Monday
8.45am: Registration
8.55am: Welcome
9-10am: Mini-course 1: Liliana Borcea, «Imaging in random media» Slides
10-10.30am: Break
10.30-11.30am: Mini-course 1: Liliana Borcea, «Imaging in random media»
11.30-12.30: Special presentation: Randy Bartels
12.30-2pm: Lunch Break
2-3pm: Mini-course 2: Mahadevan Ganesh, «Computational wave propagation» Slides
3-3.30pm: Break
3.30-4.30pm: Mini-course 2: Mahadevan Ganesh, «Computational wave propagation»
4.30-5pm: Junior session: Wei Li abstract Slides
Tuesday
9-10am: Mini-course 1: Liliana Borcea, «Imaging in random media»
10-10.30am: Break
10.30-11.30am: Mini-course 1: Liliana Borcea, «Imaging in random media»
11.30-12.30: Special presentation: Margaret Cheney « Problems in radar imaging »
12.30-2pm: Lunch Break
2-3pm: Mini-course 2: Mahadevan Ganesh, «Computational wave propagation»
3-3.30pm: Break
3.30-4.30pm: Mini-course 2: Mahadevan Ganesh, «Computational wave propagation»
4.30-5pm: Special presentation: Venkatachalam Chandrasekaran, «Propagation of dual-polarized Electromagnetic waves from S to K bands through precipitation: An experimental perspective »
Wednesday
9-10am: Mini-course 3: Lenya Ryzhik, «Weak randomness in evolution problems» abstract Lecture Notes
10-10.30am: Break
10.30-11.30am: Mini-course 3: Lenya Ryzhik, «Weak randomness in evolution problems»
11.30-12.30: Special presentation: Mark Ablowitz, « Wave Dynamics in Linear/Nonlinear Photonic Lattices and Topological Insulators » Slides
Free afternoon
Thursday
9-10am: Mini-course 3: Lenya Ryzhik, «Weak randomness in evolution problems»
10-10.30am: Break
10.30-11.30am: Mini-course 3: Lenya Ryzhik, «Weak randomness in evolution problems»
11.30-12pm: Junior session: Samuel Cogar abstract Slides
12-12.30pm: Junior session: Yongjoon Hong abstract Slides
12.30-2pm: Lunch Break
2-3pm: Mini-course 4: Guillaume Bal, «An introduction to periodic and stochastic homogenization»
3-3.30pm: Break
3.30-4.30pm: Mini-course 4: Guillaume Bal, «An introduction to periodic and stochastic homogenization»
4.30-5pm: Junior session: Rachael Keller abstract Slides
Friday
9-10am: Mini-course 4: Guillaume Bal, «An introduction to periodic and stochastic homogenization»
10-10.30am: Break
10.30-11.30am: Mini-course 4: Guillaume Bal, «An introduction to periodic and stochastic homogenization»
11.30-12pm: Junior session: Ornella Mattei abstract Slides
12-12.30pm: Junior session: Ngoc Do abstract Slides
End of the school
Mini-courses abstracts
Weak randomness in evolution problems
We will discuss some very basic questions in the study of the long time effect of weak random fluctuations on the evolution of very simple deterministic systems describing the dynamics of particles and PDE. Our focus will be on the long time scales when the effect of the fluctuations is no longer small. Depending on the speed of the lectures, the specific examples should include particles in weakly random flows, relaxation enhancement flows, the stochastic acceleration problem and the random Schroedinger equation. If we go very fast, we may have time to discuss the stochastic heat equation and its connection to the heat equation with a weak random potential.
http://inside.mines.edu/~mganesh/pp4_csu_lect1.pdf
http://inside.mines.edu/~mganesh/pp4_csu_lect2.pdf
Junior speakers
Eigenvalue problems in inverse scattering theory
We investigate the development of target signatures in order to detect changes in the material properties of an inhomogeneous medium from its measured scattering data. After an overview of this qualitative approach to inverse scattering theory, we present some recently developed eigenvalue problems arising from acoustic scattering that provide useful target signatures, and we briefly discuss the extension of these ideas to electromagnetic scattering. Our presentation includes both theoretical results and numerical experiments.
Youngjoon Hong, University of Illinois at Chicago
A
High-Order Perturbation of Surfaces Method for Electromagnetic
Scattering by multiply Layered Periodic Crossed Gratings
The
capability to rapidly and robustly simulate the scattering of
linear waves by periodic, multiply layered media in two and three
dimensions is crucial in many engineering applications. In this
regard, we present a High-Order Perturbation of Surfaces method for
linear wave scattering in a multiply layered periodic medium to
find an accurate numerical solution of the governing Helmholtz
equations. For this we restate the governing time harmonic Maxwell
equations as vector Helmholtz equations which are coupled by
transmission boundary conditions at the layer interface. We then
apply the method of Transformed Field Expansions which delivers a
Fourier collocation, Legendre–Galerkin, Boundary Perturbation
approach to solve the problem in transformed coordinates. A
sequence of numerical simulations demonstrate the efficient and
robust spectral convergence which can be achieved with the proposed
algorithm.
Wei Li, Louisiana State University
Second-harmonic
Imaging in Random Media
We consider the imaging of
small nonlinear scatterers in random media. This problem is
analyzed in weakly scattering media which respond linearly to
light. We show that for propagation distances within a few
transport mean free paths, robust images can be constructed by the
coherent interferometry (CINT) imaging functions. We also show that
imaging the quadratic susceptibility with CINT yields better
result, because that the CINT imaging function for the linear
susceptibility has noisy peaks in a region that depends on the
geometry of the aperture and the cone of incident directions.
(Joint work with Liliana Borcea and Alexander Mamonov and John
Schotland)
Ornella Mattei, University of Utah
Field patterns: A new type of wave
Abstract: Field patterns are a new type of wave propagating in one-dimensional linear media with moduli that vary both in space and time. Specifically, the geometry of these space-time materials is commensurate with the slope of the characteristic lines so that a disturbance does not generate a complicate cascade of subsequent disturbances, but rather concentrates on a periodic space-time pattern, that we call field pattern. Field patterns present spectacularly novel features. One of the most interesting ones is the appearance of a wave generated from an instantaneous source, whose amplitude, unlike a conventional wake, does not tend to zero away from the wave front. Furthermore, very interestingly, the band structure associated with these special space-time geometries is infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern.
Ngoc Do, University of Arizona
Theoretically exact solution of the inverse source problem for the wave equation with spatially and temporally reduced data
Abstract: The inverse source problem for the wave equation arises in several promising emerging modalities of medical imaging. Of special interest here are theoretically exact inversion formulas, explicitly expressing solution of the problem in terms of the measured data. Almost all known formulas of this type require data to be measured on a closed surface completely surrounding the object. This, however, is too restrictive for practical applications. I will present an alternative approach that, under certain restriction on geometry, yields explicit, theoretically exact reconstruction from the data measured on a finite open surface. Numerical simulations illustrating the work of the method will be presented.
Band Degeneracies in π/2 Rotationally Invariant, Periodic Schroedinger Operators"
Abstract : Operators with periodic potentials are central to the mathematical description of waves in periodic media, with many applications in Quantum Physics, Electromagnetics and other fields. Wave propagation properties are encoded in “band structure,'' the collection of dispersion surfaces and associated Floquet-Bloch eigenmodes. Band degeneracies are special quasi-momentum energy pairs where consecutive dispersion surfaces intersect, and these intersections are often caused by symmetries of the potential. Among the best known degeneracies are Dirac points (conical intersections) arising for honeycomb potentials on R2 (Fefferman and Weinstein, 2012; Fefferman, Lee-Thorp, and Weinstein 2016). In this work we study the class of Z2-periodic potentials which are real-valued, even and π/2-rotationally invariant. The band structure for such potentials is proved to have spectral degeneracies at certain high-symmetry quasi-momenta. We give a detailed general picture of the dispersion surfaces near such degeneracies, and applying our results, we show that the well-known conical + flat band dispersion for the tight-binding model for the Lieb lattice does not persist for finite-depth potentials.