Newton's method for nonlinear mappings into vector bundles Part II: Application to variational problems

TL;DR

This paper develops a Newton's method framework for solving variational equations on manifolds, utilizing differential geometry and damping strategies, with applications demonstrated through numerical results.

Contribution

It introduces a geometric Newton's method for variational problems on manifolds, incorporating an affine covariant damping strategy, and demonstrates its effectiveness through applications.

Findings

Successful application to variational problems on manifolds

Numerical results validate the method's effectiveness

Provides differential geometric tools for Newton's method on manifolds

Abstract

We consider the solution of variational equations on manifolds by Newton's method. These problems can be expressed as root finding problems for mappings from infinite dimensional manifolds into dual vector bundles. We derive the differential geometric tools needed for the realization of Newton's method, equipped with an affine covariant damping strategy. We apply Newton's method to a couple of variational problems and show numerical results.