Newton's method for nonlinear mappings into vector bundles

TL;DR

We address root finding for mappings $F:\mathcal{X}\to\mathcal{E}$ where $\mathcal{X}$ is a Banach manifold and $\mathcal{E}\to\mathcal{Y}$ a vector bundle. The authors develop a Newton framework using a connection map $Q$ and a retraction $R$ to define the Newton equation $Q_{F(x)}F'(x)\delta x + F(x)=0_{y(x)}$ and a manifold-valued update $x_+=R_x(\delta x)$. They prove local convergence under Newton- and strict-differentiability notions, introduce a Banach-type metric, and extend an affine covariant damping strategy along Newton paths for globalization. The method is demonstrated on generalized non-symmetric eigenvalue problems with numerical experiments and a companion paper explores broader variational problems on manifolds.

Abstract

We consider Newton's method for finding zeros of mappings from a manifold $\mathcal X$ into a vector bundle $\mathcal E$. In this setting a connection on $\mathcal E$ is required to render the Newton equation well defined, and a retraction on $\mathcal X$ is needed to compute a Newton update. We discuss local convergence in terms of suitable differentiability concepts, using a Banach space variant of a Riemannian distance. We also carry over an affine covariant damping strategy to our setting. Finally, we will illustrate our results by applying them to generalized non-symmetric eigenvalue problems and providing a numerical example.

Figures (1)

• Figure 1: Superlinear convergence of Newton's method (left) and damping factors $\alpha$ chosen by affine covariant damping strategy (right).

Theorems & Definitions (46)

• Remark 1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Remark 2
• Definition 3.1: Retraction
• Example 3
• Lemma 4.1
• proof