A number of algorithms have been proposed and analyzed for estimating a coefficient in an elliptic boundary value problem when interior data is available. Most of the analysis has been done for the simple scalar BVP
-Δ a Δ u = f in Ω,
a (∂ u / ∂ n) g on ∂ Ω
However, some methods and the associated analysis have been extended to the problem of estimating the Lamé moduli in the system of linear, isotropic elasticity. Under certain idealized conditions, convergence of estimates to the exact Lame moduli has been proved for two techniques, the output least-squares method and a variational method similar to the equation error approach.
Journal of Physics: Conference Series
Numerical analysis of elliptic inverse problems with interior data.
Journal of Physics: Conference Series,
Retrieved from: https://digitalcommons.mtu.edu/math-fp/21