Alexander Vladimirsky

Assistant Professor of Mathematics

Web Site www.math.cornell.edu/~vlad/ Contact Information Office:  430 Malott Hall Phone:  (607) 255-9871 Fax:  (607) 255-7149 Email:  vlad@math.cornell.edu Courses & Office Hours

Education

Ph.D. (2001) University of California at Berkeley

Research Area: Numerical methods, dynamical systems, nonlinear PDEs, control theory

My research is mostly focused on building fast methods for problems in which the direction of information flow can be used to speed up the computations.

For example, numerical schemes for non-linear static PDEs often require solving coupled systems of non-linear discretized equations. For the first-order PDEs, partial knowledge of characteristic directions can be used to de-couple those systems: solving the discretized equations one at a time is much more efficient. My thesis was devoted to construction of Ordered Upwind Methods (OUMs) for the PDEs arising in the anisotropic exit-time optimal trajectory problems. These methods were later extended to a wider class of problems in anisotropic (and hybrid) control and in front propagation.

My current work (joint with John Guckenheimer) concerns fast methods for approximating invariant manifolds of vector fields. This problem is numerically challenging not only because of the complicated manifold-geometry but also because of the anisotropic behavior of the vector field on that manifold. In our approach, a (co-dimension one) invariant manifold is locally modeled as a graph of some function satisfying a particular quasi-linear PDE, which can be quickly solved using yet another version of OUMs. A more recent extension allows treating manifolds of higher co-dimension by (locally) solving a system of quasi-linear PDEs.

My other projects include games & stochastic control problems on graphs, fast methods for constructing multi-valued solutions of PDEs, and dimension reduction in the context of chemical kinetics.

Selected Publications

Fast methods for the Eikonal and related Hamilton-Jacobi equations on unstructured meshes (with J. A. Sethian), Proc. Natl. Acad. Sci. USA 97 no. 11 (2000), 5699–5703.

Ordered upwind methods for static Hamilton-Jacobi equations: theory & algorithms (with J. A. Sethian), SIAM Journal on Numerical Analysis 41 no. 1 (2003), 325–363.

A fast method for approximating invariant manifolds (with J. Guckenheimer), SIAM J. on Applied Dynamical Systems 3 no. 3 (2004) 232–260.

A survey of methods for computing (un)stable manifolds of vector fields (with M. Dellnitz, E. J. Doedel, J. Guckenheimer, M. E. Henderson, O. Junge, B. Krauskopf, and H. M. Osinga), Int. J. Bifurcation and Chaos 15 no. 3 (2005), 763–791.

Static PDEs for time-dependent control problems, Interfaces and Free Boundaries 8 no. 3 (2006), 281–300.

Label-setting methods for multimode stochastic shortest path problems on graphs, Mathematics of Operations Research (to appear).

Last modified: October 31, 2006