A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.FC

Looking back on inverse scattering theory

We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these events

On the numerical solution of the three-dimensional inverse obstacle scattering problem

AbstractThe inverse problem under consideration is to determine the shape of an impenetrable sound-soft obstacle from the knowledge of a time-harmonic incident plane acoustic wave and far-field or near-field measurements of the scattered wave. We present a method for the approximate solution which avoids the solution of the corresponding direct problem and stabilizes the ill-posed inverse problem by reformulating it as a nonlinear optimization problem. The numerical implementation of the method is described and some three-dimensional examples of reconstructions are given

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.Fundação Calouste Gulbenkia

Numerical methods in inverse obstacle scattering

We consider the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic plane waves. This is a model problem for applications in radar, sonar, geophysical exploration, medical imaging and nondestructive testing. It is difficult to solve, since it is nonlinear and extremely ill-posed. Following the historical development over the last fifteen years, in this lecture we shall describe the main ideas of three different methods for the approximate numerical solution of the inverse obstacle scattering problem that acknowledge its nonlinearity and ill-posedness

Iterative and range test methods for an inverse source problem for acoustic waves

We propose two methods for solving an inverse source problem for time-harmonic acoustic waves. Based on the reciprocity gap principle a nonlinear equation is presented for the locations and intensities of the point sources that can be solved via Newton iterations. To provide an initial guess for this iteration we suggest a range test algorithm for approximating the source locations. We give a mathematical foundation for the range test and exhibit its feasibility in connection with the iteration method by some numerical examples.FCTFundação Calouste Gulbenkia

An inverse problem related to an arbitrarily shaped impedance cylinder buried in a dielectric cylinder

Ters saçılma teorisinde yanına yaklaşılamayan cisimlerin fiziksel özelliklerinin belirlenmesi önemli bir araştırma konusu olarak bilinmektedir. Bu çalışmada, iki boyutta keyfi bir şekle sahip dielektrik silindir içerisine gömülü keyfi şekilli bir cismin üzerinde tanımlanan inhomojen empedans fonksiyonunun saçılan elektromagnetik alan verisinden yararlanılarak bulunması için sınır integral denklemlerin çözümüne dayanan bir metod önerilmiştir. İntegral denklemler bir potansiyel yaklaşımı altında türetilmiştir. Bu bağlamda saçılan ve toplam alanlar, içerisinde kaynak (yoğunluk) ve Hankel fonksiyonları içeren integral gösterilimler şeklinde tek-katman potansiyelleri ve bunların kombinasyonları kullanılarak ifade edilmiştir. Fiziksel olarak tek-katman potansiyelleri üzerinde monopollerin bulunduğu bir katmana karşı gelmekle beraber homojen Helmholtz dalga denklemini ve Sommerfeld radyasyon koşulunu sağlar. Problemin çözümünde dielektrik silindirin içerisindeki ve dışarısındaki aynı türden magnetik özellik gösteren basit ortamlara ait dalga sayılarının, silindirlerin şekillerinin ve yakın/uzak saçılan alan verisinin bilindiği varsayılmıştır. Empedans fonksiyonunun bulunabilmesi için her bir bölgedeki elektromagnetik alanların hesabına yönelik yeni bir algoritma sunulmuştur. Saçılan alandan saçıcıların üzerinde tanımlı yoğunluk fonksiyonlarının kararlı ve yaklaşık çözümlerini elde edebilmek amacıyla kötü-kurulmuş integral denklemler Tikhonov anlamında değerlendirilmiştir. Yoğunluk fonksiyonlarının bulunması,  dielektrik silindir içerisindeki toplam alanı hesaplanabilir kılmış ve uygun sıçrama koşulları altında empedans fonksiyonu empedans sınır koşulundan en küçük kareler yöntemi ile elde edilmiştir. Önerilen metodun uygulanabilirliği ve etkinliği nümerik deneylerle test edilmiş ve başarılı sonuçlar elde edilmiştir. Anahtar Kelimeler: Ters saçılma, empedans sınır koşulu, sınır integral denklemleri. Main research areas of inverse scattering theory is the reconstructions of geometrical (location and shape) and / or physical (dielectric permittivity, conductivity, impedance etc.) properties of inaccessible obstacles from the knowledge of the scattered waves (electromagnetic, acoustic, elastic etc.) at large distance. Radar / sonar applications, medical tomography, geophysical exploration and non-destructive testing lead to this type of problems. Motivated by the applications researchers proposed different types of solution methods for the mentioned problems especially after the II.World War. Impedance boundary condition is used to simplify scattering problems involving complex structures. Such that in electromagnetics, imperfectly conducting scatterers, perfectly conducting objects with a penetrable or absorbing boundary layer can be modeled by an impedance boundary condition. In this context, one can study with a simpler model for complex typed structures. The aim of the direct scattering problem for an impedance cylinder is to obtain scattered near- or far-field data for given shape of the cylinder, the impedance function and the wave number of the background medium in the case of an electromagnetic wave illumination. However in the inverse problem case one recovers the impedance function defined on the cylinder from the knowledge of scattered field, shape of cylinder and wave number of the host medium. In this study, we considered an inverse scattering problem for arbitrarily shaped cylindrical objects that have inhomogeneous impedance boundaries and are buried in arbitrarily shaped cylindrical dielectrics. This consideration is realistic, since the problem will have many possible practical applications. For example, in the non-destructive testing of a coating on a wire; the coating is characterized by an inhomogeneous lossy cylinder layer and the conducting wire is modeled by an inhomogeneous surface impedance, or in biomedical applications; the bone of the arm can be modeled in terms of an impedance boundary condition while the muscular structure over it, can be considered as an inhomogeneous lossy cylindrical layer. For the sake of brevity we assume cylinders are infinitely long and illuminated by a TM polarized electromagnetic wave whose electric field vector is always parallel to -axis. Due to the symmetry and homogeneity along the -axis the total electric field vector will be polarized both inside and outside of the cylinder parallel to the -axis. Then the problem is reduced to a scalar one in terms of total fields that have to satisfy homogeneous Helmholtz equation. In order to determine inhomogeneous impedance function we have to reconstruct the field occuring in the interior domain of the dielectric cylinder. Therefore since layer potentials are the solutions of homogeneous Helmholtz equation and they satisfy Sommerfeld radiation condition we use a potential approach to represent fields in every domain via single-layer potentials. Then the far-field expression can be obtained from the asymptotic representation of the scattered field. We note that one can use these representations under proper assumptions for the wave numbers. Roughly speaking, layer potentials are the integrals defined over the boundary of the scatterers which contain Hankel and density (source) functions. In our problem we define two density functions on the boundary of the exterior cylinder and one density function on the buried obstacle. Afterwards, one of the densities on the exterior cylinder can be reconstructed from the solution of an ill-posed far-field equation with measured far field pattern as a data via Tikhonov regularization. The rest two unknown densities are found from the integral equation system obtained by using dielectric (transmission)conditions which ensure the continuity of the fields and their normal derivatives across the boundary of the exterior cylinder. However the compactness of the operators in the system expresses its ill-posedness. Therefore to obtain a stable solution of the system for the densities we apply Tikhonov regularization. Now one can read off the values of the impedance function by substituting interior total field values to the standard impedance boundary condition using jump relations. However, the reconstruction of the impedance function will be sensitive to errors. In order to obtain stable reconstructions we express the unknown impedance function in terms of basis functions and apply least squares approximation. Furthermore, we test the applicability and the effectiveness of our inversion method with noisy data and obtain satisfactory numerical results as illustrated in the last section of the paper. Keywords: Inverse scattering, impedance boundary condition, boundary integral equations

Local estimates for entropy densities in coupled map lattices

We present a method to derive an upper bound for the entropy density of coupled map lattices with local interactions from local observations. To do this, we use an embedding technique being a combination of time delay and spatial embedding. This embedding allows us to identify the local character of the equations of motion. Based on this method we present an approximate estimate of the entropy density by the correlation integral.Comment: 4 pages, 5 figures include