A Levenberg-Marquardt Method based on Sobolev gradients P. Kazemi * , R. J. Renka † June 18, 2012 Abstract We extend the theory of Sobolev gradients to include variable met- ric methods, such as Newton’s method and the Levenberg-Marquardt method, as gradient descent iterations associated with stepwise variable inner products. In particular, we obtain existence, uniqueness, and asymp- totic convergence results for a gradient flow based on a variable inner product. Keywords: Gradient system, Least squares, Levenberg-Marquardt, Sobolev gradient, Variable metric 1 Introduction A nonlinear partial differential equation may be formulated as the problem of minimizing a sum of squared residuals. Consider the numerical solution of the corresponding discretized nonlinear least squares problem. Methods for treating this problem include steepest descent, Newton, Gauss-Newton, and Levenberg-Marquardt which combines a Newton or Gauss-Newton iteration with the method of steepest descent. In order to be effective, the numerical method should emulate an iteration in the infinite-dimensional Sobolev space in which the PDE is formulated. The standard steepest descent and Levenberg- Marquardt methods use a discretized L 2 gradient rather than a Sobolev gra- dient, and thus lack integrity because they approximate iterations that fail to maintain the smoothness required of the solution. A generalization of the Levenberg-Marquardt method, along with an equivalent trust-region method, are described in [13]. The purpose of this work is to provide a theoretical basis for that method. Methods for analyzing partial differential equations may be more or less strongly connected to numerical algorithms. The fixed point theorems of Schauder * Lindauerstrasse 100, 89079 Ulm, DE (parimah.kazemi@gmail.com) † Department of Computer Science & Engineering, University of North Texas, Denton, TX 76203-1366 (renka@cs.unt.edu) 1