Computing the Bouligand derivative of a class of piecewise-differentiable flows

TL;DR

This paper presents an algorithm to compute the Bouligand derivative of certain piecewise-differentiable flows, using an exponential number of points to efficiently evaluate the derivative as a piecewise-linear homeomorphism.

Contribution

It introduces a novel method to compute the Bouligand derivative for event-selected $C^r$ vector fields, overcoming factorial complexity with a linear algebra approach.

Findings

The B-derivative can be represented as a piecewise-linear homeomorphism.

Evaluation of the B-derivative reduces to linear algebra computations.

The number of pieces in the B-derivative is factorial in the dimension.

Abstract

Event--selected $C^r$ vector fields yield piecewise-differentiable flows, which possess a continuous and piecewise-linear Bouligand (or B-)derivative; here we provide an algorithm for computing this B-derivative. The number of "pieces" of the piecewise-linear B-derivative is factorial ($d!$) in the dimension ($d$) of the space, precluding a polynomial-time algorithm. We show how an exponential number ($2^d$) of points can be used to represent the B-derivative as a piecewise-linear homeomorphism in such a way that evaluating the derivative reduces to linear algebra computations involving a matrix constructed from $d$ of these points.