On the multi-frequency inverse source problem in heterogeneous media

Acosta, Sebastián; Chow, S. S.; Taylor, James S.; Villamizar, Vianey

doi:10.1088/0266-5611/28/7/075013

Cited by 47 publications

(69 citation statements)

References 51 publications

(128 reference statements)

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“…For instance, [22] proves that the unique recovery of the source function f (x) holds true if the trace measurement u(x, k) is known for all x ∈ ∂Ω and all k ∈ K, where K is a set of the eigenvalues of the negative Laplacian in the domain Ω. Further investigation in [1] extends the uniqueness result to heterogeneous media. At the same time, by choosing K an interval or a sequence with a limit point, [11] and [2] have proved the uniqueness in homogeneous media by unique continuation of u(x, k) by relying on analyticity with respect to the wavenumber k. Downloaded 07/11/15 to 128.122.253.228.…”

Section: Inverse Source Problem and Ill-posednessmentioning

confidence: 95%

“…1 unit norm. Using the same 150 wavenumbers, the reconstructed solution using the discrepancy principle for truncating the iteration is shown in Figure 5.…”

Section: It Follows Thatmentioning

confidence: 99%

“…Meanwhile, the nonuniqueness source identification allows minimum norm (energy) solutions [19,30,29] where physical constraints are naturally included. To uniquely identify the unknown sources, multifrequency measurements are employed in [1,2,12,22]. In all these works the unknown sources are represented by a linear combination of different basis functions, for instance, the standard Fourier basis functions in [2], finite element basis functions in [12], and eigenfunctions of Dirichlet eigenvalue problems for homogeneous and nonhomogeneous media in [1,22], respectively.…”

mentioning

confidence: 99%

“…To uniquely identify the unknown sources, multifrequency measurements are employed in [1,2,12,22]. In all these works the unknown sources are represented by a linear combination of different basis functions, for instance, the standard Fourier basis functions in [2], finite element basis functions in [12], and eigenfunctions of Dirichlet eigenvalue problems for homogeneous and nonhomogeneous media in [1,22], respectively. To numerically reconstruct the unknown sources, one needs to recursively recover the coefficients by measurements taken for different frequencies.…”

mentioning

confidence: 99%

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A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Bao¹,

Lü²,

Rundell³

et al. 2015

SIAM J. Numer. Anal.

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An iterative/recursive algorithm is studied for recovering unknown sources of acoustic field with multifrequency measurement data. Under additional regularity assumptions on source functions, the first convergence result toward multifrequency inverse source problems is obtained by assuming the background medium is homogeneous and the measurement data is noise-free. Error estimates are also provided when the observation data is contaminated by noise. Numerical examples verify the reliability and efficiency of our proposed algorithm. Introduction.Identification and recovery of acoustic sources from measurable radiated waves have attracted considerable attention in both forward and inverse scattering theory. Such inverse source problems have wide applications in antenna synthesis [7], biomedical imaging [3], and, in particular, tomographic problems [6,34]. For inverse source and other scattering problems with stochastic phenomena we mention [8,9] and references therein.A theoretical understanding of the inverse source problems starts with the expansion of wave fields generated by a localized source which can be represented in the form of an integral equation, for instance, in [32] and [13] independently. A closer investigation on this integral equation yields the definition of radiating and nonradiating sources; cf. [20,31]. The latter paper collects an infinite number of sources with vanishing radiating fields outside of certain domains. Of course, such results are based on a well-known observation, as we show in subsection 2.2.Such observations further strengthen the theoretical analysis in inverse source problems where nonuniqueness within finite frequency observation is commonly accepted, i.e., [21]. Recent results in [22] prove that unique identification holds by measuring the acoustic field of an infinite and unbounded set of frequencies on a closed boundary. Further refinement is provided in [11], where the unknown source

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Section: Inverse Source Problem and Ill-posednessmentioning

confidence: 95%

“…1 unit norm. Using the same 150 wavenumbers, the reconstructed solution using the discrepancy principle for truncating the iteration is shown in Figure 5.…”

Section: It Follows Thatmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Bao¹,

Lü²,

Rundell³

et al. 2015

SIAM J. Numer. Anal.

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show abstract

“…A specific example is the magnetoencephalography (MEG) [18,22] in which it is desired to determine the location of electric activity within the brain from the induced magnetic fields outside the head. Motivated by these significant applications, the inverse source problem has been exhaustively studied in the literature [2][3][4][5]8,[11][12][13][14][18][19][20][21][22]24,25,[33][34][35][36]38,[40][41][42]. The primary difficulty is that the inverse source problem at a fixed frequency does not have a unique solution due to non-radiating sources [3,13].…”

mentioning

confidence: 99%

Inverse Electromagnetic Source Scattering Problems with MultiFrequency Sparse Phased and Phaseless Far Field Data

Ji¹,

Liu²

2019

SIAM J. Sci. Comput.

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This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multi-frequency sparse phased or phaseless far field data. With the phased data, we show that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multi-frequency far field pattern at a single observation direction. A phased direct sampling method is also proposed to reconstruct the strip. The phaseless far field data is closely related to the outward energy flux, which can be measured more easily in practice. We show that the phaseless far field data is invariant under the translation of the sources, which implies that the location of the sources can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the scattering of magnetic dipoles with one fixed source point and at most three scattering strengths. With this technique, a fast and stable phase retrieval approach is proposed based on a simple geometrical result which provides a stable reconstruction of a point in the plane from three distances to the given points. A novel phaseless direct sampling method is also proposed to reconstruct the strip. The phase retrieval approach can also be combined with the phased direct sampling method to reconstruct the strip. Finally, to obtain a source support, we just need to superpose the indicators with respect to the sparse observation directions. Extended numerical examples in three dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval approach and direct sampling methods.

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The inverse problem of dipolar source identification from incomplete boundary measurements

Mahjoub,

Mdimagh

2023

Math Methods in App Sciences

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We consider and combine two methods to solve the inverse problem of locating dipolar sources in a restricted domain from measurements of the Cauchy data on a part of the boundary for the Laplace equation. We begin by extending the boundary measurements, which are accessible only on a part of the boundary by solving an extremal bounded problem in an annulus. The inverse problem of source identification is then solved using the reciprocity gap concept applied to the completed data. In order to evaluate the effectiveness of source detection with regard to the expansion of the data, some numerical tests are provided.

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On the multi-frequency inverse source problem in heterogeneous media

Cited by 47 publications

References 51 publications

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Inverse Electromagnetic Source Scattering Problems with MultiFrequency Sparse Phased and Phaseless Far Field Data

The inverse problem of dipolar source identification from incomplete boundary measurements

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