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## Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition (1998)

Citations: | 1049 - 45 self |

### Citations

731 |
Regularization of inverse problems
- Engl, Hanke, et al.
- 1996
(Show Context)
Citation Context ...contradiction since X is infinite dimensional. Our discussion will purposefully be brief and for more information on the solution of ill-posed problems we refer the reader to Engl, Hanke and Neubauer =-=[24]-=-, Kirsch [40] and Kress[47]. JINVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 77 We restrict our attention to the case when X and Y are infinite dimensional Hilbert spaces. We denote by (σn, ϕ... |

419 |
Linear Integral Equations
- Kress
- 1999
(Show Context)
Citation Context ...finite dimensional. Our discussion will purposefully be brief and for more information on the solution of ill-posed problems we refer the reader to Engl, Hanke and Neubauer [24], Kirsch [40] and Kress=-=[47]-=-. JINVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 77 We restrict our attention to the case when X and Y are infinite dimensional Hilbert spaces. We denote by (σn, ϕn, gn) a singular system fo... |

365 |
Inverse problems for partial differential equations, volume 127 of Applied Mathematical Sciences
- Isakov
- 2006
(Show Context)
Citation Context ...3, we will focus our attention on the situation where neither the material properties of the scatterer nor its shape are known. Then it follows from the uniqueness theorems of Nachman [53] and Isakov =-=[36]-=-, [37]that for a single frequency the best that we can hope for is to determine the shape D of the scatterer. In particular, in order to determine the coefficients in (2–1b), either multi-frequency da... |

245 |
An introduction to the mathematical theory of inverse problems, volume 120 of Applied Mathematical Sciences
- Kirsch
- 1996
(Show Context)
Citation Context ... since X is infinite dimensional. Our discussion will purposefully be brief and for more information on the solution of ill-posed problems we refer the reader to Engl, Hanke and Neubauer [24], Kirsch =-=[40]-=- and Kress[47]. JINVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 77 We restrict our attention to the case when X and Y are infinite dimensional Hilbert spaces. We denote by (σn, ϕn, gn) a sing... |

146 |
Reconstructions from boundary measurements
- Nachman
- 1988
(Show Context)
Citation Context ...near sampling method [44]. In Section 4 we will first consider acoustic inverse scattering problems associated with an isotropic inhomogeneous medium and begin with the uniqueness theorems of Nachman =-=[53]-=-, Novikov [57] and Ramm [66]. In this case special problems occur in the case of scattering in R2 . We will again discuss the linear sampling method for determining the support of the inhomogeneous sc... |

139 |
A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Probl
- Colton, Kirsch
- 1996
(Show Context)
Citation Context ...rate on the case of obstacle scattering and prove the Kirsch–Kress uniqueness theorem [46] which in turn serves as motivation for the linear sampling method for determining the shape of the scatterer =-=[12]-=-,[42]. We shall in addition present a recent optimization scheme of Kirsch which has certain attractive characteristics and is closely related to the linear sampling method [44]. In Section 4 we will ... |

120 |
A multidimensional inverse spectral problem for the equation −∆ψ+(v(x
- Novikov
- 1988
(Show Context)
Citation Context ...method [44]. In Section 4 we will first consider acoustic inverse scattering problems associated with an isotropic inhomogeneous medium and begin with the uniqueness theorems of Nachman [53], Novikov =-=[57]-=- and Ramm [66]. In this case special problems occur in the case of scattering in R2 . We will again discuss the linear sampling method for determining the support of the inhomogeneous scattering objec... |

61 |
Recent development in inverse acoustic scattering theory
- Colton, Coyle, et al.
(Show Context)
Citation Context ...ned by looking for those points z where ∥ ∥ ϕ( · , z) begins to γ→0INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 87 sharply increase. Numerical examples using this procedure can be found in =-=[9]-=-, [10] and [74]. The analogue of Theorem 3.7 for the exterior Neumann problem is established in exactly the same way as Theorem 3.7 where now it is assumed that k2 is not a Neumann eigenvalue for D. I... |

61 |
An inverse boundary value problem in electrodynamics
- Ola, Päivärinta, et al.
- 1993
(Show Context)
Citation Context ...ING THEORY 105 also note that the scattering problem (5–3) corresponds to the case when the magnetic permeability µ is constant and for uniqueness results in the case when µ is no longer constant see =-=[59]-=-, [60] and [69]. Theorem 5.4. Assume that D1 and D2 are two domains such that the electric far field patterns corresponding to the scattering problem (5–1) coincide for all incident directions d ∈ Ω a... |

52 |
A simple method using Morozov's discrepancy principle for solving inverse scattering problems. Inverse Problems
- Colton, Piana, et al.
- 1477
(Show Context)
Citation Context ...d for determining the support of the inhomogeneous scattering object, leading to an investigation of the existence, uniqueness and spectral properties of the interior transmission problem [13], [14], =-=[19]-=-, [67]. We will then proceed to an extension of these results to the case of anisotropic media. In contrast to the case of isotropic media, variational methods rather than integral equation techniques... |

50 |
The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator
- Novikov
- 1992
(Show Context)
Citation Context ... the R 3 case. i.e., in R 2 , u∞(ˆx, d) is a function of two variables and n(x) is also a function of two variables. Nevertheless, there have been numerous partial results in this case due to Novikov =-=[58]-=-, Sun and Uhlmann [68], Isakov and Nachman [38], Isakov and Sun [39] and Eskin [25] among others. We content ourselves here by stating a single result in this direction due to Sun and Uhlmann [70] (se... |

39 |
Frechét derivatives in inverse obstacle scattering
- Hettlich
- 1995
(Show Context)
Citation Context ...= u∞ as F(r) = u∞. (2–13) The following basic theorem was first proved by Kirsch [41] using variational methods and subsequently by Potthast [63] using a boundary integral equation approach (see also =-=[34]-=- and [48]). We note that the validity of the following theorem for the case of the exterior Neumann problem remains an open question [50]. Theorem 2.1. The boundary to far field map F : C 2 (Ω) → L 2 ... |

34 | The direct and inverse scattering problems for partially coated obstacles - Cakoni, Colton, et al. - 2001 |

34 | Uniqueness theorems for the inverse problem of acoustic scattering - Colton, D - 1983 |

32 |
The uniqueness of a solution to an inverse scattering problem for electromagnetic waves
- Colton, Päivärinta
- 1992
(Show Context)
Citation Context ...nes D and in the case of (5–3) whether or not a knowledge of E∞for fixed k uniquely determines n(x). To this end, we have the following theorems due to Colton and Kress [14] and Colton and Päivärinta =-=[18]-=- respectively. The proofs of both results are similar to the corresponding proofs in the acoustic case already discussed. However, for the inverse scattering problem associated with (5–3), serious tec... |

32 |
Reconstruction of an obstacle from the scattering amplitude at a fixed frequency Inverse Problems 14 949–54
- Ikehata
- 1998
(Show Context)
Citation Context ... Colton and Kirsch in [12], with a subsequent second version being given by Kirsch in [42] and [43] and has become known as the linear sampling method (related methods have been considered by Ikehata =-=[35]-=-, Norris [56] and Potthast [65]). Here, for the sake of simplicity, we will assume that u∞(ˆx, d) is known for all ˆx, d ∈ Ω rather than only on a subset of Ω. For the case of the first version of the... |

31 |
On the uniqueness of the shape of a penetrable, anisotropic obstacle
- Hähner
- 2000
(Show Context)
Citation Context ...se results to the case of anisotropic media. In contrast to the case of isotropic media, variational methods rather than integral equation techniques are a more convenient tool in this case [6], [7], =-=[31]-=-. Finally, in Section 5, we consider the inverse scattering problem for Maxwell’s equations and extend some of the results in the previous sections to this situation. However, much of what is known fo... |

30 | H.: The linear sampling method for anisotropic media
- Cakoni, Colton, et al.
- 2002
(Show Context)
Citation Context ...of a unique solution to (4–13), and using the fact that (4–13) is a compact perturbation of the interior transmission problem (4–12), we can now appeal to Theorem 4.10 to deduce the following theorem =-=[5]-=-. Theorem 4.11. Assume that either Im n > 0 or ¯ ξ·(Im A)ξ < 0 in a neighborhood of a point x0 ∈ D and that γ > 1. Then (4–12) has a unique solution v, w ∈ H 1 (D) × H 1 (D) where the boundary data (4... |

29 |
A propagation–backpropagation method for ultrasound tomography, Inverse Probl
- Natterer, Wübbeling
- 1995
(Show Context)
Citation Context ...re the number depends on the wave number k, Kleinman and Van den Berg [72], who use a modified gradient method for an output least squares formulation of the problem, and Natterer and Wübbeling [54], =-=[55]-=- who employ an algebraic reconstruction technique (ART) to determine n(x). We shall conclude our brief discussion of nonlinear optimization schemes to solve inverse scattering problems by describing t... |

29 |
Electromagnetic inverse problems and generalized Sommerfeld potentials
- Ola, Somersalo
- 1996
(Show Context)
Citation Context ...EORY 105 also note that the scattering problem (5–3) corresponds to the case when the magnetic permeability µ is constant and for uniqueness results in the case when µ is no longer constant see [59], =-=[60]-=- and [69]. Theorem 5.4. Assume that D1 and D2 are two domains such that the electric far field patterns corresponding to the scattering problem (5–1) coincide for all incident directions d ∈ Ω and all... |

28 |
On uniqueness in the inverse transmission scattering problem
- Isakov
- 1990
(Show Context)
Citation Context ...will focus our attention on the situation where neither the material properties of the scatterer nor its shape are known. Then it follows from the uniqueness theorems of Nachman [53] and Isakov [36], =-=[37]-=-that for a single frequency the best that we can hope for is to determine the shape D of the scatterer. In particular, in order to determine the coefficients in (2–1b), either multi-frequency data is ... |

27 | A nonlinear inversion method for 3D electromagnetic imaging using adjoint fields, Inv. Probl
- Dorn, Bertete-Aguirre, et al.
- 1999
(Show Context)
Citation Context ...sing such an approach are discussed. An extension of this method (which is sometimes called the “adjoint field method”) to the case of time-harmonic electromagnetic waves has been done by Dorn, et.al.=-=[23]-=-. In both the weak scattering and Newton-type methods for solving the inverse scattering problem we are faced with the problem of solving a linear operator equation of the form Aϕ = f where A : X → Y ... |

26 |
Uniqueness in inverse obstacle scattering
- Kirsch, Kress
- 1993
(Show Context)
Citation Context ...thods which avoid such strong assumptions but at the expense of needing more data. In particular, we shall concentrate on the case of obstacle scattering and prove the Kirsch–Kress uniqueness theorem =-=[46]-=- which in turn serves as motivation for the linear sampling method for determining the shape of the scatterer [12],[42]. We shall in addition present a recent optimization scheme of Kirsch which has c... |

22 | The linear sampling method for solving the electromagnetic inverse scattering problem
- Colton, Haddar, et al.
(Show Context)
Citation Context ... of writing, only the linear sampling method associated with the far field operator F has been extended to the general acoustic transmission problem (2–1) [6], [7] and the case of Maxwell’s equations =-=[11]-=-, [29], [49]. Hence, in the interest of developing a unifying theme to our paper, we will restrict our attention to the first version of the linear sampling method in order to determine D. We begin ou... |

20 |
On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs
- Colton, Kress
(Show Context)
Citation Context ...ing on the one hand the fact that86 DAVID COLTON Herglotz wave functions are dense in the space of solutions to the Helmholtz equation in D with respect to the norm in the Sobolev space H 1 (D) (see =-=[16]-=-, [21]) and on the other the factorization of the far field operator F as (F g) = − 1 4π FS−1 (Hg), where S : H−1/2 (∂D) → H1/2 (∂D) is the single layer potential ∫ (Sϕ)(x) := ϕ(y)Φ(x, y) ds(y), (3–9)... |

20 |
Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems
- Kirsch
- 1999
(Show Context)
Citation Context ...thod based on Theorem 3.11. Each of these methods, under appropriate assumptions, can be extended to the inverse scattering problem associated with the inhomogeneous medium problems (2–4) [12], [19], =-=[43]-=-, [44]. However, at the time of writing, only the linear sampling method associated with the far field operator F has been extended to the general acoustic transmission problem (2–1) [6], [7] and the ... |

18 |
Applied Inverse Problems for Acoustic, Electromagnetic and Elastic Wave Scattering
- Langenberg
- 1987
(Show Context)
Citation Context ... above are based on either what is called the “weak-scattering” approximation or on nonlinear optimization techniques. For a comprehensive discussion of such methods we refer the reader to Langenberg =-=[51]-=- and Biegler, et.al. [2] respectively. Here we shall content ourselves with only a brief description of these two approaches. We begin with the weak-scattering approximation, in particular the physica... |

17 |
L.: Far-field patterns for acoustic waves in an inhomogeneous medium
- Colton, Kirsch, et al.
- 1989
(Show Context)
Citation Context ...mpling method for determining the support of the inhomogeneous scattering object, leading to an investigation of the existence, uniqueness and spectral properties of the interior transmission problem =-=[13]-=-, [14], [19], [67]. We will then proceed to an extension of these results to the case of anisotropic media. In contrast to the case of isotropic media, variational methods rather than integral equatio... |

17 |
An optimization method in inverse acoustic scattering
- Kirsch, Kress
- 1987
(Show Context)
Citation Context ...e that since F ′ is compact each step of the iteration procedure is ill-posed. Alternate optimization strategies for determining D have been proposed by numerous people in particular Kirsch and Kress =-=[45]-=-, Angell, Kleinman and Roach [1] and Maponi et al. [52]. Newton’s method can also be used to determine the coefficient n(x) in the inverse inhomogeneous medium problem [32] In this case the nonlinear ... |

16 | Regularized sampling method for solving threedimensional inverse scattering problems
- Colton, Giebermann, et al.
(Show Context)
Citation Context ...y looking for those points z where ∥ ∥ ϕ( · , z) begins to γ→0INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 87 sharply increase. Numerical examples using this procedure can be found in [9], =-=[10]-=- and [74]. The analogue of Theorem 3.7 for the exterior Neumann problem is established in exactly the same way as Theorem 3.7 where now it is assumed that k2 is not a Neumann eigenvalue for D. It is a... |

16 |
Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium
- Colton, Kress
- 1995
(Show Context)
Citation Context ...\ D and the boundary condition vi g + vs = 0 on ∂D now shows that vi g = 0 on ∂D. The proof is finished. ˜ We now want to show that the far field F is normal. To this end, we need the following lemma =-=[15]-=-. Lemma 3.3. The far field operator F satisfies 2π ( (F g, h) − (g, F h) ) = ik(F g, F h).82 DAVID COLTON Proof. If vs and ws are radiating solutions to the Helmholtz equation with far field patterns... |

16 |
Global uniqueness for a two-dimensional semilinear elliptic inverse problem
- Isakov, Nachman
- 1995
(Show Context)
Citation Context ...nction of two variables and n(x) is also a function of two variables. Nevertheless, there have been numerous partial results in this case due to Novikov [58], Sun and Uhlmann [68], Isakov and Nachman =-=[38]-=-, Isakov and Sun [39] and Eskin [25] among others. We content ourselves here by stating a single result in this direction due to Sun and Uhlmann [70] (see also [64]) which shows that the discontinuiti... |

14 |
An approximation property of importance in inverse scattering theory
- Colton, Sleeman
(Show Context)
Citation Context ... the one hand the fact that86 DAVID COLTON Herglotz wave functions are dense in the space of solutions to the Helmholtz equation in D with respect to the norm in the Sobolev space H 1 (D) (see [16], =-=[21]-=-) and on the other the factorization of the far field operator F as (F g) = − 1 4π FS−1 (Hg), where S : H−1/2 (∂D) → H1/2 (∂D) is the single layer potential ∫ (Sϕ)(x) := ϕ(y)Φ(x, y) ds(y), (3–9) ∂D Hg... |

14 |
New characterizations of solutions in inverse scattering theory
- Kirsch
(Show Context)
Citation Context ...shape of the scatterer [12],[42]. We shall in addition present a recent optimization scheme of Kirsch which has certain attractive characteristics and is closely related to the linear sampling method =-=[44]-=-. In Section 4 we will first consider acoustic inverse scattering problems associated with an isotropic inhomogeneous medium and begin with the uniqueness theorems of Nachman [53], Novikov [57] and Ra... |

13 |
Electromagnetic waves scattering : Scattering by obstacles, Scattering
- Kress
- 2001
(Show Context)
Citation Context ... only the linear sampling method associated with the far field operator F has been extended to the general acoustic transmission problem (2–1) [6], [7] and the case of Maxwell’s equations [11], [29], =-=[49]-=-. Hence, in the interest of developing a unifying theme to our paper, we will restrict our attention to the first version of the linear sampling method in order to determine D. We begin our discussion... |

11 |
Inverse scattering for shape and impedance
- Kress, Rundell
- 2001
(Show Context)
Citation Context ...ast [63] using a boundary integral equation approach (see also [34] and [48]). We note that the validity of the following theorem for the case of the exterior Neumann problem remains an open question =-=[50]-=-. Theorem 2.1. The boundary to far field map F : C 2 (Ω) → L 2 (Ω) has a Fréchet derivative F ′ . The linear operator F ′ is compact and injective with dense range. Theorem 2.1 now allows us to apply ... |

10 |
Inverse scattering from an orthotropic medium
- Colton, Kress, et al.
- 1997
(Show Context)
Citation Context ... that the matrix A is not uniquely determined by u∞ (see [27], [61]) and hence determinning D is the most that can be hoped for. To this end we have the following theorem due to Hähner [31] (see also =-=[17]-=- and [61]). Theorem 4.9. Assume γ > 1. Then D is uniquely determined by u∞(ˆx, d) for ˆx, d ∈ Ω. The proof of this theorem uses the ideas of Theorem 3.5 together with a continuous dependence result fo... |

9 |
Radar Imaging of Airborne Targets
- Borden
- 1999
(Show Context)
Citation Context ...rity that the imaging hopes expressed in the early post-war years have to a certain extent been realized, at least in the case of electromagnetic waves with the invention of synthetic aperature radar =-=[3]-=-, [8]. However, as the imaging demands have increased so have the mathematical and computational expectations and hence at this time it seems appropriate to make an attempt at describing the state of ... |

9 |
An inverse problem for the Helmholtz equation
- Gylys-Colwell
- 1996
(Show Context)
Citation Context ...e inverse scattering problem we are concerned with is to determine D from a knowledge of the far field pattern u∞(ˆx, d) for ˆx, d ∈ Ω. We note that the matrix A is not uniquely determined by u∞ (see =-=[27]-=-, [61]) and hence determinning D is the most that can be hoped for. To this end we have the following theorem due to Hähner [31] (see also [17] and [61]). Theorem 4.9. Assume γ > 1. Then D is uniquely... |

7 |
Inverse 3d acoustic and electromagnetic obstacle scattering by iterative adaption
- Haas, Rieger, et al.
- 1997
(Show Context)
Citation Context ...s only begun to be considered. Notable accomplishments in the case of nonlinear optimization techniques to solve the inverse scattering problem associated with (5–1) have been achieved by Haas, et.al =-=[28]-=- and Maponi, et.al [52] whereas the case of nonlinear optimization techniques to solve the inverse scattering problem associated with (5–3) have recently been considered by Dorn, et.al [23]. Finally, ... |

7 |
On uniqueness for anisotropic inhomogeneous inverse scattering problems Inverse Problems 14
- Piana
- 1998
(Show Context)
Citation Context ...rse scattering problem we are concerned with is to determine D from a knowledge of the far field pattern u∞(ˆx, d) for ˆx, d ∈ Ω. We note that the matrix A is not uniquely determined by u∞ (see [27], =-=[61]-=-) and hence determinning D is the most that can be hoped for. To this end we have the following theorem due to Hähner [31] (see also [17] and [61]). Theorem 4.9. Assume γ > 1. Then D is uniquely deter... |

6 |
Characterization of the shape of the scattering obstacle by the spectral data of the far operator
- Kirsch
- 1998
(Show Context)
Citation Context ...on the case of obstacle scattering and prove the Kirsch–Kress uniqueness theorem [46] which in turn serves as motivation for the linear sampling method for determining the shape of the scatterer [12],=-=[42]-=-. We shall in addition present a recent optimization scheme of Kirsch which has certain attractive characteristics and is closely related to the linear sampling method [44]. In Section 4 we will first... |

5 |
An inverse transmission problem for the Helmholtz equation
- Angell, Kleinman, et al.
- 1987
(Show Context)
Citation Context ...step of the iteration procedure is ill-posed. Alternate optimization strategies for determining D have been proposed by numerous people in particular Kirsch and Kress [45], Angell, Kleinman and Roach =-=[1]-=- and Maponi et al. [52]. Newton’s method can also be used to determine the coefficient n(x) in the inverse inhomogeneous medium problem [32] In this case the nonlinear operator F is defined by means o... |

5 |
editors. Large-scale Optimization with Applications
- Biegler, Coleman, et al.
- 1997
(Show Context)
Citation Context ...r what is called the “weak-scattering” approximation or on nonlinear optimization techniques. For a comprehensive discussion of such methods we refer the reader to Langenberg [51] and Biegler, et.al. =-=[2]-=- respectively. Here we shall content ourselves with only a brief description of these two approaches. We begin with the weak-scattering approximation, in particular the physical optics approximation f... |

5 | On the mathematical basis of the linear sampling method
- Cakoni, Colton
(Show Context)
Citation Context ...) and sound speed cD(x), x ∈ D ⊂ R 3 . We assume that the boundary ∂D is of class C 2 having unit outward normal ν (although much of the analysis which follows is also valid for Lipschitz domains-see =-=[4]-=-,[6]) and that ρD, cD ∈ C 2 ( ¯ D). Then if the host medium is homogeneous with density ρ and sound speed c, the wave number k is defined by k = ω/c, and the pressure p(x, t) is given by p(x, t) = n(x... |

5 |
Iterative method for multi-dimensional inverse scat- tering problems at fixed frequencies, Inverse Problems
- Gutman, Klibanov
- 1994
(Show Context)
Citation Context ... equation (2–11). Other methods for determining n(x) have been proposed by Colton and Monk (see Chapter 10 of [14])who use an averaging procedure to reduce the number of unknowns, Gutman and Klibanov =-=[26]-=-, who confine themselves to reconstructing a fixed number of Fourier coefficients of n where the number depends on the wave number k, Kleinman and Van den Berg [72], who use a modified gradient method... |

5 |
The use of optimization in the reconstruction of obstacles from acoustic or electromagnetic scattering data
- Maponi, Recchioni, et al.
- 1997
(Show Context)
Citation Context ...procedure is ill-posed. Alternate optimization strategies for determining D have been proposed by numerous people in particular Kirsch and Kress [45], Angell, Kleinman and Roach [1] and Maponi et al. =-=[52]-=-. Newton’s method can also be used to determine the coefficient n(x) in the inverse inhomogeneous medium problem [32] In this case the nonlinear operator F is defined by means of the Lippmann–Schwinge... |

5 |
A direct inverse scattering method for imaging obstacles with unknown surface conditions
- Norris
- 1998
(Show Context)
Citation Context ...irsch in [12], with a subsequent second version being given by Kirsch in [42] and [43] and has become known as the linear sampling method (related methods have been considered by Ikehata [35], Norris =-=[56]-=- and Potthast [65]). Here, for the sake of simplicity, we will assume that u∞(ˆx, d) is known for all ˆx, d ∈ Ω rather than only on a subset of Ω. For the case of the first version of the linear sampl... |

4 |
The inverse scattering problem in two dimensions at fixed energy
- Eskin
(Show Context)
Citation Context ...also a function of two variables. Nevertheless, there have been numerous partial results in this case due to Novikov [58], Sun and Uhlmann [68], Isakov and Nachman [38], Isakov and Sun [39] and Eskin =-=[25]-=- among others. We content ourselves here by stating a single result in this direction due to Sun and Uhlmann [70] (see also [64]) which shows that the discontinuities of n are uniquely determined from... |

2 |
Acoustic Tomography, in Inverse Problems of Acoustic and Elastic
- Devaney
- 1984
(Show Context)
Citation Context ... that k is sufficiently small and ρ = ρD in (2–1). In order to ensure injectivity we must again assume that (2–12) is valid for an interval of k values. For further developments in this direction see =-=[22]-=-.74 DAVID COLTON Although the weak scattering models discussed above have had considerable success, particularly in their extensions to the electromagnetic case and use in the development of syntheti... |

1 |
A mathematical tutorial on syntehtic aperature radar
- Cheney
(Show Context)
Citation Context ...that for acoustic waves. In particular, although methods based on the weak scattering approximation have been used extensively, particularly in problems associated with synthetic aperature radar [3], =-=[8]-=-, the nonlinear problem has only begun to be considered. Notable accomplishments in the case of nonlinear optimization techniques to solve the inverse scattering problem associated with (5–1) have bee... |

1 |
A periodic Faddeev-type soltuion operator
- Hähner
- 1996
(Show Context)
Citation Context ...[66] and is ˜INVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 93 based on the fundamental paper of Sylvester and Uhlmann [71]. Here we follow a modification of the original proof due to Hähner =-=[30]-=- which is based on the following two lemmas, where H 2 (B) denotes a Sobolev space. Lemma 4.2. Let B be an open ball centered at the origin and containing the support of m := 1 − n. Then there exists ... |

1 |
Acoustic scattering
- Hähner
- 2001
(Show Context)
Citation Context ...particular Kirsch and Kress [45], Angell, Kleinman and Roach [1] and Maponi et al. [52]. Newton’s method can also be used to determine the coefficient n(x) in the inverse inhomogeneous medium problem =-=[32]-=- In this case the nonlinear operator F is defined by means of the Lippmann–Schwinger integral equation (2–11). Other methods for determining n(x) have been proposed by Colton and Monk (see Chapter 10 ... |

1 |
Electromagnetic waves scattering in Scattering
- Hähner
- 2001
(Show Context)
Citation Context ... 2 and, in contrast to the case of acoustic waves, it is no longer true that Rζ decays to zero as |ζ| tends to infinity. For details of how this difficulty is resolved we refer the reader to [18] and =-=[33]-=-. WeINVERSE ACOUSTIC AND ELECTROMAGNETIC SCATTERING THEORY 105 also note that the scattering problem (5–3) corresponds to the case when the magnetic permeability µ is constant and for uniqueness resu... |

1 |
The inverse scattering problem at fixed energies in two dimensions
- Isakov, Sun
- 1995
(Show Context)
Citation Context ...es and n(x) is also a function of two variables. Nevertheless, there have been numerous partial results in this case due to Novikov [58], Sun and Uhlmann [68], Isakov and Nachman [38], Isakov and Sun =-=[39]-=- and Eskin [25] among others. We content ourselves here by stating a single result in this direction due to Sun and Uhlmann [70] (see also [64]) which shows that the discontinuities of n are uniquely ... |

1 |
The domain derivative and two applicaitons
- Kirsch
- 1993
(Show Context)
Citation Context ...for a positive function r ∈ C 2 (Ω). We now view the operator F as a mapping from C 2 (Ω) into L 2 (Ω) and write F(∂D) = u∞ as F(r) = u∞. (2–13) The following basic theorem was first proved by Kirsch =-=[41]-=- using variational methods and subsequently by Potthast [63] using a boundary integral equation approach (see also [34] and [48]). We note that the validity of the following theorem for the case of th... |

1 |
Acoustic scattering in Scattering
- Kress
- 2001
(Show Context)
Citation Context ...(r) = u∞. (2–13) The following basic theorem was first proved by Kirsch [41] using variational methods and subsequently by Potthast [63] using a boundary integral equation approach (see also [34] and =-=[48]-=-). We note that the validity of the following theorem for the case of the exterior Neumann problem remains an open question [50]. Theorem 2.1. The boundary to far field map F : C 2 (Ω) → L 2 (Ω) has a... |