  We study the Levenberg-Marquardt (L-M) method for solving the highly
nonlinear and ill-posed inverse problem of identifying the Robin coefficients
in elliptic and parabolic systems. The L-M method transforms the Tikhonov
regularized nonlinear non-convex minimizations into convex minimizations. And
the quadratic convergence of the L-M method is rigorously established for the
nonlinear elliptic and parabolic inverse problems for the first time, under a
simple novel adaptive strategy for selecting regularization parameters during
the L-M iteration. Then the surrogate functional approach is adopted to solve
the strongly ill-conditioned convex minimizations, resulting in an explicit
solution of the minimisation at each L-M iteration for both the elliptic and
parabolic cases. Numerical experiments are provided to demonstrate the accuracy
and efficiency of the methods.
