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# Archive: Inverse Problems and Analysis

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 Dr. Augustin Cosse, NYU Dr. Augustin Cosse, NYU EWG 336 Title: Semidefinite programming relaxations for matrix completion, inverse scattering and blind deconvolution

Abstract: The talk will discuss three instances of semidefinite programming relaxation with applications in compressed sensing, inverse scattering and imaging. In the first part, we will close the line of work on rank one matrix completion by introducing a stable algorithm based on two levels of semidefinite programming relaxation. For this algorithm, we certify recovery of any rank one matrix, at the information limit, through the construction of a dual (sum of squares) polynomial. In passing, this dual polynomial also provides a rationale for the efficiency of the trace norm in semidefinite programming.

In the second part, we introduce a fast algorithm for inverse scattering, which leverages the traditional Adjoint State method used in geophysics, by lifting the search space. The algorithm is based on a first level ofsemidefinite programming relaxation encoded through a low rank factorization which guarantees its scalability. Numerical experiments on 2D community models will be used to highlight a modest increase with respect to the basin of attraction of traditional least squares waveform inversion. A geometric intuition for this improvement will be provided.

Finally, in the last part of the talk, we discuss how blind deconvolution can be solved with high probability, in the case of a fully unknown filter, through nuclear norm relaxation, when multiple (unknown) inputs are available. Recovery is certified through the construction of a dual certificate. The certificate expands as a Neumann series and is shown to satisfy the conditions for the recovery of the matrix encoding the unknowns by controlling the terms in this series through appropriate concentration and decoupling inequalities.

(Joint work with Laurent Demanet)

10/17/2017 3:00:00 PM 10/17/2017 4:00:00 PM False Dr. Matthew Fleeman, Baylor University Dr. Matthew Fleeman, Baylor University TBA Title: Hyponormal Toeplitz operators acting on the Bergman space.

Abstract: In 1988 Carl Cowen completely characterized hyponormal Toeplitz operators acting on the Hardy space. In the Bergman space setting, similar studies have mostly focused on Toeplitz operators with harmonic symbols. In this talk, we will examine these results and give some new results on the hyponormality of Toeplitz operators acting on the Bergman space with non-harmonic symbols. This is joint work with Conni Liaw. 10/13/2017 7:30:00 PM 10/13/2017 8:30:00 PM False Dr. Yemon Choi, Lancaster University, UK Dr. Yemon Choi, Lancaster University, UK Ewing 336 Title: Directly finite algebras arising from group representations.

Abstract: Consider doubly-infinite Toeplitz matrices, regarded as operators on $\ell^2({\bf Z})$. The collection of all such matrices forms a commutative algebra, which is a familiar object in mild disguise; and structural properties of this algebra allow us to say some things about the spectral properties of such operators. We can consider analogous objects that arise if we replace ${\bf Z}$ by a noncommutative group, possibly now continuous rather than discrete, or replace $2$ by some other $p\in (1,\infty)$. These algebras of convolution operators have been intensively studied in particular cases but are still somewhat mysterious.

In this talk I will consider one particular structural property, "direct finiteness", which turns out to have a surprising application to certain dynamical systems considered in work of Deninger and Schmidt. I will give an outline of how direct finiteness can be established for discrete groups and arbitrary $p$, or unimodular groups when $p=2$, using tools from abstract functional analysis ($C^*$-algebra-theory).

10/3/2017 3:00:00 PM 10/3/2017 4:00:00 PM False Dr. Anna Mazzucato, PennState University Dr. Anna Mazzucato, PennState University EWG 336 Title: Elliptic equations on polyhedra and applications to the FEM

Abstract: I will discussed well-posedness of boundary value/interface problems for elliptic equations on polyhedral domains in weighted Sobolev spaces. The well-posedness and regularity results can be used to construct graded meshes that yields optimal convergence rates for FEMs. 9/26/2017 3:00:00 PM 9/26/2017 4:00:00 PM False