  Inspired by certain regularization techniques for linear inverse problems, in
this work we investigate the convergence properties of the Levenberg-Marquardt
method using singular scaling matrices. Under a completeness condition, we show
that the method is well-defined and establish its local quadratic convergence
under an error bound assumption. We also prove that the search directions are
gradient-related allowing us to show that limit points of the sequence
generated by a line-search version of the method are stationary for the
sum-of-squares function. The usefulness of the method is illustrated with some
examples of parameter identification in heat conduction problems for which
specific singular scaling matrices can be used to improve the quality of
approximate solutions.
