In many applications, we are faced with the problem of solving
an ODE with multiple initial conditions. Standard ODE integrators compute
the solution for each initial condition independently, which can be
computationally expensive. The phase flow method (PFM) is a novel approach
to construct phase maps for nonlinear autonomous ordinary differential
equations on their compact invariant manifolds. It first constructs the
phase map for a small time using a standard ODE integrator and then
bootstraps the process with the help of a local interpolation scheme and
the group property of the phase flow. This construction usually takes the
time of tracing a couple of solutions and the resulting approximation to
the phase map is accurate. Once the phase map is available, integrating
the ODE for an initial condition on the invariant manifold only utilizes
local interpolation, thus having constant complexity. We present the
method and prove its properties in a general setting. As an example, the
phase flow method is applied to the fields of high frequency wave
propagation and computational geometry. In high frequency wave
propagation, we concentrate on three problems: wavefront construction,
multiple arrival time and amplitude computation. We also discuss the
adaptive issues in the implementation. In computational geometry, we apply
the phase flow method to generate geodesic flow on smooth 2D surfaces.
Numerical results will be presented as well. (Joint work with Emmanuel
Candes)

Let $\mu$ be a probability measure on a metric space $(E,d)$ and $N$ a positive integer.
The {\em quantization error} $e_N$ of $\mu$ is defined as the infimum over all subsets ${\cal{E}}$
of $E$ of cardinality $N$ of the average distance w.~r.~t.~$\mu$ to the closest point in the set
${\cal{E}}$. We study the asymptotics of $e_N$ for large $N$. We concentrate on the
case of a Gaussian measure $\mu$ on a Banach space. The asymptotics of $e_N$ is closely related to
{\em small ball probabilities} which have received considerable interest in the past decade.
The quantization problem is motivated by the problem of encoding a continuous signal
by a specified number of bits with minimal distortion. This is joint work with Steffen Dereich,
Franz Fehringer, Anis Matoussi and Michail Lifschitz.

This talk explores the potential of new geometric multiscale ideas in the area of partial differential equations. We present a recently developed multiscale system - curvelets - based on parabolic scaling, in which basis functions are supported in elongated regions obeying the relation width ~length^2. This system provides optimally sparse representations of the solution operators for a large class of symmetric systems of linear hyperbolic differential equations - such as the wave propagation operator. This has important implications both for analysis, and for numerical applications, where sparsity allows for faster algorithms. In the second part of the talk, we report on preliminary calculations which suggest that it is possible to derive accurate solutions to a wide range of differential equations in O(N log N) where N is the number of voxels; this complexity holds for arbitrary initial conditions. This is joint work with Laurent Demanet (Caltech)