Global Uniqueness and Stability for an Inverse Wave Source Problem for Less Regular Data

Yamamoto, Masahiro; Zhang, Xu

doi:10.1006/jmaa.2001.7621

Cited by 23 publications

(26 citation statements)

References 13 publications

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“…Observing data over spatial sets like (3) in inverse problems is not new. [7,8] In [8] (see also [9, chap.7, sec.2]), the authors analyze the wave equation problem…”

Section: Main Results To Be Provedmentioning

confidence: 99%

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Uniqueness results in the identification of distributed sources over Germain–Lagrange plates by boundary measurements

Kawano

2015

Applicable Analysis

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We prove that the loading composed by a finite sum of time-independent sources acting over a bounded plate can be uniquely identified by knowing the normal derivative of the displacement along a subset of the boundary or the displacement of the points in a certain set near the boundary. This is true even if these data correspond to an arbitrary small interval of time. The method used is to reduce the problem to another one for the wave equation. ARTICLE HISTORY

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“…Observing data over spatial sets like (3) in inverse problems is not new. [7,8] In [8] (see also [9, chap.7, sec.2]), the authors analyze the wave equation problem…”

Section: Main Results To Be Provedmentioning

confidence: 99%

“…We are going to need the following result for the wave equation found in [7]. The proof found in [7] is based on a Calerman estimate.…”

Section: Downloaded By [University Of Otago] At 08:59 04 October 2015mentioning

confidence: 97%

Uniqueness results in the identification of distributed sources over Germain–Lagrange plates by boundary measurements

Kawano

2015

Applicable Analysis

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“…To resolve such an inverse wave source problem by using the control results of distributed systems was ever conducted by Yamamoto [21,22], Bruckner and Yamamoto [23], and Yamamoto and ZHANG [24]. In higher dimensional domains the reciprocity gap functional technique leads to the reconstruction of smoother unknown sources by using the boundary observations [17,20].…”

Section: Examplementioning

confidence: 99%

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Liu¹,

Chen²

2017

JSOE

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Abstract:The inverse problems of wave equation to recover unknown space-time dependent functions of wave speed and wave source are solved in this paper, without needing of initial conditions and no internal measurement of data being required. After a homogenization technique, a sequence of spatial boundary functions at least the fourth-order polynomials are derived, which satisfy the homogeneous boundary conditions. The boundary functions and the zero element constitute a linear space, and then a new boundary functional is proved in the linear space, of which the energy is preserved for each dynamic energetic boundary function. The linear systems and iterative algorithms used to recover unknown wave speed and wave source functions with the dynamic energetic boundary functions as bases are developed, which converge fast at each time step. The input data are parsimonious, merely the measured boundary strains and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing exact solutions with estimated results under large noises up to 20%.

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“…Puel and Yamamoto [13] gave the first Lipschitz stability result for a multidimensional inverse problem for a hyperbolic equation. Later, this method was extended for many varieties of equations and system, such as [14,15], [16][17][18][19][20][21]. Especially, [14] concerned an inverse problem for a Schrödinger equation and where the method in the proof of the Lipschitz stability is the key ingredient to prove our stability result.…”

Section: Introductionmentioning

confidence: 99%

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

2012

Math Methods in App Sciences

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In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time-dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro-differential equation with hyperbolic memory kernel., provided that u 0 , u 1 are given.Many physical phenomena are properly described by the strongly damped wave equation as (1). Such as equations of type (1) can be considered as a class of linear evolution equations governing the motion of a viscoelastic solid (for example, a bar if the space dimension N D 1 and a plate if N D 2) composed of the material of the rate type; see [1][2][3][4]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the simply Voigt model [5]. Mathematical studies on this direct problem have been extensively developed, for example, see [1,2,[6][7][8][9] and the references therein, where the global existence and other properties of solutions are obtained. However, very little work is concerned with the inverse problem for this strongly damped wave equation.The key ingredient we follow here to determine the coefficient relies on the global Carleman estimates. For the first time, the method of Carleman estimates was introduced in the field of inverse problems in the work of Bukhgeim and Klibanov [10], also for some followup publications [11,12] of these authors. A Carleman estimate is an efficient tool not only for the unique continuation, but also for

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Global Uniqueness and Stability for an Inverse Wave Source Problem for Less Regular Data

Cited by 23 publications

References 13 publications

Uniqueness results in the identification of distributed sources over Germain–Lagrange plates by boundary measurements

Uniqueness results in the identification of distributed sources over Germain–Lagrange plates by boundary measurements

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

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