A linear sampling method for inverse problems of diffraction gratings of mixed type

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.     

Uniqueness results in the inverse scattering problem for periodic structures

This paper is concerned with the inverse electromagnetic scattering by a 2D (impenetrable or penetrable) smooth periodic curve. Precisely, we establish global uniqueness results on the inverse problem of determining the grating profile from the scattered fields corresponding to a countably infinite number of quasiperiodic incident waves. For the case of an impenetrable and partially coated perfectly reflecting grating, we prove that the grating profile and its physical property can be uniquely determined from the scattered field measured above the periodic structure. For the case of a penetrable grating, we show that the periodic interface can be uniquely recovered by the scattered field measured only above the interface. A key ingredient in our proofs is a novel mixed reciprocity relation that is derived in this paper for the periodic structures and seems to be new. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Secant method under generalized Lipschitz conditions for the divided difference operator

In this work we introduce for the first time the generalized Lipschitz conditions for the divided difference operator. A positive integrable function, partial case of which is usual Lipschitz constant, is suggested. Under the given conditions the convergence of the Secant method for solving the operator equations in Banach spaces is investigated, the uniqueness ball for the solution is obtained. As a partial case the known results for the Lipschitz constants are received. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

Asymptotic behaviour of non-linear parabolic equations with monotone principal part

We consider non-linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non-Lipschitz perturbations. The case of globally Lipschitz perturbations of a maximal monotone operator has been addressed in Boll. Un. Mat. Ital. B (8) 2 (2000) 693–706. In the case of perturbations which are not globally Lipschitz, the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors. We overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed. The results developed here are applied to the case when the subdifferential operator is the p-Laplacian to obtain existence of attractors and the existence of periodic solutions.     

An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

An a priori error estimate using a so called α,β‐ periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β‐quasi periodic solution are used to transform the associated Helmholtz problem. The well‐posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β‐quasi periodic method converges at a far lower number of degrees of freedom than the α,0‐quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer. Copyright © 2012 John Wiley & Sons, Ltd.     

The numerical solution of an inverse periodic transmission problem

We consider the inverse problem of recovering a 2D periodic structure from scattered waves measured above and below the structure. We discuss convergence and implementation of an optimization method for solving the inverse TE transmission problem, following an approach first developed by Kirsch and Kress for acoustic obstacle scattering. The convergence analysis includes the case of Lipschitz grating profiles and relies on variational methods and solvability properties of periodic boundary integral equations. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

An acoustic scattering problem for periodic,inhomogeneous media

We study the problem of the scattering by a periodic, inhomogeneous, penetrable medium. Using the Dirichlet-to-Neumann operator from the classical formulation of the problem we derive a variational equation and give regularity result to show the equivalence of both formulations. We present certain uniqueness results, which by the Fredholm alternative yield existence of the solution and its continuous dependence on the incoming wave. We prove existence of a solution for special incident waves even if there is no uniqueness. A result about analytical dependence of the solution on the wave number and the incident angle is given. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

An Inverse Problem in Periodic Diffractive Optics: Reconstruction of Lipschitz Grating Profiles

We consider the problem of recovering a two-dimensional periodic structure from scattered waves measured above the structure. Following an approach by Kirsch and Kress, this inverse problem is reformulated as a nonlinear optimization problem. We develop a theoretical basis for the reconstruction method in the case of an arbitrary Lipschitz grating profile. The convergence analysis is based on new perturbation and stability results for the forward problem.     

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L²(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star‐shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star‐combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second‐kind integral operator for which convergence of the Galerkin method in L²(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star‐combined operator implies frequency‐explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high‐frequency case. The proof of coercivity of the star‐combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.     

On the existence and the uniqueness of the solution of a fluid–structure interaction scattering problem

The existence and uniqueness of the solution of a fluid–structure interaction problem is investigated. The proposed analysis distinguishes itself from previous studies by employing a weighted Sobolev space framework, the DtN operator properties, and the Fredholm theory. The proposed approach allows to extend the range of validity of the standard existence and uniqueness results to the case where the elastic scatterer is assumed to be only Lipschitz continuous, which is of more practical interest.     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

We first study the Massera problem for the existence of a τ?periodic solution for some nondensely defined partial differential equation, where the autonomous linear part satisfies the Hille‐Yosida condition and the delayed nonlinear part satisfies a locally Lipschitz condition. Second, inspired by an existing study, we prove in the dichotomic case, for τ=1, the existence‐uniqueness and conditional stability of the periodic solution. Moreover, we show the existence of a local stable manifold around such solution. Our theoretical results are finally illustrated by an application.     

Scattering Theory for CMV Matrices: Uniqueness,Helson–Szegő and Strong Szegő Theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.     

Uniqueness in the inverse transmission scattering problem for periodic media

We study the problem of the scattering by a periodic, penetrable medium. We present certain uniqueness results and give the integral equation formulation of the transmission problem which is of Fredholm type and provides the existence and continuous dependence result. Next we investigate the question of the uniqueness for the inverse transmission problem, i.e. we concentrate on the amount of information that is necessary to completely determine the profile and constitutive parameters of the scattering grating. Copyright © 1999 John Wiley & Sons, Ltd.     

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave. As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.     

Aspects of electromagnetic scattering in chiral media

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon‐type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon‐type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far‐field operator that is defined on a unit sphere and contains the far‐field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A Stirling-like method with Hölder continuous first derivative in Banach spaces

In this paper, the convergence of a Stirling-like method used for finding a solution for a nonlinear operator in a Banach space is examined under the relaxed assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. Many results exist already in the literature to cover the stronger case when the second Fréchet derivative of the involved operator satisfies the Lipschitz/Hölder continuity condition. Our convergence analysis is done by using recurrence relations. The error bounds and the existence and uniqueness regions for the solution are obtained. Finally, two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points.     

A C0 interior penalty discontinuous Galerkin Method for fourth‐order total variation flow. II: Existence and uniqueness

We prove the existence and uniqueness of a solution of a C⁰ Interior Penalty Discontinuous Galerkin (C⁰ IPDG) method for the numerical solution of a fourth‐order total variation flow problem that has been developed in part I of the paper. The proof relies on a nonlinear version of the Lax‐Milgram Lemma. It requires to establish that the nonlinear operator associated with the C⁰ IPDG approximation is Lipschitz continuous and strongly monotone on bounded sets of the underlying finite element space.     

Volume integral equations for scattering from anisotropic diffraction gratings

We analyze electromagnetic scattering of transverse magnetic polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative index dielectric material. Such scattering problems are important for the modelization of, for example, light propagation in nano‐optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integral equation, a technique that attracts continuous interest in the engineering community but has rarely received rigorous theoretic treatment. In this paper, we prove new (generalized) Gårding inequalities in weighted and unweighted Sobolev spaces for the strongly singular integral equation. These inequalities also hold for materials for which the real part of the material parameter takes negative values inside the diffraction grating, independently of the value of the imaginary part. Copyright © 2012 John Wiley & Sons, Ltd.