Tilo Arens - Research Interests


Contents

Scattering from a sound soft obstacle, total field
See also my overview over time-harmonic scattering theory.

Research Interests

My research focusses on the following areas:

List of Publications

  1. The Scattering of Plane Elastic Waves by a One-dimensional Periodic Surface
    T. Arens
    Math. Meth. Appl. Sci. 22, 55 - 72 (1999) (abstract)
  2. A New Integral Equation Formulation for the Scattering of Plane Elastic Waves by Diffraction Gratings
    T. Arens
    J. Int. Equ. Appl. 11, 275-297 (1999) (abstract)
  3. A Nyström Method for a Class of Integral Equations on the Real Line with Applications to Scattering by Diffraction Gratings and Rough Surfaces
    A. Meier, T. Arens, S.N. Chandler-Wilde and A. Kirsch
    J. Int. Equ. Appl. 12, 281-321 (2000) (abstract).
  4. Uniqueness for Elastic Wave Scattering by Rough Surfaces
    T. Arens
    SIAM J. Math. Anal. 33, 461--476 (2001) (abstract)
    A zipped postscript version of this paper is available for download (200 kB).
  5. Linear Sampling Methods for 2D Inverse Elastic Wave Scattering
    T. Arens
    Inverse Problems 17, 1445-1464 (2001) (abstract).
  6. Existence of Solution in Elastic Wave Scattering by Unbounded Rough Surfaces
    T. Arens
    Math. Meth. Appl. Sci. 25, 507-528 (2002) (abstract).
  7. Solvability and Spectral Properties of Integral Equations on the Real Line: I. Weighted Spaces of Continuous Functions
    T. Arens, K. Haseloh and S.N. Chandler-Wilde
    J. Math. Anal. Appl. 272, 276-302 (2002).
  8. Solvability and Spectral Properties of Integral Equations on the Real Line: II. Lp-Spaces and Applications
    T. Arens, K. Haseloh and S.N. Chandler-Wilde
    J. Int. Equ. Appl. 15, 1-35 (2003).
  9. An Approximation Property of Elastic Herglotz Wave Functions and its Application in the Linear Sampling Method
    T. Arens
    J. Inverse Ill-Posed Probl. 11, 219-233 (2003) (abstract).
  10. The Factorization Method in Inverse Scattering from Periodic Structures
    T. Arens and A. Kirsch
    Inverse Problems 19, 1195-1211 (2003) (abstract).
  11. Uniqueness, Existence and Integral Equation Formulations for Interface Scattering Problems
    D. Natroshvili, T. Arens and S.N. Chandler-Wilde
    Memoirs on Differential Equations and Mathematical Physics 30, Georgian Academy of Sciences, 105-146 (2003) (abstract).
  12. Why Linear Sampling Works!
    T. Arens
    Inverse Problems 20 , 163-173 (2004) (abstract).
  13. A complete Factorization Method for scattering by periodic surfaces
    T. Arens and N. Grinberg,
    Computing 75, 111-132 (2005).
  14. On radiation conditions for rough surface scattering problems
    T. Arens and T. Hohage
    IMA J. Appl. Math. 70, 839-847 (2005).
  15. On integral equation and least squares methods for scattering by diffraction gratings
    T. Arens, S.N. Chandler-Wilde and J.A. DeSanto
    Communications in Computational Physics 1, 1010-1042 (2006).
  16. Variational Formulations for Scattering in a 3-Dimensional Acoustic Waveguide
    T. Arens, D. Gintides und A. Lechleiter
    Math. Meth. Appl. Sci. 31, 821-847 (2008).
  17. Mathematik
    T. Arens, F. Hettlich, C. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel
    Spektrum Akademischer Verlag, Heidelberg, 2008.
    Bonus material and problem solutions available at http://www.matheweb.de (German).
  18. The Linear Sampling Method Revisited
    T. Arens and A. Lechleiter
    J. Int. Equ. Appl. 21, 179-202 (2009).
    Available for download at EVASTAR.
  19. MUSIC for Extended Scatterers as an Instance of the Factorization Method
    T. Arens, A. Lechleiter and R.L. Luke
    SIAM J. Appl. Math. 70, 1283-1304 (2009).
  20. Scattering by Biperiodic Layered Media: The Integral Equation Approach
    T. Arens
    Habilitation Thesis, Karlsruhe Institue of Technology, (2010).
    Available for download at EVASTAR.
  21. Analysing Ewald's Method for the Evaluation of Green's Functions for Periodic Media
    T. Arens, K. Sandfort, S. Schmitt and A. Lechleiter
    submitted for publication.
    Available for download at EVASTAR.

Abstracts of Recent Papers

The texts of these abstracts require the Symbol font to be installed on your system.

[1] T. Arens - The Scattering of Plane Elastic Waves by a One-dimensional Periodic Surface
Math. Meth. Appl. Sci. 22, 55 - 72 (1999)
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 utilised 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.

[2] T. Arens - A New Integral Equation Formulation for the Scattering of Plane Elastic Waves by Diffraction Gratings
J. Int. Equ. Appl. 11, 275-297 (1999)
The scattering of plane elastic waves by a rigid periodic surface is considered. The Green's tensor for a half-space with a rigid surface is introduced and its properties, notably its asymptotic decay rate in horizontal layers above the plane, are analysed. The Green's tensor is then used to define single and double layer potentials for a periodic surface, making use of the generalised stress tensor. Subsequently, a novel integral equation formulation for the scattering of plane waves by a diffraction grating is derived using a Brakhage/Werner type ansatz for the solution. Employing the Fredholm alternative, existence of solution is proved for all angles of incidence and all wave-numbers.

[3] A. Meier, T. Arens, S.N. Chandler-Wilde and A. Kirsch - A Nyström Method for a Class of Integral Equations on the Real Line with Applications to Scattering by Diffraction Gratings and Rough Surfaces
J. Int. Equ. Appl. 12 , 281-321 (2000)
We propose a Nyström/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established, with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behaviour for the case when f is periodic (the diffraction grating case). The Nyström method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

[4] T. Arens - Uniqueness for Elastic Wave Scattering by Rough Surfaces
SIAM J. Math. Anal. 33, 461--476 (2001).
A zipped postscript version of this paper is available for download (200 kB).
We consider a 2D elastic wave scattering problem for an unbounded surface represented as the graph of a C1,a function. The total displacement is assumed to vanish on the surface. We present a new radiation condition, the upwards propagating radiation condition, for such problems based on a similar condition recently introduced for acoustic scattering problems. The relation between this radiation condition and more commonly used conditions is discussed. Subsequently we prove uniqueness of solution to the scattering problem under this radiation condition for a general class of incident fields, including plane and cylindrical waves.

[5] T. Arens - Linear Sampling Methods for 2D Inverse Elastic Wave Scattering
Inverse Problems 17, 1445-1464 (2001).
This paper won a 2nd prize at the 2001 Leslie Fox Prize competition.
The problem of determining the shape of a rigid body from the knowledge of the far field patterns of incident plane compressional and shear waves in two dimensional elasticity is considered. We discuss the application of the original linear sampling method and of the related (F*F)^1/4-method to tackle this problem. It is established that the far field operator is compact and normal. A theoretical basis for both reconstruction methods is developed and numerical results are shown, illustrating the excellent quality of reconstructions attainable.

[6] T. Arens - Existence of Solution in Elastic Wave Scattering by Unbounded Rough Surfaces
Math. Meth. Appl. Sci. 25, 507-528 (2002).
We consider the two-dimensional problem of the scattering of a time-harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function f C1,1(R). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all Lp spaces, p [1, ] and hence existence of solution to the elastic wave scattering problem.

[9] T. Arens - An Approximation Property of Elastic Herglotz Wave Functions and its Application in the Linear Sampling Method
J. Inverse Ill-Posed Probl. 11, 219-233 (2003)
Herglotz wave functions play an important role in a class of reconstruction methods for inverse scattering problems known as \textsl{linear sampling methods}. We here consider these functions in the setting of linearized elasticity and derive representations in terms of eigenfunctions to the Navier operator in two spatial dimensions. We then show the important property that the elastic Herglotz Wave functions are dense in the space of solutions to the Navier equation with respect to the [H1(D)]2 norm for any bounded Lipschitz domain D. The proof of this property in three-dimensions, not essentially different from the 2D argument, is also outlined. The paper is concluded with an application of the approximation property in the mathematical foundation of the linear sampling method for the reconstruction of rigid obstacles from the knowledge of the far field operator.

[10] T. Arens and A. Kirsch - The Factorization Method in Inverse Scattering from Periodic Structures
Inverse Problems 19, 1195 - 1211 (2003)
The application of the Factorization Method, the refined version of the Linear Sampling Method, to scattering by a periodic surface is considered. Central to this method is the Near Field Operator N, mapping incident fields to the corresponding scattered fields on a horizontal line. A factorization of N forms the basis for the method. It is shown that the middle operator in this factorization, the adjoint of the single layer operator, is coercive if the wave number is small enough. Thus this operator can, for general wave number, always be written as the sum of a coecive and a compact operator. We use this property to define an auxiliary positive operator N# which can be constructed directly from N and which makes it possible to reconstruct the scattering surface directly using a simple numerical algorithm.

[11] D. Natroshvili, T. Arens and S.N. Chandler-Wilde - Uniqueness, Existence and Integral Equation Formulations for Interface Scattering Problems
Memoirs on Differential Equations and Mathematical Physics 30, Georgian Academy of Sciences, 105-146 (2003)
We consider a two-dimensional transmission problem in which Hemlotz equations with different wave numbers hold in adjacent non-locally perturbed half-planes having a common boundary which is an infinite, one-dimensional, rough interface line. First a uniqueness theorem for the interface problem is proved provided that the scatterer is a lossy obstacle. Afterwards, by potential methods, the non-homogeneous interface problem is reduced to a system of integral equations and existence results are established.

[12] T. Arens - Why Linear Sampling Works!
Inverse Problems 20 , 163-173 (2004)
The Linear Sampling Method is a method to reconstruct the shape of an obstacle in time-harmonic inverse scattering without a priori knowledge on either the physical properties or the number of disconnected components of the scatterer. Although it has numerically proven to be a fast and reliable method in many situations, no mathematical argument has yet been found to prove why this is so. Using results obtained by Kirsch in deriving the related Factorization Method, we show in this paper that for a large class of scattering problems, the Linear Sampling can be interpreted rigorously as a numerical method to reconstruct the shape of an obstacle, or in other words that Linear Sampling must work for problems in this class.


Doctoral Thesis

T. Arens - The Scattering of Elastic Waves by Rough Surfaces
Brunel University, June 2000

Abstract.We consider the scattering of elastic waves by an unbounded surface on which the displacement vanishes. The wave field is assumed to be time-harmonic and the propagation medium to be homogeneous and isotropic. The scattering surface is assumed to be given as a graph of a bounded function f C1,a, but otherwise no assumptions are made.

The problem is formulated as a boundary value problem for the scattered field in the unbounded domain above the scattering surface. This boundary value problem formulation includes a novel radiation condition characterising upward propagating waves. The way in which this radiation condition generalises other radiation conditions commonly employed in elastic wave scattering problems is discussed in detail. It is then shown that the boundary value problem, and thus the scattering problem, admits at most one solution for a general class of incident fields including plane and cylindrical waves.

Existence of solution is established via the boundary integral equation method. The properties of elastic single- and double-layer potentials on rough surfaces are studied with an emphasis on obtaining estimates uniformly for classes of such surfaces. An equivalent formulation of the scattering problem as a boundary integral equation of the second kind is obtained. Since the scatterer is unbounded the integral operator in this equation is not compact, and nor is the equation of a standard singular type previously studied. Thus, a new solvability theory is developed, which establishes solvability of the integral equation in the space of bounded and continuous functions, and also in all Lp-spaces, 1 p .

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