Multidimensional Manifold Continuation for Adaptive Boundary-Value Problems

Abstract

Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.