This is a continuation of Hu and Kirsch’s previous work (2024) on well-posedness of time-harmonic scattering by locally perturbed periodic curves of Dirichlet kind. The scattering interface is supposed to be given by a non-self-intersecting Lipschitz curve. We study properties of Green’s function and prove new well-posedness results for scattering of plane waves at a propagative number. In such a case, there exist guided waves to the unperturbed problem, which are also known as bound states in the continuum (BICs) in physics. In this paper, the uniqueness of the forward scattering follows from an orthogonal constraint condition enforced on the total field to the unperturbed scattering problem. This constraint condition, which is also valid under the Neumann boundary condition, is derived from the singular perturbation arguments and also from the approach of approximating a plane wave by point source waves. For the inverse problem of determining the defect, we prove several uniqueness results using a finite or infinite number of point sources and plane waves, depending on whether a priori information on the size and height of the defect is available.

The Rayleigh expansion is widely used as a formal radiation condition in the analysis and numerical treatment of grating diffraction problems for incoming plane waves. However, the Rayleigh expansion does not always lead to uniqueness of open waveguide scattering problems, due to the existence of surface/guided waves (in other words, Bound States in the Continuum (BICs)) which exponentially decay in the direction perpendicular to the periodicity. In this paper we suppose that a bi-periodic inhomogeneous medium supports BICs at some real-valued incident wavenumber. Based on singular perturbation arguments, we justify the Limiting Absorption Principle (LAP) for both time-harmonic acoustic and electromagnetic scattering of plane waves from bi-periodic structures. Replacing the wavenumber k with k+iε, we prove that the unique solution with ε>0 converges to a solution of the original diffraction problem that additionally satisfies an orthogonal identity. This constraint condition together with the classical Rayleigh expansion leads to a sharp radiation condition to ensure uniqueness of time-harmonic scattering of plane waves by BIC-supporting bi-periodic materials.

The perfectly matched layers method is a well known truncation technique for its efficiency and convenience in numerical implementations of wave scattering problems in unbounded domains. In this paper, we study the convergence of the perfectly matched layers (PML) for wave scattering from a local perturbation of an open waveguide in the half space above the real line, where the refractive index is a function which is periodic along the axis of the waveguide and equals to one above a finite height. The problem is challenging due to the existence of guided waves, and a typical way to deal with the difficulty is to apply the limiting absorption principle. Based on the Floquet-Bloch transform and a curve deformation theory, the solution from the limiting absorption principle is rewritten as the integral of a coupled family of quasi-periodic problems with respect to the quasi-periodicity parameter on a particularly designed curve. By comparing the Dirichlet-to-Neumann maps on a straight line above the locally perturbed periodic layer, we finally show that the PML method converges exponentially with respect to the PML parameter. Finally, the numerical examples are shown to illustrate the theoretical results.

The paper is concerned with well-posedness of TE and TM polarizations of time-harmonic electromagnetic scattering by perfectly conducting periodic surfaces and periodically arrayed obstacles with local perturbations. The classical Rayleigh Expansion radiation condition does not always lead to well-posedness of the Helmholtz equation even in unperturbed periodic structures. We propose two equivalent radiation conditions to characterize the radiating behavior of time-harmonic wave fields incited by a source term in an open waveguide under impenetrable boundary conditions. With these open waveguide radiation conditions, uniqueness and existence of time-harmonic scattering by incoming point source waves, plane waves and surface waves from locally perturbed periodic structures are established under either the Dirichlet or Neumann boundary condition. A DtN operator without using the Green's function is constructed for proving well-posedness of perturbed scattering problems.

We study the Helmholtz equation with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and solve the radiation problem in a self‐contained exposition. In this context, we simplify the non‐degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied, and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.

We consider the scattering of a plane wave by a locally perturbed periodic (with respect to x_1) medium. If there is no perturbation it is usually assumed that the scattered wave is quasi-periodic with the same parameter as the incident plane wave. As it is well known, one can show existence under this condition but not necessarily uniqueness. Uniqueness fails for certain incident directions (if the wavenumber is kept fixed), and it is not clear which additional condition has to be assumed in this case. In this paper we will analyze three concepts. For the Limiting Absorption Principle (LAP) we replace the refractive index n=n(x) by n(x)+iε in a layer of finite width and consider the limiting case when ε tends to zero. This will give an unsatisfactory condition. In a second approach we require continuity of the field with respect to the incident direction. This will give the same satisfactory condition as the third approach where we approximate the incident plane wave by an incident point source and let the location of the source tend to infinity.

In this paper, we consider the propagation of waves in an open waveguide in where the index of refraction is a local perturbation of a function which is periodic along the axis of the waveguide (which we choose to be the x1 axis) and equal to one for |x2| > h0 for some h0 > 0. Motivated by the limiting absorption principle (proven in an earlier paper by the author), we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution under the assumption that no bound states exist. In the second part, we determine the order of decay of the radiating part of the solution in the direction of the layer and in the direction orthogonal to it. Finally, we show that it satisfies the classical Sommerfeld radiation condition and allows the definition of a far field pattern.

In this chapter, we study a different approach to regularizing operator equations of the form Kx = y, where x and y are elements of certain function spaces. This approach is motivated by the fact that for the numerical treatment of such equations one has to discretize the continuous problem and reduce it to a finite system of (linear or nonlinear) equations. We see in this chapter that the discretization schemes themselves are regularization strategies in the sense of Chap. 2.

In the previous chapters, we considered linear problems which we wrote as \(Kx=y\), where K was a linear and (often) compact operator between Hilbert spaces. Needless to say that most problems in applications are nonlinear. For example, even in the case of a linear differential equation of the form \(-u^{\prime \prime }+cu=f\) for the function u the dependence of u on the parameter function c is nonlinear; that is, the mapping \(c\mapsto u\) is nonlinear. In Chapters 5, 6, and 7 we will study particular nonlinear problems to determine parameters of an ordinary or partial differential equation from the knowledge of the solution.

This second part of a pair of papers complements the first part (see Kirsch 2018 ( 35 104004)) but can be read independently. Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. As in the first part we consider a medium described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We formulate a proper radiation condition which allows the existence of traveling modes (and is motivated by the limiting absorption principle proven in the first part) and prove uniqueness and existence.

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter . We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair . This abstract result allows us to derive explicitly a representation for the limiting absorption solution as a sum of a decaying part (along the axis of the cylinder) and a finite sum of propagating modes.

In this follow-up of [A. Kirsch and A. Rieder, Inverse Problems, 32 (2016), 085001] we generalize our previous abstract results so that they can be applied to the viscoelastic wave equation which serves as a forward model for full waveform inversion (FWI) in seismic imaging including dispersion and attenuation. FWI is the nonlinear inverse problem of identifying parameter functions of the viscoelastic wave equation from measurements of the reflected wave field. Here we rigorously derive rather explicit analytic expressions for the Fréchet derivative and its adjoint (adjoint state method) of the underlying parameter-to-solution map. These quantities enter crucially Newton-like gradient decent solvers for FWI. Moreover, we provide the second Fréchet derivative and a related adjoint as ingredients to second-degree solvers.

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These nonzero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply nonuniqueness of any solution to the scattering problem. In this paper, we consider a medium that is defined in the upper two-dimensional half-space by a penetrable and periodic contrast. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. Our method of proof seems to be new: By the Floquet-Bloch transform we first reduce the scattering problem to a finite-dimensional one that is set in the linear space spanned by all surface waves. In this space, we then compute explicitly which modes propagate along the periodic structure to the left or to the right. This finally yields a representation for our limiting absorption solution which leads to a proper extension of the well known upward propagating radiation condition. Finally, we prove the uniqueness of a solution under this radiation condition.

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.

This workshop addressed specific inverse problems for the time-harmonic Maxwell's equations, resp. special cases of these, such as the Helmholtz equation or quasistatic approximations like in impedance tomography. The inverse problems considered include the reconstruction of obstacles and/or their material properties in a known background, given various kinds of data, such as near or far field measurements in the scattering context and boundary measurements in the quasistatic case.

In this paper, we compare the far-field operators for the full nonlinear inverse scattering problem with the Born approximation as its linearization. The Factorization Method shows that both operators share the same behavior with respect to illposedness of the inverse problem. The results are derived for acoustic and electromagnetic scattering problems and the corresponding problem in electrical impedance tomography.