Inverse Scattering

In this paper, the 3-D inverse scattering series (ISS) internal multiple attenuation algorithm is modified for a one-dimensional subsurface to incorporate a 3-D point source in multiple predictions, for improved realism and effectiveness. The new algorithm, which assumes the earth is only varying in the z-direction (1-D subsurface/earth, reasonable in many circumstances in Central North sea , on-shore Canada, and the Middle East), represents more than a small increase in effectiveness of predicting the shape and amplitude of multiples, compared to a frequently employed 1.5-D ISS internal multiple attenuator (assuming a 2-D line source and a 1-D earth). The numerical tests are performed on a 3-D source synthetic data set from a 1-D subsurface. The results demonstrate that the new algorithm incorporating a 3-D point source can change the prediction from 'causing harm' to 'providing benefit' in comparison to an internal multiple attenuation algorithm that assumes a 1-D earth and a line source.

This paper describes recent progress in attenuating free surface and internal multiples for marine and on-shore plays. While there is much to celebrate within the multiple attenuation toolbox, with recent progress and improved capability, there are also significant fundamental open issues and practical challenges that remain to be addressed.

The attenuation of internal multiple energy on land data is still one of the most challenging tasks in seismic data preprocessing. Low data quality and lack of velocity information of complicated structure (especially in near surface) on land data often result in poor predictions in many cases. Inverse Scattering Series (ISS) internal multiple attenuation is a very promising algorithm to attenuate internal multiple energy on land seismic exploration data. The key characteristic of this ISS-based methods is that they do not require any information about the subsurface, i.e., they are fully data driven. Internal multiples from all possible generators are predicted simultaneously from the input data. In this paper we apply Inverse Scattering Series (ISS) internal multiple attenuation algorithms on a land seismic data from Canada.

We report on a way-point in the development of a formalism for the recovery of absorptive and dispersive medium parameters based on the inverse scattering series. We pursue such a methodology because of its theoretical promise to reduce/eliminate the need for prior information and medium structure assumptions in Q estimation and compensation procedures; to perform Q-compensation, in other words, in the absence of an accurate knowledge of Q. The first step in this development is a detailed understanding of the linear inverse problem. In this paper we present a linear inverse procedure for two parameters (wavespeed and Q) that are permitted arbitrary variation in depth, given a single shot-record as input. We demonstrate that the problem is over-determined, and recommend using incident angle as a final degree of freedom over which we perform averaging; we consider a normal incidence example to demonstrate a mitigation of the attenuation (transmission) error. This linear inverse, therefore, in addition to its main task as input to higher order terms in the inverse scattering series, has value on its own as a stand-alone parameter estimation procedure. A particular aspect of the Q-estimation and Q-compensation procedures implied by this formalism is the drastically different way in which the data is interrogated for Q information, namely through analysis of the characteristic absorptive/dispersive reflectivity.

VSP experiments provide a much greater opportunity to estimate local reflectivity information than do surface-constrained experiments. In this paper we describe a simple, data driven means by which the reflection coefficient associated with an interface at depth, uncontaminated by transmission losses, may be determined, regardless of the origins of these losses or of the overburden parameters associated with them. An amplitude correction operator is formed through a comparison of the direct and reflected waves just above a generating interface. Error grows as the distance above the generating interface at which the two events are compared grows. The formulation of the problem is in the plane-wave domain, but with a slight additional error the approach can be applied to data associated with fixed zero or nonzero offset. The method can be applied to generate scalar reflectivity values in acoustic/elastic environments, or phase and amplitude spectra of reflectivities in anacoustic/attenuative environments. A field data example from the Ross Lake heavy oil field in Saskatchewan, Canada illustrates the method. Our sense is that these results indicate applicability to more complex geometries, walkway or 3-D VSP surveys, assisting with the construction of AVO/AVA panels.

Since the classic paper by Gelfand and Levitan [Am. Math. Soc. Transl., Ser. 2, 1, 253–304 (1955)], much has been published on the inverse scattering problem. Assuming no bound states, there are several well-known solutions that reconstruct one-dimensional or spherically symmetric potentials from farfield scattering information. Moses [Phys. Rev. 102, 559–567 (1956)] and Newton [Phys. Rev. Lett. 4, 541 (1979)] have generalized these procedures for three-dimensional potentials. In this paper, we show how the work of Moses and Newton can be extended to provide a nearfield inverse scattering solution to the acoustic wave equation. In particular, we describe a procedure for obtaining a nearfield inverse scattering solution when the incident probes are arbitrary, unknown, and not necessarily reproducible.

We present a computationally efficient algorithm which combines the finite element method with Padé approximation. The combination is used to solve the problem of transverse magnetic and transverse electric scattering from homogeneous lossy dielectric cylinders over a continuous range of the complex permittivity variable by requiring the finite element solution only at a single complex permittivity value. The proposed method is based on (1) assuming a power series expansion for the unknown solution vector, the excitation vector, and the system matrix, (2) substituting this into the system matrix equation, and (3) finding the recursion relation for the solution vectors.

We simulate a total internal reflection tomography experiment in which an unknown object is illuminated by evanescent waves and the scattered field is detected along several directions. We propose a full-vectorial threedimensional nonlinear inversion scheme to retrieve the map of the permittivity of the object from the scattered far-field data. We study the role of the solid angle of illumination, the incident polarization, and the position of the prism interface on the resolution of the images. We compare our algorithm with a linear inversion scheme based on the renormalized Born approximation and stress the importance of multiple scattering in this particular configuration. We analyze the sensitivity to noise and point out that using incident propagative waves together with evanescent waves improves the robustness of the reconstruction.

Figure 3 Reconstructed relative dielectric permittivity for different values of the original permittivity. Comparison among the exact Ž . Ž . solution, the BA and the RA d s 1.6 . Noisy input data SrN s 20 dB Žwith a Gaussian noise with zero mean value and a signal-to-Ž . . noise ratio SrN corresponding to 20 dB ; see Figure . In this case, the reconstruction is unstable and strongly dependent on parameters. Nevertheless, the RA works better than the BA for a range of permittivity values of at least 1.7 F r F 2.8. Finally, for SrN F 10 dB, no significant advantages of the RA over the BA have been observed. The use of the RA is proposed for 3D imaging purposes. As applied to a numerical solution for the inverse-scattering problem in the space domain, it is proved to involve the same computation of the first-order BA, and it has been shown that, at least in simple cases, the range of applicability in Ž terms of the contrast is larger almost double, in the noisy . case for the RA than for the BA.

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et al [Stud. Appl. Math., 53 (1974) 249-315] and one by Hirota and Satsuma [J. Phys. Soc. Japan, 40 (1976) 611-612]. A catalogue of classical and nonclassical symmetry reductions, and a Painlevé analysis, are given. Of particular interest are families of solutions found containing a rich variety of qualitative behaviours. Indeed we exhibit and plot a wide variety of solutions all of which look like a two-soliton for t > 0 but differ radically for t < 0. These families arise as nonclassical symmetry reduction solutions and solutions found using the singular manifold method. This example shows that nonclassical symmetries and the singular manifold method do not, in general, yield the same solution set. We also obtain symmetry reductions of the shallow water equation solvable in terms of solutions of the first, third and fifth Painlevé equations. We give evidence that the variety of solutions found which exhibit "nonlinear superposition" is not an artefact of the equation being linearisable since the equation is solvable by inverse scattering. These solutions have important implications with regard to the numerical analysis for the shallow water equation we study, which would not be able to distinguish the solutions in an initial value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours.

This article examines the (2 + 1)-dimensional variable coefficient KdV-type equation that arises in oceanography. Oceanography is the science of studying the oceans. It is an earth science that addresses many different subjects, such as the dynamics of ecosystems, ocean currents, waves, and geophysical fluid dynamics. Fluid dynamics is a nonlinear science that studies the flow of liquids and gases in physics, physical chemistry, and engineering. The assistance of the Hirota bilinear method secures different forms of multiple solitons and M-lump waves, such as one-, two-, and three-soliton, by using different functions of variable coefficients. Moreover, the long-wave technique is under consideration when analyzing hybrid solutions, namely mixed soliton-lump, two-soliton-lump, and soliton-two-lump solutions. The results of this exploration exhibit the physical characteristics of solutions and collision-related aspects within the variety of nonlinear systems. The physical significance of the extracted solutions is meticulously analyzed by presenting a variety of profiles that display the behavior of the solutions for specific parameter values. Our results indicate their potential use in future endeavors to discover unique and assorted solutions for nonlinear evolution equations encountered in engineering and mathematical physics. The reported results in this study are the first of their kind to be reported in the literature. These results may be helpful in explaining the physical behavior of various nonlinear physical models. 1 | Introduction Mathematical modeling of complex time-varying processes requires examining several types of nonlinear partial differential equations. Since the mid-1800 s, experts have tried distilling complicated physical phenomena into nonlinear partial differential equations. Special modeling simulates many nonlinear aspects, including quantum mechanics, fluid mechanics, nonlinear optics, neural networks, and solid-state physics, to name a

We demonstrate an inverse scattering algorithm for reconstructing the structure of lossy fiber Bragg gratings. The algorithm enables us to extract the profiles of the refractive index and the loss coefficient along the grating from the grating transmission spectrum and from the reflection spectra, measured from both sides of the grating. Such an algorithm can be used to develop novel distributed evanescent-wave fiber Bragg sensors that measure the change in both the refractive index and the attenuation coefficient of the medium surrounding the grating. The algorithm can also be used to analyze and to design fiber Bragg gratings written in fiber amplifiers. A novel method to overcome instability problems in extracting the parameters of the lossy grating is introduced. The new method also makes it possible to reduce the spectral resolution needed to accurately extract the grating parameters.

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the form Under certain conditions the functions ε -2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and then the limit S 0 of S ε might be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ ′ .

This is the first in a series of papers on scattering theory for onedimensional Schrödinger operators with highly singular potentials q ∈ H -1 loc (R). In this paper, we study Miura potentials q associated to positive Schrödinger operators that admit a Riccati representation q = u ′ + u 2 for a unique u ∈ L 1 (R) ∩ L 2 (R). Such potentials have a well-defined reflection coefficient r(k) that satisfies |r(k)| < 1 and determines u uniquely. We show that the scattering map S : u → r is realanalytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.

We study one-dimensional Schrödinger operators S with real-valued dis-tributional potentials q in W −1 2,loc (R) and prove an extension of the Povzner-Wienholtz theorem on self-adjointness of bounded below S thus providing additional information on its domain. The results are further specified for q ∈ W −1 2,unif (R).

Solving Maxwell's equation and Schrodinger's equation is usually completed in the frequency domain. In the frequency domain, the two equations can be solved by Green's function method. Maxwell equation and Schrodinger equation can be simplified into Helmholtz equation, and then the Helmholtz equation is solved by Green's function method. Both Green's function methods require Sommerfeld radiation conditions. Sommerfeld radiation condition requires that the solution of the equation is equivalent to an attenuated plane wave on an infinite boundary. However, when we solve the Helmholtz equation, we often want to obtain the asymptotic behavior of the far field. The Sommerfeld radiation condition directly specifies the far-field asymptotic behavior of the field, rather than the asymptotic behavior obtained by calculation. In this way, the Sommerfeld radiation condition makes the process of solving differential equations unreasonable. If there is no other better methods, then we must use Sommerfeld radiation conditions. In addition, the Sommerfeld radiation condition is a strict frequency domain condition. As long as the frequency of the solution deviates from the frequency of the Green's function, the effectiveness of the solution is greatly reduced. The electromagnetic field signals we actually encounter are usually signals with specific frequency bands. Green's function is a function with a fixed frequency. In this way, the frequency of electromagnetic field signal may deviate from the frequency of Green's function, so it can not fully meet the Sommerfeld radiation condition. So we can't prove that we got the right solution. In this paper, Maxwell's equations and Schrodinger's equation are solved directly in time domain. A new method similar to Green's function method is introduced in time domain. In this method, the new Green's function and electromagnetic field adopt different waves. For example, if the electromagnetic signal is a retarded wave, the Green's function uses an advanced wave. If the electromagnetic signal is an advanced wave, the Green's function uses a retarded wave. Therefore, at the infinite boundary, the retarded wave and the advanced wave do not arrive at the same time, so they are not zero at the same time, so their inner product is zero. That is, the surface integral is zero, which avoids Sommerfeld and similar radiation conditions. Another advantage of this new method is that it is insensitive to frequency, which means that the effectiveness of the solution is guaranteed even if the frequency of the solution is different from that of the Green's function.

We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT -symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.

We consider canonical systems and investigate the Szegő class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory gem in the sense proposed by Barry Simon.

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuous (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [-2, 2]. Moreover, Killip and Simon described explicitly the spectral properties for such A + B. Jointly with Damanik they generalized this result to the case of perturbations of periodic Jacobi matrices in the non-degenerated case. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko posed in a review dedicated to F. Gesztesy (2013) the following question: "is there an extension of the Damanik-Killip-Simon theorem to the general finite system of intervals case?" In this paper we solve this problem completely. Our method deals with the Jacobi flow on GMP matrices. GMP matrices are probably a new object in the spectral theory. They form a certain Generalization of matrices related to the strong Moment Problem, the latter ones are a very close relative of Jacobi and CMV matrices. The Jacobi flow on them is also a probably new member of the rich family of integrable systems. Finally, related to Jacobi matrices of Killip-Simon class, analytic vector bundles and their curvature play a certain role in our construction and, at least on the level of ideology, this role is quite essential.

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title.

An exact time-domain method is proposed to time reverse a transient scalar wave using only the field measured on an arbitrary closed surface enclosing the initial source. Under certain conditions, a timereversed field can be approximated by retransmitting the measured signals in a reversed temporal order. Exact reconstruction for three-dimensional broadband diffraction tomography (a linearized inverse scattering problem) is proposed by time-reversing the measured field back to the time when each secondary source is excited. The algorithm is verified by a numerical simulation. Extension to the case using Green's function in a heterogeneous medium is discussed.

The following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from global EM scattering measurements, at various incidences and wave frequencies. This large scale ill-posed inverse problem is explored by an intensive exploitation of an efficient 2D Maxwell solver, distributed on high performance computing machines. Applied to a large training data set, a statistical analysis reduces the problem to a simpler probabilistic metamodel, on which Bayesian inference can be performed. Considering the radioelectric properties as a hidden dynamic stochastic process, that evolves in function of the frequency, it is shown how advanced Markov Chain Monte Carlo methods, called Sequential Monte Carlo (SMC) or interacting particles, can take benefit of the structure and provide local EM property estimates.

This paper applies the minimum gradient method (MGM) to denoise signals in engineering problems. The MGM is a novel technique based on the complexity control, which defines the learning as a bi-objective problem in such a way to find the best trade-off between the empirical risk and the machine complexity. A neural network trained with this method can be used to pre-process data aiming at increasing the signal-to-noise ratio (SNR). After training, the neural network behaves as an adaptive filter which minimizes the cross-validation error. By applying the general singular value decomposition (GSVD), we show the relation between the proposed approach and the Wiener filter. Some results are presented, including a toy example and two complex engineering problems, which prove the effectiveness of the proposed approach.

The large N limit of fermionic vectors models is studied using bilocal variables, in the framework of a collective field theory approach. The large N configuration is determined completely using only classical solutions of the model. Further, the Bethe-Salpeter equations of the model are cast as a Green's function problem. One of the main results of this work is to show that this Green's function is in fact the large N bilocal itself.

The large-N limit of fermionic vectors models is studied using bilocal variables in the framework of a collective field theory approach. The large-N configuration is determined completely using only classical solutions of the model. Further, the Bethe–Salpeter equations of the model are cast as a Green&#39;s function problem. One of the main results of this work is to show that this Green&#39;s function is in fact the large-N bilocal itself.

We derive a sharp upper bound for the first eigenvalue λ1,p of the p-Laplacian on asymptotically hyperbolic manifolds for 1 < p < ∞. We then prove that asymptotically CMC submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold Y k+1 within H n+1 (-1), λ1,p(Y ) = k p p . We then obtain lower bounds on λ1,2(Y ) in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant βY for each such submanifold, enabling us to generalize a result due to Cheung-Leung in .

in 2003. Currently, he is a researcher and PhD student in the Department of Civil Engineering of the University of Minho, Guimar ã es, Portugal. His main research areas are the inspection and diagnosis of concrete and masonry structures using GPR and sonic techniques. He is an active member of the ' Sustainable Bridges ' European Project.

The Sturm spirals which can be introduced as those plane curves whose curvature radius is equal to the distance from the origin are embedded into oneparameter family of curves. Explicit parametrization of the ordinary Sturmian spirals along with that of a wider family of curves are found and depicted graphically.

In this paper, we study the inverse scattering problem for energydependent Schrödinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions, we transform this scattering problem to the one for non-canonical Dirac systems and show that, in turn, the latter can be placed within the known scattering theory for ZS-AKNS systems. This allows us to give a complete description of the corresponding scattering functions S for the class of problems under consideration and justify an algorithm of reconstructing the problem from S.

This is the first in a series of papers on scattering theory for onedimensional Schrödinger operators with highly singular potentials q ∈ H -1 loc (R). In this paper, we study Miura potentials q associated to positive Schrödinger operators that admit a Riccati representation q = u ′ + u 2 for a unique u ∈ L 1 (R) ∩ L 2 (R). Such potentials have a well-defined reflection coefficient r(k) that satisfies |r(k)| < 1 and determines u uniquely. We show that the scattering map S : u → r is realanalytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.

This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u ′ + u 2 for some u ∈ L 2 (R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L 1 `R, (1 + |x|)dx ´generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.

We study the Yangian symmetry of the multicomponent Quantum Nonlinear Schrödinger hierarchy in the framework of the Quantum Inverse Scattering Method. We give an explicit realization of the Yangian generators in terms of the deformed oscillators algebra which naturally occurs in this framework. Dans le cadre de la méthode de diffusion quantique inverse, nous étudions le yangien comme symétrie de la hiérarchie quantique de Schrödinger à plusieurs composantes. Nous donnons une réalisation explicite des générateurs du yangien en fonction de l'algèbre déformée d'oscillateurs qui apparaît naturellement dans cette approche.

In this article we study the presence of multiple critical points in the usual topology optimization formulation of the classical inverse scattering problem. We consider a very simple example of the two-dimensional problem: the scatterer is a disk composed of a homogeneous material that is within a homogeneous medium of different material properties. The example considers the most favorable conditions for solving the inverse problem: measurements are taken on a whole circle surrounding the scatterer, for incident plane waves coming from all directions. In addition, these measurements have no error. However, in the case of medium frequencies (wavelengths from one-third to one scatterer diameter), we show that multiple critical points exist when we consider the optimality criteria provided by the shape gradient and also when we consider the optimality criteria provided by the topological derivative of the usual objective functional of the problem. These critical points have the same radial symmetry as the circular scatterer, which allows the application of a very accurate semi-analytical method based on series expansions to obtain the solutions of the forward and adjoint problems, as well as to compute the shape and topological derivatives. The critical points are obtained by solving the nonlinear system of equations that provide the optimality conditions, and are reported for different values of the material properties that define the inverse problem.

Reconstruction of inhomogeneous dielectric objects from microwave scattering is a nonlinear and ill-posed inverse problem. In this paper, we develop a new class of weakly convex discontinuity adaptive (WCDA) models as a regularization for quantitative microwave tomography. We show that this class includes the Huber regularizer and we show how to combine these methods with electromagnetic solvers operating on the complex permittivity profile. 2D reconstructions of objects from the Institute Fresnel database and experimental data at a single frequency demonstrate the effectiveness of the proposed regularization even when employing far less transmitters and receivers than available in the database.

In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schrödinger equation to prove local uniqueness results in the setting of inverse metric problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients T (λ, n) and reflection coefficients L(λ, n) and R(λ, n) of a Dirac wave having a fixed energy λ and angular momentum n. For instance, the reflection coefficients L(λ, n) correspond to the scattering experiment in which a wave is sent from the left end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed λ = 0, the knowledge of the reflection coefficients L(λ, n) (resp. R(λ, n)) -up to a precise error term of the form O(e -2nB ) with B > 0 -determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude B of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.

In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) -that the Dirac equation can be separated into radial and angular ordinary differential equations -to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitely by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on R n , n ≥ 2. First, we consider the class A of potentials q(r) which can be extended analytically in 2 . If q and q are two such potentials and if the corresponding phase shifts δ l and δl are super-exponentially close, then q = q. Secondly, we study the class of potentials q(r) which can be split into q(r) = q1(r) + q2(r) such that q1(r) has compact support and q2(r) ∈ A. If q and q are two such potentials, we show that for any fixed a > 0, when l → +∞ if and only if q(r) = q(r) for almost all r ≥ a. The proofs are close in spirit with the celebrated Borg-Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in ℜz ≥ 0 with | q(z) |≤ C (1+ | z |) -ρ , ρ > 1 , we show that the Regge poles are confined in a vertical strip in the complex plane.

In this paper, we study inverse scattering of massless Dirac fields that propagate in the exterior region of a Reissner-Nordström black hole. Using a stationary approach we determine precisely the leading terms of the high-energy asymptotic expansion of the scattering matrix that, in turn, permit us to recover uniquely the mass of the black hole and its charge up to a sign.

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter-Reissner-Nordström black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ = 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ = 0 is the fixed energy and n ∈ N * denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M , the square of the charge Q 2 and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T (λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ∈ L where L is a subset of N * that satisfies the Müntz condition n∈L 1 n = +∞. Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the "unphysical" scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities 1 T (λ,z) , R(λ,z) T (λ,z) and L(λ,z) T (λ,z) belong to the Nevanlinna class in the region {z ∈ C, Re(z) > 0} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of 2 × 2 matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse

In this paper a new parameterization method for logarithmic image processing (LIP) model is presented. This method is based on nonlinear transformation with using bijection. The use of this method provide means for construction of different logarithmic type algebras for image processing.

A genetic programming-based geometry optimization method for inverse scattering that uses a tree data structure to encode Boolean combinations of convex shapes has recently been shown to outperform other genetic algorithm-based techniques. Nonetheless, the genetic algorithms are still inefficient for inverse scattering. This letter treats the inefficiency with local search techniques. First, an implementation using greedy search is discussed. Next, a local search method is added to the genetic algorithm used in the previous approach. Numerical results show that the number of function evaluations necessary to image conducting cylinders can be reduced by several orders of magnitude.

The inverse problem under consideration is to reconstruct the characteristic of scatterer from the scattering E field. Dynamic differential evolution (DDE) and selfadaptive dynamic differential evolution (SADDE) are stochastic-type optimization approach that aims to minimize a cost function between measurements and computer-simulated data. These algorithms are capable of retrieving the location, shape, and permittivity of the dielectric cylinder in a slab medium made of lossless materials. The finite-difference timedomain (FDTD) is employed for the analysis of the forward scattering. The comparison is carried out under the same conditions of initial population of candidate solutions and number of iterations. Numerical results indicate that SADDE outperforms the DDE a little in terms of reconstruction accuracy.

We have used coherent X-ray diffraction experiments to characterize both the 1-D and 2-D foci produced by nanofocusing Kirkpatrick-Baez (K-B) mirrors, and we find agreement. Algorithms related to ptychography were used to obtain a 3-D reconstruction of a focused hard X-ray beam waist, using data measured when the mirrors were not optimally aligned. Considerable astigmatism was evident in the reconstructed complex wavefield. Comparing the reconstructed wavefield for a single mirror with a geometrical projection of the wavefront errors expected from optical metrology data allowed us to diagnose a 40 μrad misalignment in the incident angle of the first mirror, which had occurred during the experiment. Good agreement between the reconstructed wavefront obtained from the Xray data and off-line metrology data obtained with visible light demonstrates the usefulness of the technique as a metrology and alignment tool for nanofocusing X-ray optics.

We are concerned with the numerical solution of linear parameter identification problems for parabolic PDE, written as an operator equation $Ku=f$. The target object $u$ is assumed to have a sparse expansion with respect to a wavelet system $Psi={psi_lambda}$ in space-time, being equivalent to a priori information on the regularity of $u=mathbf u^ opPsi$ in a certain scale of Besov spaces $B^s_{p,p}$. For the recovery of the unknown coefficient array $mathbf u$, we miminize a Tikhonov-type functional begin{equation*} min_{mathbf u}|Kmathbf u^ opPsi-f^delta|^2+alphasum_{lambda}omega_lambda|u_lambda|^p end{equation*} by an associated thresholded Landweber algorithm, $f^delta$ being a noisy version of $f$. Since any application of the forward operator $K$ and its adjoint involves the numerical solution of a PDE, perturbed versions of the iteration have to be studied. In particular, for reasons of efficiency, adaptive applications of $K$ and $K^*$ are indispensable cite{Ra07}. By a suit...