Computation of high frequency solutions to wave equations is important
in many applications, and notoriously difficult in resolving wave
oscillations. Gaussian beams are asymptotically valid high frequency solutions
concentrated on a single curve through the physical domain, and superposition
of Gaussian beams provides a powerful tool to generate more general high
frequency solutions to PDEs. In this talk I will present a recovery theory of
high frequency wave fields from phase space based measurements. The
construction use essentially the idea of Gaussian beams, level set description
in phase space as well as the geometric optics. Our main result asserts that
the kth order phase space based Gaussian beam superposition converges to the
original wave field in L2 at the rate of $\epsilon^{k/2-n/4} in dimension $n$.
The damage done by caustics is accurately quantified. Though some calculations
are carried out only for linear Schroedinger equations, our results and
main arguments apply to more general linear wave equations. This work is in
collaboration with James Ralston (UCLA).

The unique solvability of second order elliptic and parabolic equations (in either divergence form or non-divergence form) in Sobolev spaces is well known if the leading coefficients are, for example, uniformly continuous.
However, in general, it is not possible to solve equations in Sobolev spaces unless the coefficients have some regularity assumptions.
In this talk we will discuss some possible classes of discontinuous coefficients with which elliptic and parabolic equations are uniquely solvable in Sobolev spaces.
Especially, as our main results, we will focus on the unique solvability of equations with coefficients only measurable in one spatial variable and having small mean oscillations in the other variables (called partially BMO coefficients).
We will also discuss some applications of our results as well as a new approach to a priori L_p estimates.
Most of the talk is based on joint work with Nicolai Krylov and with Hongjie Dong.

In processes involving multi-phase flow in highly heterogeneous media, such as oil recovery by gas injection, mobile phases will seek high-permeability flow paths. Therefore, it is essential that models for such processes effectively account for these paths. For this purpose, we have developed a computational framework for flow solvers based on adapted Cartesian grids that are equipped with multi-point flux approximations obtained with specialized transmissibility upscaling methods.

For gridding, we propose using Cartesian Cell-based Anisotropically Refined (CCAR) grids, which inherit the ease of Cartesian grids while providing rapid transition between coarse and fine scales to resolve fine-scale features accurately and efficiently. We present an iterative algorithm for automatically generating such grids based on geological data and information from global coarse-scale flow simulations.

For upscaling, we discuss a local transmissibility upscaling method, called Variable Compact Multi-Point (VCMP), that uses spatially varying and compact multi-point flux stencils. The stencil weights are chosen so as to reproduce generic local flow problems accurately, while remaining as close as possible to a two-point flux for the sake of robustness. The inherent flexibility of VCMP can also be exploited to ensure that the solution of the resulting system satisfies a discrete maximum principle.

We conclude with application of these gridding, upscaling and discretization methods, originally designed for single-phase flow, to two-phase flow, which requires enhancing our adaptive mesh refinement scheme in order to accurately resolve rapidly advanacing saturation fronts. We show that adaptivity allows such accurate resolution by upscaling only single-phase parameters, thus avoiding the significant computational expense of multi-phase upscaling.

In this lecture, we present a method which has broad applicability to studies of nonlinear stability of periodic traveling-wave solutions for equations of KdV-type. In particular we obtain the existence and stability of a family of periodic traveling-wave solutions for the Benjamin-Ono equation via the classical Poisson summation theorem and positivity properties of the Fourier transform.

This is a report on a joint work with Isabelle Catto, Norbert Mauser and Saber Trabelsi. The Multiconfiguration time dependent Hartree Fock Method (MCTDHF) is a nonlinear approximation of a linear system of /N/ quantum particles with binary interaction. It combines the principle of the Hartree Fock and the Galerkin approximation. The main difficulty is the introduction of a global (in space) density matrix $\Gamma(t) $ which may degenerate. By construction this approximation formally preserves the mass and the energy of the system. The conservation of energy can be used to balance the singularities Coulomb potential and to provide sufficient conditions for the global in time invertibility of $\Gamma(t)$.

In numerical computations this matrix is very often regularized (changed into $\Gamma(t) +\epsilon(t)$). In this situation the energy is no more conserved
and the mathematical analysis done in $L^2$ relies on Strichartz type estimates.