  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.
