A Homotopy Method Based on Theory of Functional Connections

TL;DR

This paper introduces a novel homotopy method based on the Theory of Functional Connections, enabling more flexible and reliable path tracking for zero-finding problems by implicitly defining and selecting promising homotopy paths.

Contribution

It develops a TFC-based homotopy method with a two-layer continuation algorithm that improves path selection and failure recovery over traditional methods.

Findings

The TFC-based method effectively tracks homotopy paths in numerical simulations.

It offers a flexible path switching mechanism compared to pseudo-arclength methods.

Numerical results demonstrate the method's robustness and efficiency.

Abstract

A method for solving zero-finding problems is developed by tracking homotopy paths, which define connecting channels between an auxiliary problem and the objective problem. Current algorithms' success highly relies on empirical knowledge, due to manually, inherently selected homotopy paths. This work introduces a homotopy method based on the Theory of Functional Connections (TFC). The TFC-based method implicitly defines infinite homotopy paths, from which the most promising ones are selected. A two-layer continuation algorithm is devised, where the first layer tracks the homotopy path by monotonously varying the continuation parameter, while the second layer recovers possible failures resorting to a TFC representation of the homotopy function. Compared to pseudo-arclength methods, the proposed TFC-based method retains the simplicity of direct continuation while allowing a flexible path switching. Numerical simulations illustrate the effectiveness of the presented method.