Variational Integrators

In my early days as an undergraduate student, I had a lot of fun writing C code to integrate ODE’s numerically, so I could simulate physical systems in a computer (integrating the pendulum ODE is a classic). However, I was not very happy to realize that traditional numerical methods very easily give results with no physical meaning.

For example, think of a simple pendulum oscillating on a vertical plane. Assuming that there are no damping forces, there’s energy conservation. However, traditional numerical methods (such as Runge-Kutta methods) do not, in general, preserve energy. The numerical simulation of the pendulum may thus exhibit non-physical energy dissipation (or amplification).

[ image courtesy of Ari Stern ]

When we integrate an ODE numerically, we obtain a discrete trajectory which we hope is “close enough” to the exact flow of the differential equation. In some problems in Physics, the continuous flow lies on a manifold, and we would like to preserve this geometric property: it would be wonderful if the discrete trajectory lied on the same manifold as the continuous flow. While thinking about numerical methods that preserve certain invariants, I became interested in the field of geometric integration.

One particular class of geometric integrators I have been interested in over the past year is the class of variational integrators. Many physical laws are formulated as variational principles from which differential equations can be derived. Instead of discretizing the differential equations, why not discretize the variational principles instead? This rather sexy thought lies at the basis of the nascent field of variational integrators.

Here’s an overview on variational integrators, by Prof. Matthew West. I have been reading some papers on the topic. This field seems to have been explored rather vigorously at Caltech, as there are/were a lot of people working on it:

• Prof. Jerrold Marsden

• Prof. Mathieu Desbrun

• Prof. Houman Owhadi

• Prof. Matthew West (now at Stanford)

• Prof. Adrian Lew (now at Stanford)

• Prof. Melvin Leok (now at Purdue)

• Prof. Eva Kanso (now at USC)

• Ari Stern

• Liliya Kharevych

• Nawaf Bou-Rabee

and their collaborators. I have compiled a list of lecture slides, theses and papers on the area of variational integrators. Hope you find the list useful (I sure do!).

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Some lecture slides for a gentle introduction:

• Discrete Mechanics, Variational Integrators, and Optimization (by Jerrold Marsden)

• Discrete Geometric Mechanics and Variational Integrators (by Ari Stern)

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A few PhD theses on variational integrators and their application:

• Variational integrators – by Matthew West (Caltech 2003)

• Variational time integrators in computational solid mechanics – by Adrian Lew (Caltech 2003)

• Foundations of computational geometric mechanics – by Melvin Leok (Caltech 2004)

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Some papers on the foundations of variational integrators:

• An overview of variational integrators (by A. Lew, J. E. Marsden, M. Ortiz and M. West)

• Variational time integrators (by A. Lew, J. E. Marsden, M. Ortiz and M. West)

• Asynchronous variational integrators (by A. Lew, J. E. Marsden, M. Ortiz and M. West)

• Generalized Galerkin Variational Integrators (by Melvin Leok)

• Multisymplectic geometry, variational integrators, and nonlinear PDEs (Jerrold E. Marsden, George W. Patrick, Steve Shkoller)

Applications to Computational Mechanics:

• Discrete mechanics and variational integrators (by J. E. Marsden and M. West)

• Discrete Geometric Mechanics for Variational Integrators (by Ari Stern, and Mathieu Desbrun) -> a rather nice introduction!

• Geometric, Variational Integrators for Computer Animation (by L. Kharevych, Weiwei, Y. Tong, E. Kanso, J. E. Marsden, P. Schröder, and Mathieu Desbrun)

• Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems (by C. Kane, J. E. Marsden, M. Ortiz and M. West)

• Discrete variational Hamiltonian mechanics (by Sanjay Lall, Matthew West)

• Lagrangian Mechanics and Variational Integrators on Two-Spheres (by Taeyoung Lee, Melvin Leok, N. Harris McClamroch)

• Polyhedral Potential and Variational Integrator Computation of the Full Two Body Problem (by Eugene G. Fahnestock, Taeyoung Lee, Melvin Leok, N. Harris McClamroch, Daniel J. Scheeres)

• Variational integrators and time-dependent lagrangian systems (by M. de Leon, D. Martin de Diego)

Applications to Optimal Control Theory:

• Discrete Control Systems (by Taeyoung Lee, Melvin Leok, N. Harris McClamroch)

• Discrete mechanics and optimal control (by Junge, O., J. E Marsden, and S. Ober-Blöbaum)

• Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics (by Leyendecker S., S. Ober-Blöbaum, J.E. Marsden, and M. Ortiz)

• A Discrete Geometric Optimal Control Framework for Systems with Symmetries (by Marin Kobilarov, Mathieu Desbrun, Jerrold E. Marsden, and Gaurav S. Sukhatme)

• Discrete Mechanics and Optimal Control Applied to the Compass Gait Biped (by Pekarek D., A. D. Ames, and J. E. Marsden)

• A Discrete Variational Integrator for Optimal Control Problems on SO(3) (by Islam I. Hussein, Melvin Leok, Amit K. Sanyal, Anthony M. Bloch)

• Discrete variational integrators and optimal control theory (by M. de Leon, D. Martin de Diego, A. Santamaria-Merino)

Applications to Computational Electromagnetism:

• Variational Integrators for Maxwell’s Equations with Sources (by Ari Stern, Yiying Tong, Mathieu Desbrun, Jerrold E. Marsden)

• Computational Electromagnetism with Variational Integrators and Discrete Differential Forms (by Ari Stern, Yiying Tong, Mathieu Desbrun, Jerrold E. Marsden)

Applications to Computational Physics:

• Some applications of semi-discrete variational integrators to classical field theories (by Manuel de Leon, Juan Carlos Marrero, David Martin de Diego)

• Variational Integrators for the Gravitational N-Body Problem (by Will M. Farr, Edmund Bertschinger)

The papers I have listed so far refer to deterministic integrators. However, quite recently Prof. Houman Owhadi and Nawaf Bou-Rabee have been working on stochastic variational integrators:

• Stochastic Variational Integrators (by Nawaf Bou-Rabee, Houman Owhadi)

• Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems (by Nawaf Bou-Rabee, Houman Owhadi)

I wonder if stochastic variational integrators could be used in quantitative finance. Just a wild thought…

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If you know some more papers, please let me know! Last but not least, Ganesh Swami once wrote a brief post on variational integrators.

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Tags: Ari Stern, Calculus of Variations, Computational Electromagnetism, Computational Mechanics, Computational Physics, Discrete Calculus of Variations, Discrete Lagrangian Mechanics, Discrete Mechanics, Geometric Integrators, Jerrold Marsden, Mathieu Desbrun, Matthew West, Melvin Leok, Numerical Methods, PhD theses, Stochastic Variational Integrators, Variational Integrators