Full identification of acoustic sources with multiple frequencies and boundary measurements

Alves, Carlos J.S.; Martins, Nuno F. M.; Roberty, Nilson C.

doi:10.3934/ipi.2009.3.275

Cited by 36 publications

(38 citation statements)

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“…In this case, however, the source can be identified from an infinite number of boundary measurements by varying the wavenumber (cf. [8]). …”

Section: Introductionmentioning

confidence: 93%

An iterative shape reconstruction of source functions in a potential problem using the MFS

Martins

2012

Inverse Problems in Science and Engineering

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In this work, we address the reconstruction of characteristic source functions in a potential problem, from the knowledge of full and partial boundary data. The inverse problem is formulated as an inverse obstacle problem and two iterative methods are applied. A decomposition method based on the Kirsch-Kress method (requires Cauchy data reconstruction) and a Newton-type method based on the domain derivative (requires the resolution of direct transmission problems) has been applied. For the reconstruction of Cauchy data we use the method of fundamental solutions (MFS) and we show that, for partial data, we can consider only one exterior artificial boundary. We test the domain derivative method using the MFS (for transmission problems) and present theoretical results that justifiy this numerical approximation. The feasibility of these methods will be illustrated by numerical simulations for both full and partial data.

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“…In this case, however, the source can be identified from an infinite number of boundary measurements by varying the wavenumber (cf. [8]). …”

Section: Introductionmentioning

confidence: 93%

An iterative shape reconstruction of source functions in a potential problem using the MFS

Martins

2012

Inverse Problems in Science and Engineering

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“…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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“…One serious issue here is the uniqueness of solution from a single pair of Cauchy data (7) and (8). Unlike the inverse conductivity problem of electrical impedance tomography (EIT) in which the knowledge of the Dirichlet-to-Neumann map suffices to determine uniquely an isotropic conductivity, in the problem under investigation a single pair of Cauchy data contains all the information necessary for reconstruction, see Alves et al (2009), in the sense that any other Cauchy data pair does not add any further information. Thus, one could take, without loss of generality, homogeneous Dirichlet data f ≡ 0 in (7) and non-identically zero Neumann data 0 ≡ g ∈ H −1/2 (∂Ω) in (8).…”

Section: Mathematical Formulationmentioning

confidence: 99%

Reconstruction of a source domain from boundary measurements

Bin‐Mohsin

Lesnic

2017

Applied Mathematical Modelling

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Inverse and ill-posed problems which consist of reconstructing the unknown support of a source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable, e.g. potential, temperature or pressure, may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free-space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free-term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknown source domain is assumed to be starshaped. This in turn enables its boundary to be parametrised by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed. Since the inverse problem is still ill-posed with respect to small errors in the data and possibly additional ill-conditioning introduced by the spectral feature of the MFS approximation, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data.

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Full identification of acoustic sources with multiple frequencies and boundary measurements

Cited by 36 publications

References 0 publications

An iterative shape reconstruction of source functions in a potential problem using the MFS

An iterative shape reconstruction of source functions in a potential problem using the MFS

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Reconstruction of a source domain from boundary measurements

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