Nonlinear optimization in Hilbert space using Sobolev gradients with applications

Details Nonlinear optimization in Hilbert space using Sobolev gradients with applications Nonlinear optimization in Hilbert space using Sobolev gradients with applications

2011-12-14

arxiv.org/abs/1112.3150v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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