Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Hu, Guanghui; Kian, Yavar

doi:10.3934/ipi.2020022

Cited by 12 publications

(18 citation statements)

References 38 publications

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“…Our uniqueness proof seems new and leads straightforwardly to a numerical algorithm. Finally, the argument for recovering source terms independent of one spatial variable has simplified the corresponding proof in linear elasticity contained in [18]. Note that, although the measurement data are taken on a spherical surface, our results carry over to other non-spherical surfaces straightforwardly.…”

Section: Introductionmentioning

confidence: 83%

“…(i) Assuming that c ∈ C 1 (R n ), one can apply the local unique continuation results of [38,Theorem 1] in order to derive a global Holmgren uniqueness theorem similar to [27,Theorem 3.16] (see also [28,Theorem A.1.]). Combining this with the arguments used in [18,Theorem 2] it is possible to prove Theorem 2.1 in a more straightforward way. However, for more general coefficients c ∈ L ∞ (R n ), it is not clear that [38, Theorem 1] holds true and we can not apply such arguments.…”

Section: Spatial-dependent Source Terms In An Inhomogeneous Backgrounmentioning

confidence: 91%

“…The proof of this result can be carried out using the elliptic regularity properties of the Laplace operator (see [18,17,33,34,35]). The goal of this section is to recover both the source term F (x, t) and the embedded obstacle D from the boundary surface data {u(x, t) : |x| = R, t > 0} over an infinite time period.…”

Section: Simultaneous Determination Of Sound-soft Obstacles and Sourcmentioning

confidence: 99%

“…for j = 1, 2. By using standard argument for deriving energy estimates, we can prove that u j (x, t) (j = 1, 2) has a long time behavior which is at most of polynomial type (see e.g., [18,Proposition 9]). This allows us to define the Laplace transform of u := u 1 − u 2 with respect to the time variable as following:…”

Section: Spatial-dependent Source Terms In An Inhomogeneous Backgrounmentioning

confidence: 99%

“…However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated.…”

mentioning

confidence: 99%

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Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Kian

Zhao

2019

Acta Math. Appl. Sin. Engl. Ser.

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This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f (x)g(t) and f (x 1 , x 2 , t)h(x 3 ), where g(t) and h(x 3 ) are given and x = (x 1 , x 2 , x 3 ) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.problem for doubly hyperbolic equations arising from the nucleation rate reconstruction in the three-dimensional time cone model was analyzed in [32]. A Lipschitz stability result was proved for recovering the spatial component of the source term using interior data and an iterative thresholding algorithm (see also [26] with the final observation data) was tested. However, most of the above mentioned works dealt with recovery of time independent source terms. We refer to [9,36,2,18] where specific time-dependent source terms for hyperbolic equations were considered and to [29] for the recovery of some class of space-time-dependent source terms in the parabolic equation on a wave guide. In the time-harmonic case, inverse source problems with multi-frequency data have been extensively investigated. The increasing stability analysis in recovering spatial-dependent source terms has been carried out from both theoretical and numerical points of view (see e.g., [3,6,4,5,7,11,31,40]).In the time domain, it is very natural to transform the wave scattering problem governed by hyperbolic equations into elliptic inverse problems in the Fourier or Laplace domain with multi-frequency data; see e.g. [24] for determining sound-hard and impedance obstacles in a homogeneous background medium. In [7], the time-domain analysis helps for deriving an increasing stability to time-harmonic inverse source problems via Fourier transform. The same idea was used in [2,20,19] for recovering spatial-dependent sources as well as moving source profiles and orbits in elastodynamics and electromagnetism. The aim of this paper is to analyze the acoustic counterpart with new uniqueness results. Specially, this paper concerns the following four inverse problems with a single boundary surface data:1. Simultaneous determination of sound-soft obstacles and separable source terms in an inhomogeneous medium (Subsection 2.1).2. Simultaneous determination of sound-soft obstacles and general time-dependent source terms from a family of solutions (Subsection 2.2).3. Inverse moving point source pro...

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Section: Introductionmentioning

confidence: 83%

Section: Spatial-dependent Source Terms In An Inhomogeneous Backgrounmentioning

confidence: 91%

Section: Simultaneous Determination Of Sound-soft Obstacles and Sourcmentioning

confidence: 99%

Section: Spatial-dependent Source Terms In An Inhomogeneous Backgrounmentioning

confidence: 99%

mentioning

confidence: 99%

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Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Kian

Zhao

2019

Acta Math. Appl. Sin. Engl. Ser.

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Materials chemistry at Nankai University: A special issue dedicated to the 100th anniversary of Nankai University

Zhao

et al. 2019

Sci. China Mater.

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Proactive failure detection of instances is vitally essential to microservice systems because an instance failure can propagate to the whole system and degrade the system's performance. Over the years, many single-modal (i.e., metrics, logs, or traces) data-based anomaly detection methods have been proposed. However, they tend to miss a large number of failures and generate numerous false alarms because they ignore the correlation of multimodal data. In this work, we propose AnoFusion, an unsupervised failure detection approach, to proactively detect instance failures through multimodal data for microservice systems. It applies a Graph Transformer Network (GTN) to learn the correlation of the heterogeneous multimodal data and integrates a Graph Attention Network (GAT) with Gated Recurrent Unit (GRU) to address the challenges introduced by dynamically changing multimodal data. We evaluate the performance of AnoFusion through two datasets, demonstrating that it achieves the 𝐹 1 -score of 0.857 and 0.922, respectively, outperforming the state-of-the-art failure detection approaches. CCS CONCEPTS• Software and its engineering → Maintaining software; • Computing methodologies → Failure Detection; Graph Neural Networks.

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Uniqueness for time-dependent inverse problems with single dynamical data

Aïcha

Vashisth

et al. 2021

Journal of Mathematical Analysis and Applications

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Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

Cited by 12 publications

References 38 publications

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

Materials chemistry at Nankai University: A special issue dedicated to the 100th anniversary of Nankai University

Uniqueness for time-dependent inverse problems with single dynamical data

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