Acoustic source identification using multiple frequency information

Eller, Matthias; Valdivia, Nicolas

doi:10.1088/0266-5611/25/11/115005

Cited by 108 publications

(108 citation statements)

References 23 publications

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“…A similar uniqueness result is derived in [21] but for a chosen unbounded set of frequencies (the Dirichlet eigenvalues of the Laplacian). Our uniqueness result corresponds to the applications mentioned in the introduction, and the frequency continuation method developed in [5][6][7][8]10,11] where the measurements are taken on a bounded band of frequency.…”

Section: Remark 33mentioning

confidence: 62%

“…Since the singular eigenvalues of the forward linear problem are exponentially decreasing [20], it is expected that these algorithms have a logarithmic convergence. More recently, in [21], Eller and Valdivia considered the inverse problem of identifying the shape and location of a finitely supported source function from measurements of the acoustic field on a closed surface for an infinite unbounded (chosen) frequencies. They showed the uniqueness of the solutions when the set of frequencies coincides with the Dirichlet eigenvalues of the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

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A multi-frequency inverse source problem

Bao

Lin

Triki

2010

Journal of Differential Equations

187

172

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International audienceThis paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies. Rigorous stability estimates are established when the background medium is homogeneous. It is shown that the ill-posedness of the inverse problem decreases as the frequency increases. Under some regularity assumptions on the source function, it is further proven that by increasing the frequency, the logarithmic stability converts to a linear one for the inverse source problem

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Section: Remark 33mentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

A multi-frequency inverse source problem

Bao

Lin

Triki

2010

Journal of Differential Equations

187

172

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show abstract

“…We finally gave numerical evidence, that the error between the exact and regularized solution is smaller for higher wave numbers, hence, the problem shows an increasing stability (although it is of course still ill-posed). A numerical evidence of increasing stability in inverse problems can be found in [3,7,21,22]. Finally, we think that a similar analysis can be done and is useful for other problems in partial differential equations, like the lateral Cauchy problem and inverse problems for parabolic and hyperbolic equations and systems.…”

mentioning

confidence: 79%

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Isakov

Kindermann

2011

Methods and Applications of Analysis

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Abstract. We study the stability in the Cauchy Problem for the Helmholtz equation in dependence of the wave number k. For simple geometries, we show analytically that this problem is getting more stable with increasing k. In more detail, there is a subspace of the data space on which the Cauchy Problem is well posed, and this subspace grows with larger k. We call this a subspace of stability. For more general geometries, we study the ill-posedness by computing the singular values of some operators associated with the corresponding well-posed (direct) boundary value problems. Numerical computations of the singular values show that an increasing subspace of stability exists for more general domains and boundary data. Assuming the existence of such subspaces we develop a theory of (Tikhonov type) regularization for the Cauchy Problem, giving convergence rates and regularization parameter choice which improve with growing k.

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“…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…”

Section: Introductionmentioning

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.…”

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confidence: 99%

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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Acoustic source identification using multiple frequency information

Cited by 108 publications

References 23 publications

A multi-frequency inverse source problem

A multi-frequency inverse source problem

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

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