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A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

by H. Ammari and E. Bretin and P. Millien and L. Seppecher

(Report number 2018-20)

Abstract
The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an \(L^2\)-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.

Keywords: Elastography, Inverse Problem, Shear Modulus Imaging.

@Techreport{ABMS18_774,
  author = {H. Ammari and E. Bretin and P. Millien and L. Seppecher},
  title = {A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-20.pdf },
  year = {2018}
}

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