Hamilton-Jacobi type (HJ) PDEs arise in optimal control, dynamic
implicit surfaces for fluid animation and simulation, image
processing, and many other fields. There are two broad classes of
equations: time-dependent and stationary.

Level set methods are a group of finite difference algorithms for
dynamic implicit surfaces and the time-dependent class of equations.
I will describe the Toolbox of Level Set Methods, a publicly
available collection of Matlab routines providing high order accurate
finite difference approximations on Cartesian grids in any number of
dimensions (although computational cost and visualization make
dimensions four and higher a challenge). The modular design of the
toolbox makes it easy to try out new level set algorithms, as will be
shown by the simple addition of a collection of explicit RK
integrators and monotone approximations for degenerate second order
spatial terms. I will also demonstrate how the toolbox permits quick
and easy experiments with state of the art level set algorithms, and
some of the extensive set of examples that are included with the
software release. The toolbox and all of its source code is
available from my web site.

While the level set algorithms for time-dependent HJ PDEs evolved
from those used to approximate conservation laws, the algorithms for
stationary HJ PDEs have more of a dynamic programming flavor that
befits their close connection to shortest path problems. I will
describe some algorithms and results for continuous shortest path
problems in which the cost depends on the direction of travel and
problems involving multiple cost metrics.

The focus of this talk is on the optimal control of nonlinear systems subject to mixed state and control constraints, a difficult core problem in the history of control theory and system engineering. Pseudospectral (PS) methods for optimal control will be introduced. Originally developed as a computational method for partial differential equations, PS methods have become an emerging approach in solving optimal control problems with highly nonlinear dynamics and mixed state-control constraints. Following the introduction of PS algorithms, in this talk I will address three fundamental questions, namely the existence of feasible trajectories, the convergence of the optimal solutions, and the rate of convergence in the approximation of the optimal control. The problems will be addressed for both continuous and discontinuous optimal controls. In addition, illustrative examples of optimal control using the Legendre PS method will also be presented.

Seismic imaging is a technology for creating images of the Earth's
interior by using recordings of reflected seismic waves. While
depth-domain imaging attempts to produce images in true depth
coordinates, time-domain imaging takes a shortcut by using distorted
coordinates and more robust and efficient numerical algorithms.
Mathematically, this involves approximations of the wave equation
Green's functions that appear in seismic imaging operators. In this
presentation, I will describe the latest developments in time-domain
imaging algorithms and in transforming time-domain images to depth.

Multiscale and stochastic approaches play a crucial role in faithfully
capturing the dynamical features and making insightful predictions of
cellular reacting systems involving gene expression. A Genetic
Regulatory Networks (GRN), describing all the reacting channels and
species involved in gene expression, consists of a set of genes,
proteins, small molecules and their mutual regulatory
interactions. From the point of view of modeling, Genetic Regulatory
Networks, unlike metabolism networks, involve fewer
species and lower concentrations of molecules in a small volume within a
cell; therefore stochastic effects have a significant
impact on the system. Despite their accuracy, the standard stochastic
simulation algorithms are necessarily inefficient for most of the
realistic problems with a multiscale nature characterized by 1.) Rare
events arising from the metastability
of the system, 2.) Multiple time scales induced by widely disparate
reactions rates, and 3.) Multiple well separated concentration scales of
the reacting species. In this talk, I will discuss some recent progress
on using asymptotic techniques for probability theory, e.g. Random
Homogenization and Large Deviation Theory, to simplify the complex
networks and help to design efficient numerical schemes.