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, CMAP, Ecole Polytechnique. Title: Looking for the hot spot: homogenization and localization of a convection-diffusion equation in a bounded domain. Abstract: This is a joint work with I. Pankratova and A. Piatnitski. We consider the homogenization of a non-stationary convection-diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the ``hot spot'', i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion the position of the ``hot spot'' is not always intuitive as exemplified in some numerical tests. |
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, Ecole Normale Supérieure, Paris
. Title: Enhancement of Near Cloaking Using Generalized Polarization and Scattering Tensors Vanishing Structures. Abstract: The aim of this talk is to provide an original method of constructing very effective near-cloaking structures for the conductivity problem and the Helmholtz equation. These new structures are such that their first Generalized Polarization Tensors or Scattering Tensors vanish. We show that this in particular significantly enhances the cloaking effect. We then present some numerical examples of Generalized Polarization and Scattering Tensors vanishing structures. This is a joint work with H. Kang, H. Lee, and M. Lim. |
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Columbia University. Title: Inverse Problems with Internal Functionals. Abstract: Hybrid inverse problems (also known as multi-wave or coupled-physics inverse problems) aim at combining the high contrast of one imaging modality (such as e.g. Electrical Impedance Tomography or Optical Tomography in medical imaging) with the high resolution of another modality (such as e.g. based on ultrasound or magnetic resonance). Mathematically, these problems often take the form of inverse problems with internal information. This talk will review several results of uniqueness and stability obtained recently in the field of hybrid inverse problems. |
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, Chalmers University of Technology and
Gothenburg University. Title: Approximate globally convergent numerical method and adaptivity technique for inverse problems with experimental data. Abstract: In this talk we will summarize results of our research group during 2007- 2011 on the approximate globally convergent numerical method with combination of the adaptivity technique for solution of hyperbolic coefficient inverse problems , which we call two-stage numerical procedure. We will also briefly discuss the framework of the functional analysis for the adaptivity technique. At the end of the talk we will present verification of the two-stage numerical procedure on the experimental data. |
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, LMS, Ecole Polytechnique. Title: Topological derivative of elastodynamic energy functionals. |
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IMJ, Université Paris-Diderot.
Title: Long-time asymptotics for integrable nonlinear waves: the Riemann-Hilbert approach. Abstract: I will present recent results on long-time asymptotics of integrable nonlinear waves obtained by the inverse scattering method in terms of matrix Riemann-Hilbert problems. I will give examples of Cauchy problems on the line with step initial conditions. I will also give examples of initial-boundary value problems on the half-line with time decaying initial data and periodic boundary conditions. Joint work with V. Kotlyarov, D. Shepelsky et al. |
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CMAP, Ecole Polytechnique.
Title: Crack kinking and energy estimates at the tip of a 2D fracture. Abstract:This is a joint work with G. Francfort, J.-J. Marigo and A. Lemenant. We discuss a stability criterion for 2D linearized elasticity, at the tip of a straight crack, and its consequences. In particular, we show that in theory, both classical criteria (local symmetry, and maximal energy release) for the selection of the kinking angle seem to lead to unstable evolutions. Our approach suffers the drawbacks that it relies on energy estimates at the tip which are not so-well known for arbitrary (nonsmooth) fractures. In the 2D antiplane (scalar) setting, we also show how more general fractures could be considered. |
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, SHFJ, CEA, Orsay. Title: The role of image reconstruction algorithms in positron emission tomography. Abstract: Positron emission tomography (PET) is a biomedical imaging technique that allows for the visualization and the measurement of biological processes. PET is based on the injection to the patient of a molecular probe for the process of interest. The probe is labeled with radio-isotopes that decay by emitting a positron. The emitted positron annihilates almost immediately with an electron, resulting in the emission of two photons that are detected in detector rings surrounding the patient. In a first approximation, data collected in such a way can be modeled as a Poisson inhomogeneous process, with mean given by the X-ray transform of the probe spatial distribution. PET reconstruction is the inverse problem associated: finding the spatial distribution of the probe from a noisy realization. The reconstructed images suffer from two main drawbacks: limited spatial resolution and high level of statistical noise due to the low number of photons detected. Spatial regularization during reconstruction is often used, which further degrades spatial resolution. To obtain an optimal trade-off between resolution and statistical noise, it is necessary to accurately model during reconstruction the detection process and the noise in the data. Therefore, the choice of the reconstruction algorithm plays an important role in the quality of the reconstructed images, hence in their diagnostic utility. The development of new reconstruction algorithms for PET is an active field of research. The various strategies that are used for PET image reconstruction will be presented, with emphasis on iterative techniques. The impact of the modeling approximations will be discussed in regard to the use of the reconstructed images by the physician. |
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, Landau Institute for Theoretical Physics. Title: On the Ground Level of Purely Magnetic Algebro-Geometric 2D Pauli Operator (spin 1/2). Abstract: This is a joint work with A.Mironov and S.Novikov. Full manifold of the complex Bloch-Floquet eigenfunctions is investigated for the ground level of the purely magnetic 2D Pauli operators. This operator is factorisable, i.e. it is a direct sum of two 2D pure magnetic Schrodinger operators, connected by the Laplace transform. It is shown that such Schrodinger operators are associated with factorisable complex Fermi-curves. This spectral problem is associated with the 2D analog of the ''Burgers Nonlinear Hierarchy''. No Algebro-Geometric operators where known earlier in the case of nonzero flux. We found periodic operators with zero flux, singular magnetic fields and Aharonov-Bohm phenomenon. After removing the integer Aharonov-Bohm term we obtain regular operators with integer non-zero magnetic flux through an elementary cell. |
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, LMJL, Université de Nantes. Title: Analytic and iterative reconstructions in SPECT. Abstract: We consider analytic and iterative reconstructions in the single-photon emission computed tomography (SPECT). As analytic techniques we use, in particular, Chang's approximate inversion formula and Novikov's exact inversion formula for the attenuated ray transform, on one hand, and Wiener type filters for data with strong Poisson noise, on other hand. As iterative techniques we consider the least square and expectation maximization iterative reconstructions. Different comparaisons are given. This talk is based on joint work with R. G. Novikov. |
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CMAP, Ecole Polytechnique.
Title: A Factorization Method for a Far-Field Inverse Scattering Problem in the Time Domain. Abstract:This is a joint work with A. Lechleiter. We consider a far-field inverse obstacle scattering problem for the scalar wave equation in the time domain. We prove that certain test functions given as far-fields of pulse solutions to the wave equation characterize the obstacle by a range criterion: If the source point of the pulse is inside the obstacle, then the test function belongs to the range of the ``square root'' of the time derivative of the far-field operator. If the source point is outside the obstacle, then the test function does not belong to this range. This is hence an explicit characterization of the obstacle by far-field measurements of time-dependent scattered waves. The proof relies on an operator factorization related to the Factorization method for inverse scattering in the frequency domain, and on the positivity of the time derivative of the inverse of the retarded single-layer operator. |
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, LPTM, Université de Cergy-Pontoise. Title: Stability in inverse scalar transport theory. Abstract: We give stability estimates for the reconstruction of the optical parameters in the stationary or non stationary Boltzmann transport equation from angularly and spatially resolved measurements and angularly and spatially resolved boundary sources. Then for the nonstationary case we give uniqueness and stability results for the reconstruction of those parameters from angularly averaged and spatially resolved boundary measurements and isotropic spatially resolved boundary sources. These results are of interest in optical tomography. The problem of the reconstruction of the optical parameters from internal angularly averaged measurements in the stationary case will be also addressed and is of interest for photoacoustic tomography. |
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, University of North Carolina at Charlotte. Title: Approximate Global Convergence of Some Numerical Methods for Coefficient Inverse Problems. Abstract: Coefficient Inverse Problems are important for many applications. However, they are very challenging to compute too. Challenges take place because of the nonlinearity and ill-posedness combined. An important question in their numerical treatment is this: How to construct such an algorithm which would be guaranteed to provide a point in a small neighborhood of the solution without any a priori knowledge of this neighborhood? Because of the above challenges, it is unlikely that this question would be effectively addressed without some approximations, even though these approximations would be justified intuitively rather than completely rigorously. Purely roughly speaking, we call a numerical method providing the above desired point via an approximate mathematical model "approximately globally convergent method". Naturally, this method should be not only constructed analytically but also verified numerically. In the past three years our group (Beilina, Kuzhuget, Pantong, Klibanov) has developed approximately globally convergent methods for a Coefficient Inverse Problems. The first work was of L. Beilina and M.V. Klibanov in SIAM J. Sci. Comp., 2008. These methods will be presented. Theorems ensuring the approximate global convergence property will be formulated. Verification for both computationally simulated and experimental data will be presented. |
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, St.-Petersburg University. Title: Inverse problems, trace formulas and resonances for multidimensional Schrodinger operators on the lattice. Abstract: In this talk (based on joint work with H. Isozaki) we consider the multidimensional Schrodinger operator H on the lattice with a finitely supported potential. Our goal is the following: 1) inverse scattering problems, the reconstruction of the potential from the scattering matrix, 2) the computation of trace formula using the spectral shift function, 3) the construction the analytical continuation of the resolvent from the spectral (the first) sheet on some Riemann surface. |
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, CMLS, Ecole Polytechnique. Title: Action-angles for integrable PDEs: results, problems and conjectures. Abstract: The only way to contruct the action-angle variables for an integrable PDE under periodic boundary conditions is through study of inverse and direct spectral problems for the corresponding L-operator (for the KdV equations the L-operator is the Shturm-Liouville operator, while for the NLS equation this is the Dirac operator, etc). In my talk I will discuss recent progress in the theory of action-angle variables and relate it with results and problems from the spectral theory of the corresponding L-operators. In particular, with some recent results of E.Korotyayev. |
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, CMAP, Ecole Polytechnique. Title: Exponential instability in inverse problems for the Schrodinger equation. Abstract: We consider inverse scattering and inverse boundary value problems for the multidimensional Schrodinger equation. In this talk we give a short review of old and new results on stability and instability estimates for these problems. |
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, CMAP, Ecole Polytechnique. Title: Large time asymptotics for solutions of the Novikov-Veselov equation. Abstract: Novikov-Veselov equation is a (2+1)-dimensional analog of the classic KdV equation integrable via the inverse scattering method for the 2-dimensional Schrodinger equation. In this talk we give a short review of new results concerning the behavior of solutions to this equation at large times. |
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, CMAP, Ecole Polytechnique. Title: Uniqueness, stability and reconstruction for the Gel'fand-Calderon inverse problem in two dimensions. Abstract: We consider the (multi-channel) Gel'fand-Calderon inverse problem on a two-dimensional bounded domain. In this talk we give in particular a short review of recent global results on uniqueness, stability and reconstruction for this problem. |