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,CMAP, Ecole Polytechnique. Title: Geometry and topology optimization of laminated composite structures Abstract:This is a joint work with G. Delgado EADS IW. We propose a level set algorithm for multi-layered composite design in structural optimization. The level set method for shape and topology optimization of structures has became an efficient and popular numerical algorithm, able to treat various mechanical models and objective functions. We extend its range of applicability to the optimal design of composite structures, built by lamination of a sequence of unidirectional reinforced layers or plies, which appear on the fuselages or on the wings of an airplane. Given a stacking sequence and fiber orientation of a composite structure, we determine the optimal shape of each layer which is made of a linear orthotropic elastic material in a membrane model. We minimize the weight of the composite structure, under a rigidity constraint, like compliance or maximum stress (reserve factor). We use a numerical algorithm for shape and topology optimization based on the level set method coupled with the shape and topological derivatives. This requires new ingredients for computing accurately elastic shape derivatives and anisotropic topological derivatives. A 2-d airplane fuselage section test case is discussed. |
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, INRIA, Sophia Antipolis. Title: Inverse magnetization problems on thin plates. Abstract: This is a joint work with D. Hardin, E. Lima, E. Saff and B. Weiss. A ${\bf R}^3$-valued magnetization ${\bf m}=(m_1,m_2,m_3)$ supported on the horizontal plane $x_3=0$ in ${\bf R}^3$ generates a magnetic potential above the plane which is the Poisson extension of $R_1(m_1)+R_2(m_2)+m_3$ where $R_j$ indicate the Riesz transforms. The inverse magnetization problem we consider is to recover ${\bf m}$ from the obeservation of the field (i.e. the gradient of the potential) on a plane $x_3=h$. Using a new Hardy-Hodge decomposition for ${\bf R}^3$-valued vector fields on a plane, we characterize equivalent thin-plate magnetizations. We show that, under fairly general assumptions, there exist unidimensional magnetizations equivalent to a given one, and we discuss how they can be used to recover a compactly supported magnetization. |
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,Chalmers University of Technology and Gothenburg University. Title: Quantitative imaging technique using an approximately globally convergent method Abstract: We will present a new model for an approximately globally convergent numerical method to the solution of a Coefficient Inverse Problem (CIP) for a hyperbolic equation. An approximate globally convergent algorithm rigorously guarantee obtaining at least one point in a small neighborhood of the exact solution without any advanced knowledge of that neighborhood. The data in our CIP are generated by either a single location of the point source or by a single direction of the incident plane wave which is a special case of interest in military applications or airport security. We will present results of reconstruction on simulated and experimental data in 3D using efficient finite element implementation (C++/PETSc) of new numerical method in the software package WavES, www.waves24.com Numerical verification of an approximately globally convergent method implemented in WavES is performed on the blind experimental data provided by the Optical Center of UNCC, Charlotte, USA, |
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, POEMS, ENSTA ParisTech Title: On Lipschitz stability for a class of inverse problems Abstract: We prove an abstract theorem on Lipschitz stability for a general class of inverse problems and give some examples of application, for example the inverse impedance problem for the Laplace equation or the inverse medium problem for the Helmholtz equation. |
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,CMAP, Ecole Polytechnique. Title: Short-time existence results for the (mostly 2D) crystalline curvature flow Abstract: The evolution by crystalline mean curvature of sets is, in some sense, the "gradient flow" of an anisotropic perimeter whose surface tension is neither elliptic nor smooth. Existence and uniqueness results for such evolutions of sets are known in very few situations. I will discuss some results on this topic, on regular evolutions with or without forcing term. While some comparison results exists, existence is mostly unknown except in 2D or for the non forced motion of convex crystals, in simple settings. |
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, Universite Paris Diderot Title: Enhanced stability of correlation-based imaging in random media Abstract: Sensor array imaging in a randomly scattering medium is usually limited because coherent signals recorded at the array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. We will see in this talk how correlation-based imaging techniques (using cross correlations that are quadratic forms in the data) can mitigate or even sometimes benefit from the multiple scattering of waves. |
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, Université Paris-Sud. Title: The cubic Szeg\H{o} equation and the solution of inverse spectral problems for Hankel operators Abstract: The cubic Szeg\H{o} equation is an infinite dimensional Hamiltonian system on the Hardy space of the unit disc, which admits a Lax pair involving Hankel operators. I shall describe new results about inverse spectral problems for these operators, and their connection to the dynamics of this equation. This talk is based on a joint work with Sandrine Grellier. |
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, University of California, Berkeley. Title: A nonlinear inverse problem for quantum walks Abstract: Consider a quantum walk with discrete time, such as in arXiv 1202.3903 CMP Jan 2013 or arXiv 1302.7286. For a given initial state $\psi$ consider the probabilities $p_n$ of a return to $\psi$ in n steps as well as $q_n$ , the probabilities of a first return to $\psi$ in n steps. If one only knows the values of $p_n$ for $n=1,2,3,...$ the problem of recovering the spectral measure of the walk is equivalent to the famous "phase problem in crystallography" and the measure is not uniquely determined. Is the extra information considered here enough to determine the measure uniquely? If the answer is positive, is there a good reconstruction algorithm? This is an open problem relating complex analysis, spectral theory and quantum mechanics. |
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, CMAP, Ecole Polytechnique. Title: Stability and instability for the Gel'fand inverse boudary value problem Abstract: In this talk we discuss new global Holder-logarithmic stability estimates for the Gel'fand inverse boudary value problem. These estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. Instability results showing the optimality of estimates of such a type will be also discussed. In addition, we consider examples of some other inverse problems, where similar estimates hold. This talk is based , in particular, on recent works [ M. Isaev, R. Novikov, 2012] and [ M. Isaev, 2012]. |
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, LPTM, Université de Cergy-Pontoise. Title: Inverse scattering at high energies for the multidimensional relativistic Newton equation in a long-range electromagnetic field Abstract: We consider in $\mathbb R^n$, $n\ge 2$, classical relativistic particles moving in a smooth and long-range static electromagnetic field $F$ decomposed as a sum of a known long range tail $F^l$ and an unknown short range part $F^s$. We define a scattering map and then give estimates and asymptotics for this map and for the scattering solutions in the regime of small scattering angles compared to the dynamics generated by $F^l$. We show that at high energies the first component of the scattering map uniquely determines the x-ray transform of $F^s$ on a big enough set of lines so that it uniquely determines $F^s$. In addition we show that at high energies the second component of the scattering map uniquely determines the short range part of the electric potential up to radial potentials as well as the short range part of the magnetic field when $n\ge 3$ (up to radial magnetic fields when $n=2$). Similar results are also given when one changes the definition of the scattering map. |
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, CMAP, Ecole Polytechnique. Title: Absence of sufficiently localized solitons for the Novikov-Veselov equation at nonzero energy 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 will show that, unlike its (1+1)-dimensional counterpart (for which exponentially localized solitons are well-known), the Novikov-Veselov equation does not possess sufficiently localized solitons at nonzero energy. In particular, Grinevich–Zakharov potentials have almost the strongest localization possible for the solitons of the Novikov-Veselov equation. The talk is based on the recent work [A.Kazeykina, 2013]. |
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, University of North Carolina at Charlotte. Title: Uniqueness of some 3-d phaselss inverse scattering problems Abstract: Conventional uniqueness theorems for inverse problems in frequency domain work only with the case when the total complex valued wave field is known at a certain set. However, there are a number of applications, e.g. neutron specular reflection, when only the modulus of the scattered wave can be measured, while the phase is unknown. We will present uniqueness theorems for four 3d different inverse scattering problems for the latter case and will outline the main idea of the proof. The unknown coefficient is the compactly supported potential in the 3d Schrodinger equation. In the proof a new idea of the theory of complex variables is combined with some properties of the solution of the associated hyperbolic PDE as well as with the method of Carleman estimates for the inverse problem for that PDE. The preprint is available online at arxiv: 1303.0923v1 [math-ph] 5 Mar 2013 The result can be extended to the case of the 3d acoustic equation. The single previous similar result the author is aware of the one in 1d of Klibanov and Sacks (1992). There were also a number of uniqueness theorems of the author for the problem of the reconstruction of a compactly supported complex valued function from the modulus of its Fourier transform (1985-2006). |
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, CMAP, Ecole Polytechnique. Title: Exact and approximate reconstruction methods for a multi-channel inverse problem Abstract:In this talk some reconstruction results for the multi-channel Gel'fand-Calderon inverse problem in two dimensions are presented. This is the problem of the recovery of a matrix-valued potential in the Schrodinger equation from boundary data (Dirichlet-to-Neumann map) at fixed energy. The principal motivation for studying the multi-channel 2D problem is the approximation of the 3D scalar problem, which is formally overdetermined. We will first present an exact reconstruction procedure, based on modified Faddeev-type functions inspired by some results of Buckheim. This exact reconstruction yields directly an uniqueness result, but it satisfies a log-type stability estimates which is too weak for practical application. The next result is an approximate reconstruction algorithm for the same problem, which is Lipschitz stable and rapidly converging at high energies. The algorithm admits practical applications, namely for ocean acoustic tomography. It is based on the theory of inverse quantum scattering and it exploits a stabilizing phenomena when the energy is sufficiently large. This talk is based, in particular, on recent works [R. Novikov, M. Santacesaria, 2011, 2013] and [M. Santacesaria, 2012]. |
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, Université Paris-Sud. Title: Inverse scattering versus PDE’s Abstract:We will compare for classical 2D integrable equations and systems (KP, Davey-Stewartson, Novikov-Veselov...) the results obtained via inverse scattering methods and via pure PDE methods. |
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, Laboratoire Jean Kuntzmann, Grenoble. Title: Weighted stability estimates for some electro-acoustic inverse problems Abstract:A major problem in solving hybrid inverse problems is the presence of critical points where the collected data vanishes. The set of these critical points depends on the choice of the boundary conditions, and can be directly determined from the data itself. In most existing stability results, the boundary conditions are assumed to be close to a set of CGO solutions where the citical points are controlled. In this talk I will present new stability estimates for some electro-acoustic inverse problems without assumption on the boundary conditions. |
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, LPTM, Université de Cergy-Pontoise. Title: Geometric aspects of the invertibility of some Radon transform on circular arcs in the plane. Abstract: Some inverse problems arising in imaging processes are solved by inverting related Radon transforms. By starting from the classical Radon transform and its inversion via the A M Cormack method of circular harmonic components, we show that the interior/exterior Radon problem on some classes of circular arcs in the plane has a solution obtained by special geometric radial transformations. Such Radon transforms may find applications in Compton scatter tomography. |
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, Loughborough, UK. Title: Spectral theory and classical integrable systems: variations on a theme of Moser Abstract: In 1978 in his CIME lectures at Bressanone Moser described a remarkable connection between the 1D periodic Schroedinger operators with finite-gap spectrum and classical Neumann system of motion on the sphere with quadratic potential. In the talk I will discuss this correspondence (known now as Moser-Trubowitz isomorphism) as well as some generalizations, related to the Jacobi geodesic problem on ellipsoid. |