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 through 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.

本文的目的是提出并分析一种解决线性弹性反问题的直接方法。给定介质内部某些位移场的测量结果，我们表明，即使对于不连续的参数且没有边界信息，弹性参数的稳定重建也是可能的。我们基于刚度到力算子的弱定义提供了一种通用方法，这有助于将问题视为线性系统。我们证明在剪切模量重建的情况下，我们有一个\（L ^ 2 \）在最小平滑度假设下仅进行一次测量即可获得稳定性。该稳定性结果是通过证明线性算子求反具有封闭范围而获得的。然后，我们描述了直接离散化，它提供了各向同性和各向异性刚度张量的稳定重建。