  The problem of finding roots or solutions of a nonlinear partial differential
equation may be formulated as the problem of minimizing a sum of squared
residuals. One then defines an evolution equation so that in the asymptotic
limit a minimizer, and often a solution of the PDE, is obtained. The
corresponding discretized nonlinear least squares problem is an often met
problem in the field of numerical optimization, and thus there exist a wide
variety of methods for solving such problems. We review here Newton's method
from nonlinear optimization both in a discrete and continuous setting and
present results of a similar nature for the Levernberg-Marquardt method. We
apply these results to the Ginzburg-Landau model of superconductivity.
