A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation

AbstractThis paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method

Direct Imaging Methods for Inverse Obstacle Scattering

Direct imaging methods recover the presence, position, and shape of the unknown obstacles in time-harmonic inverse scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer, i.e., on the boundary condition. However, most of these methods require multi-static data and only obtain partial information about the obstacle. These qualitative methods are based on constructing indicator functions defined on the domain of interest, which help determine whether a spatial point or point source lies inside or outside the scatterer. This paper explains the main themes of each of these methods, with emphasis on highlighting the advantages and limitations of each scheme. Additionally, we will classify each method and describe how some of these methods are closely related to each other.Comment: 39 page

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

Point Source Approximation Methods in Inverse Obstacle Reconstruction Problems

Wir untersuchen verschiedene Punktquellenverfahren zur Lösung inverser Objektrekonstruktionsprobleme für die Laplace- und Helmholtz-Gleichung. Dabei stellen wir einen Zweischritt-Algorithmus vor, der durch eine geeignete Wahl von Approximationsgebieten zunächst die Umgebung des Objekts rekonstruiert. In einem zweiten Schritt wird durch Adaption des Approximationsgebietes das unbekannte Gebiet selbst rekonstruiert. Wir formulieren den Zweischrittalgorithmus für die Punktquellenmethode, die Methode singulärer Quellen und die Probe Methode. Hierbei zeigen wir Rekonstruktionsergebnisse für die Laplace und die Helmholtzgleichung in zwei bzw. drei Dimensionen. Schließlich vergleichen wir die Punktquellenverfahren mit der Faktorisierungs- und der Linear Sampling Methode sowohl für exakte als auch für fehlerbehaftete Daten.We consider point source approximation methods for the solution of inverse object reconstruction problems for the Laplace and the Helmholtz equation. We present a two-step algorithm to reconstruct the neighbourhood of the obstacle by a proper choice of approximation domains, first. Then we reconstruct the obstacle in the second step of the algorithm by varying the approximation domains adaptively. We formulate this two-step algorithm for the point source method, the singular sources method and the probe method. Moreover we show numerical examples both for the Laplace and the Helmholtz equation in two and three dimensions, respectively. Finally, we compare the point source approximation methods with the factorisation and the linear sampling method both for exact data and noisy data

DIRECT SAMPLING METHODS FOR INVERSE SCATTERING PROBLEMS

Recently, direct sampling methods became popular for solving inverse scattering problems to estimate the shape of the scattering object. They provide a simple tool to directly reconstruct the shape of the unknown scatterer. These methods are based on choosing an appropriate indicator function f on Rd, d=2 or 3, such that f(z) decides whether z lies inside or outside the scatterer. Consequently, we can determine the location and the shape of the unknown scatterer. In this thesis, we first present some sampling methods for shape reconstruction in inverse scattering problems. These methods, which are described in Chapter 1, include Multiple Signal Classification (MUSIC) by Devaney, the Linear Sampling Method (LSM) by Colton and Kirsch, the Factorization Method by Kirsch, and the Direct Sampling Method by Ito et al. In Chapter 2, we introduce some direct sampling methods, including Orthogonality Sampling by Potthast and a direct sampling method using far field measurements for shape reconstruction by Liu. In Chapter 3, we generalize Liu\u27s method for shape reconstruction in inverse electromagnetic scattering problems. The method applies in an inhomogeneous isotropic medium in R3 and uses the far field measurements. We study the behavior of the new indicator for the sampling points both outside and inside the scatterer. In Chapter 4, we propose a new sampling method for multifrequency inverse source problem for time-harmonic acoustics using a finite set of far field data. We study the theoretical foundation of the proposed sampling method, and present some numerical experiments to demonstrate the feasibility and effectiveness of the method. Final conclusions of this thesis are summarized in Chapter 5. Recommendations for possible future works are also given in this chapter

Linear sampling type methods for inverse scattering problems: theory and applications.

Dai, Lipeng.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 73-75).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.ivChapter 1 --- Introduction --- p.1Chapter 1.0.1 --- Linear sampling method --- p.2Chapter 1.0.2 --- choice of cut-off values --- p.5Chapter 1.0.3 --- Underwater image problem --- p.7Chapter 2 --- Mathematical justification of LSM --- p.10Chapter 2.1 --- Some mathematical preparations --- p.11Chapter 2.2 --- Well-posedness of an interior transmission problem --- p.13Chapter 2.3 --- Linear sampling method: full aperture --- p.20Chapter 2.4 --- Linear sampling method: limited aperture --- p.23Chapter 3 --- Strengthened linear sampling method --- p.28Chapter 3.1 --- Proof of theorem 1.0.3 --- p.28Chapter 3.2 --- Several estimates in theory for strengthened LSM --- p.33Chapter 4 --- Underwater imaging problem --- p.38Chapter 4.1 --- Boundary integral method --- p.38Chapter 4.2 --- Approximation of the Integral Kernel in (4.12) --- p.40Chapter 4.3 --- Numerical solution of (4.12) --- p.44Chapter 4.4 --- Underwater image problem --- p.45Chapter 4.5 --- Imaging scheme without a reference object --- p.48Chapter 4.6 --- Numerical examples without a reference object --- p.49Chapter 4.7 --- Imaging scheme with a reference object --- p.59Chapter 4.8 --- Numerical examples with a reference object --- p.6

Survey on numerical methods for inverse obstacle scattering problems.

Deng, Xiaomao.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 98-104).Chapter 1 --- Introduction to Inverse Scattering Problems --- p.6Chapter 1.1 --- Direct Problems --- p.6Chapter 1.1.1 --- Far-field Patterns --- p.10Chapter 1.2 --- Inverse Problems --- p.16Chapter 1.2.1 --- Introduction --- p.16Chapter 2 --- Numerical Methods in Inverse Obstacle Scattering --- p.19Chapter 2.1 --- Linear Sampling Method --- p.19Chapter 2.1.1 --- History Review --- p.19Chapter 2.1.2 --- Numerical Scheme of LSM --- p.21Chapter 2.1.3 --- Theoretic Justification --- p.25Chapter 2.1.4 --- Summarize --- p.38Chapter 2.2 --- Point Source Method --- p.38Chapter 2.2.1 --- Historical Review --- p.38Chapter 2.2.2 --- Superposition of Plane Waves --- p.40Chapter 2.2.3 --- Approximation of Domains --- p.42Chapter 2.2.4 --- Algorithm --- p.44Chapter 2.2.5 --- Summarize --- p.49Chapter 2.3 --- Singular Source Method --- p.49Chapter 2.3.1 --- Historical Review --- p.49Chapter 2.3.2 --- Algorithm --- p.51Chapter 2.3.3 --- Far-field Data --- p.54Chapter 2.3.4 --- Summarize --- p.55Chapter 2.4 --- Probe Method --- p.57Chapter 2.4.1 --- Historical Review --- p.57Chapter 2.4.2 --- Needle --- p.58Chapter 2.4.3 --- Algorithm --- p.59Chapter 3 --- Numerical Experiments --- p.61Chapter 3.1 --- Discussions on Linear Sampling Method --- p.61Chapter 3.1.1 --- Regularization Strategy --- p.61Chapter 3.1.2 --- Cut off Value --- p.70Chapter 3.1.3 --- Far-field data --- p.76Chapter 3.2 --- Numerical Verification of PSM and SSM --- p.80Chapter 3.3 --- Inverse Medium Scattering --- p.83Bibliography --- p.9

On some open problems in the implementation of the linear sampling method

Not availabl

Numerical methods for high frequency scattering by multiple obstacles

For problems of time harmonic wave scattering, standard numerical methods (using piecewise polynomial approximation spaces) require the computational cost to grow with the frequency of the problem, in order to maintain a fixed accuracy. This can make many problems of practical interest difficult or impossible to solve at high frequencies. The Hybrid Numerical Asymptotic Boundary Element Method (HNA BEM) overcomes this by enriching the approximation space with oscillatory basis functions, in such a way that accuracy may be maintained with a computational cost that grows only modestly with frequency. HNA methods have previously been developed for a range of problems, including screens in two and three dimensions, also convex, non-convex and penetrable polygons in two dimensions. To date, all HNA methods have been developed for problems of plane wave scattering by a single obstacle. The key aim of this thesis is to extend the HNA method to multiple obstacles. A range of extensions to the HNA method are made in this thesis. Previous HNA methods for convex polygons use an approximation space on two overlapping meshes, here we use HNA on a single mesh. This single-mesh approach is easier to implement, and we prove that the frequency-dependence of the size of the approximation space is the same as for the overlapping mesh. We generalise HNA theory to provide a priori error estimates for a broader range of incident fields than just the plane wave, including point sources, beam sources, and Herglotz-type incidence. We also extend the HNA ansatz to include multiple obstacles. In addition to the development of HNA methods, we also consider other ideas and developments related to multiple scattering problems. This includes the first (to the best knowledge of the author) mesh and frequency explicit condition for wellposedness of Galerkin BEM for multiple scattering. We investigate numerical implementation of Embedding Formulae, which provide the far-field pattern for any incident plane wave, given the far-field patterns induced by a small (frequency independent) number of plane waves. We establish points at which a naive implementation of the theory can cause numerical instabilities and present alternative, numerically stable Embedding Formulae. We also extend the Embedding Formulae to produce the far-field pattern of any Herglotz-type wave. The recently developed Tmatrom method, a numerically stable T-matrix method, is explored as an alternative means of extending the HNA method from single to multiple obstacles. Tmatrom typically requires a number of single scattering problems to be solved, and this number grows (more than) linearly with the frequency of the problem. Using the numerically stable Embedding Formulae, we show that Tmatrom can be applied by solving a number of problems that depends only on the geometry of the obstacle, and not the frequency of the incident wave

Monotonicity methods for inverse scattering problems

We consider two inverse scattering problems in unbounded free space. On the one hand, we investigate an inverse acoustic obstacle scattering problem governed by the time-harmonic Helmholtz equation. On the other hand, we examine an inverse electromagnetic medium scattering problem modeled by the time-harmonic Maxwell equations. In both cases, our goal is to recover the position and the shape of compactly supported scatterers D from far field observations of scattered waves. For the acoustic scattering problem, we assume that the scatterers are impenetrable obstacles that carry mixed Dirichlet and Neumann boundary conditions. For the electromagnetic scattering problem, the media are supposed to be penetrable, non-magnetic and non-absorbing but the electric permittivity may be inhomogeneous inside the scattering objects. We approach both shape identification problems utilizing a monotonicity-based reconstruction ansatz. First, we establish monotonicity relations for the eigenvalues of the far field operators which map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered fields. In addition, we discuss the existence of localized wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy in some other prescribed region. Combining the monotonicity relations and the localized wave functions leads to rigorous characterizations of the support of the scattering objects. More precisely, we develop criteria that allow us to evaluate whether certain probing domains B are contained inside the unknown scatterer D or not and vice versa. Therefore, we introduce probing operators corresponding to the probing domains B and show that the number of positive or negative eigenvalues of suitable linear combinations of the far field operator corresponding to D and these probing operators is finite if and only if B is contained within D or if and only if B contains D. Finally, we complement our theoretical findings with numerical reconstruction algorithms and give some examples to illustrate the reconstruction procedure