Numerical methods, dynamical systems, nonlinear PDEs, control theory
I study the effects of anisotropy and inhomogeneity on analytic properties of differential equations and the computational efficiency of numerical methods. I am interested in a variety of discrete and continuous nonlinear problems that have some causal properties defining the direction of “information flow.” Much of my work also exploits the often underutilized connections between computer science/operations research and “continuous” numerical analysis.
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 joint work with J.A. Sethian was devoted to construction of Ordered Upwind Methods (OUMs) for the PDEs arising in anisotropic (and hybrid) optimal control and in front propagation.
My joint work with John Guckenheimer introduced a new fast method 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, an 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.
My other projects include games and stochastic control problems on graphs, numerical homogenization, multi-objective and randomly-terminated optimal control, approximation of invariant manifolds of delay-differential equations, fast methods for constructing multi-valued solutions of PDEs, and dimension reduction in the context of chemical kinetics.
- Optimal control with budget constraints and resets (with R. Takei, W. Chen, Z. Clawson, and S. Kirov), SIAM J. on Control and Optimization 53 no.2 (2015), 712–744.
- Fast two-scale methods for Eikonal equations (with A. Chacon), SIAM J.on Scientific Computing 34 no.2 (2012), A547-A578.
- An efficient method for multiobjective optimal control and optimal control subject to integral constraints (with A. Kumar), Journal of Computational Mathematics 28 no. 4 (2010), 517–551.
- Homogenization of metric Hamilton-Jacobi equations (with A. M. Oberman and R. Takei), Multiscale Modeling and Simulation 8 no. 1 (2009), 269–295.
- Label-setting methods for multimode stochastic shortest path problems on graphs, Mathematics of Operations Research 33 no. 4 (2008), 821–838.
- A fast method for approximating invariant manifolds (with J. Guckenheimer), SIAM J. on Applied Dynamical Systems 3 no. 3 (2004), 232–260.
- 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.
- 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.