# PDE and Applied Mathematics Seminar

## Fall 2012

### Tuesday, September 4 at 4:30pm in Olin Hall 119A
Itai Shafrir
(Technion) will speak on
"Asymptotic behavior of critical points of an energy involving a
"circular-well" potential".

####
Abstract.
We study a generalization of the Ginzburg-Landau functional in which
the potential $(1-|u|^2)^2$ is replaced by a more general
"circular-well" potential, i.e., a nonnegative function whose zeros
set consists of a closed curve in the plane. We study the limit of
critical points of the energy when the parameter epsilon goes to zero.
This is a joint work with Petru Mironescu (Lyon I).

### Thursday, September 20 at 4:30pm in CAS 107
Malena Espanol
will speak on
"Multilevel Methods for Image Deblurring".

####
Abstract.
In this talk, I will introduce multilevel methods for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In particular, I will present wavelet-based multilevel methods for signal and image restoration problems as well as for blind deconvolution problems. In these methods, orthogonal wavelet transforms are used to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. I will show results that indicate the promise of these approaches on restoration of signals and images with edges as well as restoration of blurring operator in the case of the blind deconvolution problem.

### Thursday, October 4 at 4:30pm in Leigh Hall 305
Daniela Calvetti
will speak on
"Bayesian source separation in MEG".

####
Abstract.
Magnetoencephalography (MEG) is a completely non-invasive brain-mapping modality which uses measurements of the magnetic field outside the head induced by electrical brain activity to localize and characterize the activity inside the brain. Potentially, it is particularly useful in the study of epilepsy as a tool for localizing the focii of the onset of seizures. A key issue in MEG is the separation of sources of a different nature. Non-focal sources from both inside and outside of the brain produce interference, making the inverse problem of identifying the focal source signal extremely difficult. In this talk we show how Bayesian methods can be used to address this issue. In particular, we illustrate how a mixed prior distribution is able to separate sources which are statistically different from each other. Furthermore, we propose using a depth scan to identify activity from deep focal sources. Numerical simulations are used to generate controlled data in order to validate the model.

### Thursday, October 18 at 2:30 in CAS 136
Cyrill Muratov
will speak on
"Gamma-convergence for pattern forming systems with competing interactions".

###

####
Abstract.
I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions. This is joint work with D. Goldman and S. Serfaty.

### Thursday, November 15 at 2:30 in CAS 143
Robert Pego
will speak on
"Coagulation dynamics of uniform growth and random shock waves".

###

####
Abstract.
Smoluchowski's coagulation equation is related in remarkable ways to certain
`solvable' models of ballistic annihilation and aggregation. I plan to discuss
(a) preliminary work with Jack Carr concerning clustering due to uniform domain
growth in 1D, and (b) how kinetic coagulation models are tied to Menon and
Srinivasan's recent discoveries that indicate complete integrability for
Burgers turbulence.