamilton-Jacobi theorem of Abraham and Marsden. These results are readily applied to discrete optimal control setting, and some well-known results in discrete optimal control theory, such as the Bellman equation (discrete-time Hamilton-Jacobi-Bellman equation) of dynamic programming, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case of the discrete Hamilton-Jacobi equation.

Resources for young mathematicians

I have collected below some resources for aspiring and young mathematicians:

Advice to a Young Mathematician

Career advice from Terrence Tao

NSF CAREER Proposal Writing Tips

A particularly insightful comment from Prof. Michael Atiyah is "one could say that all the really creative aspects of mathematical research precede the proof stage."

Prospective and current UCSD Math graduate students should consult the UCSD Mathematics Graduate Handbook, which is full of useful practical advice.

Computational Geometric Mechanics at San Diego

The computational geometric mechanics group has relocated to the University of California, San Diego, where it is housed in the Department of Mathematics within the Division of Physical Sciences, and it is affiliated with the Center for Computational Mathematics, the Program in Computational Science, Mathematics, and Engineering, and the Cymer Center for Control Systems and Dynamics.

It focuses on developing a self-consistent discretization of geometry and mechanics to enable the systematic construction of geometric structure-preserving numerical schemes based on the approach of geometric mechanics, with a view towards obtaining more robust and accurate numerical implementations of feedback and optimal control laws arising from geometric control theory.