The electromagnetic inverse scattering problem of determining the locations and polarization tensors of a collection of small scatterers is investigated. The locations of scatterers are determined by the multiple signal classification (MUSIC) method and the polarization tensors are retrieved by the two-step least squares method. Multiple scattering effect is taken into account and the inverse scattering problem is nonlinear. However, the proposed method does not involve iterative evaluations of the corresponding forward scattering problem. In addition, the method provides better imaging resolution than the standard MUSIC does, and applies to degenerate scatterers to which the standard MUSIC does not apply. The underlying mathematical principle and physical insight are discussed in detail.

Abstract:  In order to fight infection or heal injury, living cells have to be able to move in response to chemical cues around them. As a first step, an internal chemical "map" is rapidly induced in the cell, leading to its polarization, reorganization of structural proteins (cytoskeleton), shape change, and crawling motility. In my talk, I will summarize some of the work done in my group addressing these processes. I will explain the basic properties of the signalling proteins (small GTPases) and their role in cell polarization. I will motivate a sequence of mathematical models that we have studied to understand the underlying mechanisms. One theme in my talk will be the cross-fertilization of the complex biological problem and its simplified mathematical caricatures. Another theme is the effect of cell shape on internal chemical dynamics. I will conclude with recent developments on computations of cell shape dynamics in 2D simulations.

In 1966, Almgren showed that any immersed minimal surface in
S^3 of genus 0 is totally geodesic, hence congruent to the equator. In
1970, Blaine Lawson constructed many examples of minimal surfaces in S^3
of higher genus; he also constructed numerous examples of immersed minimal
tori. Motivated by these results, Lawson conjectured that any embedded
minimal surface in S^3 of genus 1 must be congruent to the Clifford
torus.

In this talk, I will describe a proof of Lawson's conjecture. The proof
involves an application of the maximum principle to a function that depends
on a pair of points on the surface.

The Stokes-Darcy model arises in many interesting real world applications, including groundwater flows in karst aquifers, interaction between surface and subsurface flows, industrial filtrations, oil reservoir in vuggy porous medium, and so on. This model describes the free flow of a liquid by the Stokes or Navier-Stokes equation and the confined flow in a porous media by the Darcy equation; the two flows are coupled through interface conditions. For the problems mentioned, the resulting coupled Stokes-Darcy model has higher fidelity than either the Darcy or Stokes systems on their own. However, coupling the two constituent models leads to a very complex system.

This presentation discusses multi-physics domain decomposition methods for solving the coupled Stokes-Darcy system. Robin boundary conditions based on the physical interface conditions are utilized to decouple the Stokes and Darcy parts of the system. A parallel iterative domain decomposition method is first constructed for the steady state Stokes-Darcy model with the Beavers-Joseph interface condition. Then two parallel non-iterative domain decomposition methods are proposed for the time-dependent Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. Numerical examples are presented to illustrate the features of these methods and verify the theoretical results.