A Factorization Method for Multifrequency Inverse Source Problems with Sparse Far Field Measurements

Griesmaier, Roland; Schmiedecke, Christian

doi:10.1137/17m111290x

Cited by 44 publications

(57 citation statements)

References 30 publications

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“…is the test function defined by (4.9). To establish a lower bound for our indicator function for z ∈ M m=1 int(K S Dm (θ j )), we need to use the following lemmas in the factorization method for multifrequency inverse source problems with sparse far field measurements discussed in [31].…”

Section: Theoretical Foundation Of the Proposed Sampling Methodsmentioning

confidence: 99%

“…By combining Theorem 3.6 and Theorem 3.7 in [31], we have the following lemma which is defined in (4.9). For any z ∈ R d , z ∈ M m=1 int(K S Dm (θ j )) if and only if there…”

Section: Theoretical Foundation Of the Proposed Sampling Methodsmentioning

confidence: 99%

“…In this chapter we propose a new sampling method for multifrequency inverse source problems for time-harmonic acoustic waves with a finite set of far field data. The method is based on the factorization method for multifrequency inverse source problems with sparse far field measurements waves discussed by Griesmaier and Scmiedecke in 2017 [31]. We approximate the position and the convex geometry Furthermore, u also satisfies the asymptotic behavior…”

Section: Stability Statementmentioning

confidence: 99%

“…Note that, for D = m=M m=1 D m , we have Recently, Griesmaier [31] shows that under certain conditions on the source f , the middle set K s D (Θ) can be reconstructed by the data given in (4.5). We study the behaviors of the indicator function for each single observation direction.…”

Section: Stability Statementmentioning

confidence: 99%

“…, θ J } = Θ ⊆ S d−1 , where d = 2, 3 denotes the dimension.The main feature of this method is that the indicator function is based on the inner product and therefore the method is very simple to implement. With the help of the factorization of the corresponding far field operator[31], a lower bound estimate will be established for the sampling points inside the support of the source. Moreover, we will show that the indicator function decays as the sampling point moves away from the support of the source, and thus gives a characterization of the support of the source.In order to reduce the number of sensor locations required to obtain a useful reconstruction of the support of the sources[31], we develop a reconstruction method that efficiently utilizes multifrequency information.…”

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confidence: 99%

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Section: Theoretical Foundation Of the Proposed Sampling Methodsmentioning

confidence: 99%

“…By combining Theorem 3.6 and Theorem 3.7 in [31], we have the following lemma which is defined in (4.9). For any z ∈ R d , z ∈ M m=1 int(K S Dm (θ j )) if and only if there…”

Section: Theoretical Foundation Of the Proposed Sampling Methodsmentioning

confidence: 99%

Section: Stability Statementmentioning

confidence: 99%

Section: Stability Statementmentioning

confidence: 99%

mentioning

confidence: 99%

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The reconstruction of multipolar acoustic or electromagnetic sources from their farfield radiation patterns plays a crucial role in numerous practical applications. Most of the existing techniques for source reconstruction require dense multi-frequency data at the Nyquist sampling rate. Accessibility of only sparse data at a sub-sampled grid contributes to the null space of the inverse source-to-data operator, which causes significant imaging artifacts. For this purpose, additional knowledge about the source or regularization is required. In this article, we propose a novel two-stage strategy for multipolar source reconstruction from the sub-sampled sparse data that takes advantage of the sparsity of the sources in the physical domain. The data at the Nyquist sampling rate is recovered from sub-sampled data and then a conventional inversion algorithm is used to reconstruct sources. The data recovery problem is linked to a spectrum recovery problem for the signal with the finite rate of innovations (FIR) that is solved using an annihilating filter-based structured Hankel matrix completion approach (ALOHA). For an accurate reconstruction of multipolar sources, we employ a Fourier inversion algorithm. The suitability of the suggested approach for both noisy and noise-free measurements is supported by numerical evidence.

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