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Applicable Analysis
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Volume 103, 2024 - Issue 2
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Research Article

Elastic shear modulus and density profiles inversion: Lipschitz stability results

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Pages 445-460 | Received 29 May 2022, Accepted 11 Mar 2023, Published online: 18 Mar 2023

References

  • Aki K. Quantitative seismology. Theory Methods. 1980;1:557.
  • Doyley MM. Model-based elastography: a survey of approaches to the inverse elasticity problem. Phys Med Biol. 2012;57(3):R35–R73.
  • Gennisson J-L, Deffieux T, Fink M, et al. Ultrasound elastography: principles and techniques. iagn Interv Imaging. 2013;94(5):487–495.
  • Parker KJ, Doyley MM, Rubens DJ. Imaging the elastic properties of tissue: the 20 year perspective. Phys Med Biol. 2010;56(1):R1.
  • Yuan H, Guzina BB, Sinkus R. Application of topological sensitivity toward tissue elasticity imaging using magnetic resonance data. J Eng Mech. 2014;140(3):443–453.
  • Alessandrini G. Stable determination of conductivity by boundary measurements. Appl Anal. 1988;27(1–3):153–172.
  • Harrach B. Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes. Inverse Probl. 2019;35(2):Article ID 024005.
  • Harrach B, Meftahi H. Global uniqueness and Lipschitz stability for the inverse robin transmission problem. SIAM J Appl Math. 2019;79(2):525–550.
  • Ikehata M. Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection. SIAM J Appl Math. 1990;50(6):1635–1644.
  • Imanuvilov OY, Yamamoto M. Global uniqueness in inverse boundary value problems for the Navier–Stokes equations and Lamé system in two dimensions. Inverse Probl. 2015;31(3):Article ID 035004.
  • Nakamura G, Uhlmann G. Global uniqueness for an inverse boundary problem arising in elasticity. Invent Math. 1994;118(1):457–474.
  • Beretta E, Francini E, Vessella S. Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements; 2013. arXiv preprint arXiv:1303.2443.
  • Akamatsu M, Nakamura G, Steinberg S. Identification of Lamé coefficients from boundary observations. Inverse Probl. 1991;7(3):335–354.
  • Beretta E, Francini E, Vessella S. Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements. Inverse Probl Imaging. 2014;8(3):611–644.
  • Eskin G, Ralston J. On the inverse boundary value problem for linear isotropic elasticity. nverse Probl. 2002;18(3):907–921.
  • Nakamura G, Uhlmann G. Identification of Lamé parameters by boundary measurements. Am J Math. 1993;115:1161–1187.
  • Nakamura G, Uhlmann G. Inverse problems at the boundary for an elastic medium. SIAM J Math Anal. 1995;26(2):263–279.
  • Nakamura G, Uhlmann G. Global uniqueness for an inverse boundary value problem arising in elasticity. Invent Math. 2003;152(1):205–207.
  • Alberti GS, Santacesaria M. Calderón's inverse problem with a finite number of measurements. In: Giovanni S. Alberti, Matteo Santacesaria, editors. Forum of mathematics, sigma Vol. 7. Cambridge University Press; 2019.
  • Alessandrini G, de Hoop MV, Gaburro R, et al. Lipschitz stability for a piecewise linear Schrödinger potential from local cauchy data. Asymptot Anal. 2018;108(3):115–149.
  • Alessandrini G, Vessella S. Lipschitz stability for the inverse conductivity problem. Adv Appl Math. 2005;35(2):207–241.
  • Beretta E, De Hoop MV, Qiu L. Lipschitz stability of an inverse boundary value problem for a Schrödinger-type equation. SIAM J Math Anal. 2013;45(2):679–699.
  • Beretta E, Francini E. Lipschitz stability for the electrical impedance tomography problem: the complex case. Commun Partial Differ Equ. 2011;36(10):1723–1749.
  • Gaburro R, Sincich E. Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities. Inverse Probl. 2015;31(1):Article ID 015008.
  • Arridge SR, Lionheart WR. Nonuniqueness in diffusion-based optical tomography. Opt Lett. 1998;23(11):882–884.
  • Harrach B. Simultaneous determination of the diffusion and absorption coefficient from boundary data. Inverse Probl Imaging. 2012;6(4):663–679.
  • Meftahi H. Uniqueness, Lipschitz stability, and reconstruction for the inverse optical tomography problem. SIAM J Math Analysis. 2021;53(6):6326–6354.
  • Mandache N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Probl. 2001;17(5):1435–1444.
  • Eberle S, Harrach B, Meftahi H, et al. Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity. Inverse Probl Sci Eng. 2021;29(3):396–417.
  • Harrach B, Lin Y-H, Liu H. On localizing and concentrating electromagnetic fields. SIAM J Appl Math. 2018;78(5):2558–2574.
  • Harrach B, Seo JK. Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J Math Anal. 2010;42(4):1505–1518.
  • Harrach B, Ullrich M. Local uniqueness for an inverse boundary value problem with partial data. Proc Am Math Soc. 2017;145(3):1087–1095.
  • De Hoop MV, Qiu L, Scherzer O. Local analysis of inverse problems: Hölder stability and iterative reconstruction. Inverse Probl. 2012;28(4):Article ID 045001.
  • Maarten V, Qiu L, Scherzer O. An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in banach spaces subject to stability constraints. Numer Math. 2015;129(1):127–148.
  • Alberti GS, Santacesaria M. Infinite-dimensional inverse problems with finite measurements; 2019. arXiv preprint arXiv:1906.10028.
  • Uhlmann G, Lin C-L, Nakamura G, et al. Quantitative strong unique continuation for the Lamé system with less regular coefficients. Methods Appl Anal. 2011;18(1):85–92.
  • Gebauer B. Localized potentials in electrical impedance tomography. Inverse Probl Imaging. 2008;2(2):251–269.

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