Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Isakov, Victor; Kindermann, Stefan

doi:10.4310/maa.2011.v18.n1.a1

Cited by 30 publications

(36 citation statements)

References 26 publications

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“…The Lipschitz stability estimate for the inverse source problems is also addressed in [11] under sufficiently high-frequency information. This increasing stability for Helmholtz Cauchy problems and wave imaging by measurements under high frequencies was also observed in [25,5], respectively.…”

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confidence: 61%

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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mentioning

confidence: 61%

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Bao¹,

Lü²,

Rundell³

et al. 2015

SIAM J. Numer. Anal.

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show abstract

“…Ill-posedness occurs at the stage of the continuation of solutions of partial differential equations from observation set toward an obstacle. Several rigorous justifications of the increasing stability phenomena in the Cauchy (or continuation) problem in different settings were obtained by Isakov et al [HI,I2,IK,AI]. These justifications are in form of conditional stability estimates which are getting nearly Lipschitz when the wave number k is getting large.…”

Section: Introductionmentioning

confidence: 99%

Increasing stability for near field from the scattering amplitude

Isakov¹

2015

Contemporary Mathematics

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Abstract. We obtain stability estimates for the near field of a radiating solution of the Helmholtz equation from the far field (scattering amplitude). This estimates contain a best possible Lipschitz term, a Hölder term, and terms which decay as powers of the frequency k for large k under some a priori bounds. These estimates contain only explicit constants and show increasing stability of recovery of the near field from scattering amplitude with growing k. Proofs are elementary and are based on new explicit bounds for Hankel functions. We give first applications to increasing stability in (linearized) inverse scattering by obstacles.

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“…These formulas show that the exponentially ill-posed characteristics of the inverse medium scattering problem at a fixed frequency [4,19,22] is due to the exponential decay of the scattering coefficients. Moreover, they clearly indicate the stability of the reconstruction from multifrequency measurements [16,17,20,21]. Based on the decay property of the inhomogeneous scattering coefficients, a resolution analysis analogous to the one in [6] can be easily derived.…”

Section: Introductionmentioning

confidence: 90%

The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering

Ammari

Chow

Zou

2014

SIAM J. Math. Anal.

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This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems. Mathematics Subject Classification (MSC2000): 35R30, 35B30

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Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Cited by 30 publications

References 26 publications

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Increasing stability for near field from the scattering amplitude

The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering

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