  A robust algorithm is proposed to reconstruct the spatial support and the
Lam\'e parameters of multiple inclusions in a homogeneous background elastic
material using a few measurements of the displacement field over a finite
collection of boundary points. The algorithm does not require any linearization
or iterative update of Green's function but still allows very accurate
reconstruction. The breakthrough comes from a novel interpretation of
Lippmann-Schwinger type integral representation of the displacement field in
terms of unknown densities having common sparse support on the location of
inclusions. Accordingly, the proposed algorithm consists of a two-step
approach. First, the localization problem is recast as a joint sparse recovery
problem that renders the densities and the inclusion support simultaneously.
Then, a noise robust constrained optimization problem is formulated for the
reconstruction of elastic parameters. An efficient algorithm is designed for
numerical implementation using the Multiple Sparse Bayesian Learning (M-SBL)
for joint sparse recovery problem and the Constrained Split Augmented
Lagrangian Shrinkage Algorithm (C-SALSA) for the constrained optimization
problem. The efficacy of the proposed framework is manifested through extensive
numerical simulations. To the best of our knowledge, this is the first
algorithm tailored for parameter reconstruction problems in elastic media using
highly under-sampled data in the sense of Nyquist rate.
