Dmitry Pavlov (CV, Research and Teaching Statements)

 

circa 2008

Graduate Student in Mathematics
dmitry@caltech.edu
Dept of Computer Science, MS 256-80
California Institute of Technology
1200 E. California Boulevard
Pasadena, CA 91125
Tel : (626) 395 8334
Fax : (626) 792 4257

Advisors:  Jerrold E. Marsden
                  Mathieu Desbrun

 

 

Research Focus

My PhD work has been focused on a geometric discretization of Euler equations, to provide differential, yet readily-discretizable foundations to fluid dynamics. In particular, I proposed a finite dimensional Lie group to approximate the diffeomorphism group, and defined a Lagrangian on this Lie group such as the dynamics approximates the Euler equation. Due to the variational nature of the discretization, this approach preserves the basic geometric structures; consequently, one gets an exact discrete version of Kelvin circulation theorem, as well as a wealth of properties mimicking the continuous setting. This method can provide numerical schemes on arbitrary grids with good energy/momenta behavior. Also, our construction can be considered as an approximation to Brenier-type generalized flows and may help to study generalized solutions to the Euler equation and geometry of the diffeomorphism group.

 

 

 

Education

2003-2009:      Ph.D. in Mathematics at the California Institute of Technology.

                        Advisors: Prof. J.E.Marsden and Prof. M. Desbrun.

                        Expected completion: June 2009.

1998-2003:      M.S. in Mathematics, Moscow State University.

                        Department of Mechanics and Mathematics.

                        Diploma with distinction. Advisor: Prof. Yu.S. Ilyashenko

 

 

 

 

 

Publications

• Structure-Preserving Variational Discretization of Incompressible Ideal Fluids, D. Pavlov, P. Mullen, E. Kanso, Y. Tong, J.E. Marsden, M., Desbrun, In progress.

• Variational Integrators for Computational Fluid Dynamics, P. Mullen, D. Pavlov, Y. Tong, E. Kanso, J.E. Marsden, M., Desbrun, In progress.

• Discrete Lie advection of differential k-forms, P. Mullen, A. McKenzie, D. Pavlov, Y. Tong, E. Kanso, J.E. Marsden, M. Desbrun, Submitted (2007).

• Bifurcation of a twisted homoclinic surface of a saddle-node cycle in general case, M.S. Thesis, (2003).

 

 

 

 

Collaborations

My local collaborators at Caltech include:

·   Jerrold E. Marsden (advisor)

·   Mathieu Desbrun (coadvisor)

·   Patrick Mullen (CS grad student)

 

I also have active collaborations with:

Eva Kanso (Assistant Professor in Aerospace and Mechanical Engineering, USC)

Yiying Tong (Assistant Professor in Computer Science, MSU)

 

 

Sponsors

We wish to acknowledge the generous and constant support of:

·  Federal funding: National Science Foundation, Department of Energy.

·  Corporate funding: Pixar Animation Studios, NVidia, Microsoft Research.