Classical potential theory converts linear constant-coefficient elliptic
problems in complex domains into integral equations on interfaces, and
generates robust, efficient numerical methods. The conversion is
usually carried out for a particular situation such as the Poisson
equation in dimension 2, and the efficiency of the resulting methods
then depends on detailed analysis of the appropriate special functions.

We present a general conversion scheme which leads naturally to a fast
general algorithm: arbitrary elliptic problems in arbitrary dimension
are converted to first-order systems, a periodic fundamental solution is
mollified for convergence, and the mollification is locally corrected
via Ewald summation. Local linear algebra and the elementary theory of
distributions yield a simple boundary integral equation. With the aid
of a new nonequidistant fast Fourier transform for piecewise polynomial
functions, the resulting numerical methods provide highly accurate
solutions to general elliptic systems in complex domains.

We will discuss several projects with the general goal of
designing second-order accurate numerical methods for
the motion of a viscous fluid with a moving interface of
zero thickness which exerts a force in response to its
stretching. The interfacial force results in jumps in
the fluid quantities at the interface. In recent work with
Anita Layton we have found that the problem of the Navier-Stokes
equations with an elastic interface can be simplified by decomposing
the velocity at each time into a ``Stokes'' part, determined
by the (equilibrium) Stokes equations, with the interfacial
force, and a ''regular'' remainder which can be calculated
on a rectangular grid without special treatment at the interface.
For the Stokes part we use the immersed interface method; for the
regular part we use the semi-Lagrangian method. Smaller time
steps can be used to advance the interface with Stokes flow,
using boundary integrals, if needed, to handle the boundary force.
This decomposition exhibits second-order accuracy in
simple test problems. Analytical issues of accuracy and some related
error estimates for the immersed interface method will be
described. We allow for the possibility of more general
boundary motion in work with John Strain for the case of Stokes flow,
in which we use Strain's semi-Lagrangian contouring method
to move the interface. We represent the velocity, on or off
the interface, as a singular integral, and calculate it using
Ewald splitting. The smooth or regularized part is computed as a
Fourier series, while the local part is approximated analytically.

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.