Increasing Stability in Acoustic and Elastic Inverse Source Problems

Entekhabi, Mozhgan; Isakov, Victor

doi:10.1137/19m1279885

Cited by 19 publications

(9 citation statements)

References 23 publications

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“…Many studies have been on the inverse scattering of second-order differential equations, such as acoustic waves, 10-12 elastic waves, 10,[13][14][15][16][17][18] and electromagnetic waves. 19 The increasing stability estimate has been established in previous studies.…”

Section: Introductionmentioning

confidence: 99%

Stability of inverse scattering problem for the damped biharmonic plate equation

Liu

2022

Math Methods in App Sciences

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This paper is concerned with the stability of the inverse scattering problem for the damped biharmonic plate equation in a bounded domain with the Cauchy data. A sharp estimate for the mass density is established using a priori information concerning Sobolev norms and a priori information about the support of the inhomogeneity. Our results improve previous estimates and explicitly depend on the damping coefficient. The proof mainly relies on the complex geometric optics solution method and the Fourier analysis.

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Section: Introductionmentioning

confidence: 99%

Stability of inverse scattering problem for the damped biharmonic plate equation

Liu

2022

Math Methods in App Sciences

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“…The first stability result was obtain in [5] for the inverse source problem of the Helmholtz equation by using multifrequency data. The increasing stability was studied for the inverse source problems of the acoustic, elastic and electromagnetic wave equations [4,6,9,10,19,20]. A topic review can be found in [3] on the general inverse scattering problems with multifrequency.…”

Section: Introductionmentioning

confidence: 99%

Stability for an inverse source problem of the biharmonic operator

Li¹,

Yao²,

Zhao³

2021

Preprint

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In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in R 3 . Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr: odinger operator ∆ 2 `V pxq on a ball which contains the support of the potential V . We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions φ and their Laplacian ∆φ corresponding to the bi-Schr: odinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr: odinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr: odinger operator and prove the existence of a resonance-free region and an estimate of L 2 comp ´L2loc type for the resolvent. As an application, we prove a bound of the analytic continuation of the data from the given data to the higher frequency data. Finally, we derive the stability estimate which consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.

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“…The first stability result was obtained in [4] for the inverse source problem of the Helmholtz equation by using multi-frequency data. Later on, the increasing stability was studied for the inverse source problems of various wave equations including the acoustic, elastic, and electromagnetic wave equations, as well as the wave equation with the biharmonic Schrödinger operator [5,12,17,18,[22][23][24]. A more recent study on the stability for the inverse medium problem can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide

Li,

Yao,

Zhao

2021

Preprint

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In this paper, we study the meromorphic continuation of the resolvent for the Schrödinger operator in a three-dimensional planar waveguide. We prove the existence of a resonance-free region and an upper bound for the resolvent. As an application, the direct source problem is shown to have a unique solution under some appropriate assumptions. Moreover, an increasing stability is achieved for the inverse source problem of the Schrödinger operator in the waveguide by using limited aperture Dirichlet data only at multiple frequencies. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.

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Increasing Stability in Acoustic and Elastic Inverse Source Problems

Cited by 19 publications

References 23 publications

Stability of inverse scattering problem for the damped biharmonic plate equation

Stability of inverse scattering problem for the damped biharmonic plate equation

Stability for an inverse source problem of the biharmonic operator

Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide

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