An Open Newton Method for Piecewise Smooth Functions

TL;DR

This paper introduces an open Newton method for piecewise smooth functions that relaxes the need for local bijectivity, broadening applicability and demonstrating robustness through theoretical analysis and an application in cardiovascular mathematics.

Contribution

It develops a new Newton-type algorithm based on local openness rather than bijectivity, supported by a weak implicit function theorem and structural analysis of PS functions.

Findings

The method converges under weaker conditions than classical semismooth Newton.

Existence of PS functions where all Clarke Jacobian elements are singular at roots.

Application to cardiovascular mathematics illustrates practical utility.

Abstract

Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. For local bijectivity of the linearization at a root, a radius of quadratic convergence was explicitly calculated in terms of local Lipschitz constants of the underlying PS function. In the present work we relax the criterium of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions $f:\mathbb R^2\rightarrow\mathbb R^2$ satisfying the weaker criterium where every neighborhood of the root of $f$ contains a point $x$ such that all elements of the Clarke Jacobian at $x$ are singular. In such neighborhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given.