Three different singularly perturbed eigenvalue problems in perforated
domains, or in domains with perforated boundaries, with direct
biological applications, are studied asymptotically. In the context
of cellular signal transduction, a common scenario is that a diffusing
surface-bound molecule must arrive at a localized signalling region,
or trap, on the cell membrane before a signalling cascade can be
initiated. In order to determine the time-scale for this process,
asymptotic results are given for the mean first passage time (MFPT) of
a diffusing particle confined to the surface of a sphere that has
absorbing traps of small radii. In addition, asymptotic results are
given for the related narrow escape problem of calculating the MFPT
for a diffusing particle inside a sphere that has small traps on an
otherwise reflecting boundary. The MFPT for this narrow escape problem
is shown to be minimized for particular trap configurations that
minimize a certain discrete variational problem (DVP). This DVP is
closely related to the classic Fekete point problem of determining the
minimum energy configuration for repelling Coulomb charges on the unit
sphere. Finally, in the context of spatial ecology, a long-standing
problem is to determine the persistence threshold for extinction of a
species in a heterogeneous spatial landscape consisting of either
favorable or unfavorable local habitats. For a 2-D spatial landscape
consisting of such localized patches, the persistence threshold is
calculated asymptotically and the effects of both habitat
fragmentation and habitat location on the persistence threshold is
examined. From a mathematical viewpoint, the persistence threshold
represents the principal eigenvalue of an indefinite weight singularly
perturbed eigenvalue problem, resulting from a linearization of the
diffusive logistic model.

The analysis of these three PDE eigenvalue problems is based on the
development of a common singular perturbation methodology to treat
localized patches or traps in combination with some detailed
analytical properties of the Neumann Green's function for the
Laplacian. With this asymptotic framework, the problem of optimizing
the principal eigenvalue for the each of these three problems is
reduced to the simpler task of determining optimal configurations for
certain discrete variational problems.

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Speaker: Michael Ward (UBC). Joint work with Dan Coombs (UBC), Alexei
Chekhov (U. Sask), Alan Lindsay (UBC), Anthony Peirce (UBC), Samara
Pillay (JP Morgan), Ronny Straube (Max Planck, Magdeburg).

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.