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.