Polynomial and Parallelizable Preconditioning for Block Tridiagonal Positive Definite Matrices
Abstract
The efficient solution of moderately large-scale linear systems arising from the KKT conditions in optimal control problems (OCPs) is a critical challenge in robotics. With the stagnation of Moore's law, there is growing interest in leveraging GPU-accelerated iterative methods, and corresponding parallel preconditioners, to overcome these computational challenges. To improve the performance of such solvers, we introduce a parallel-friendly, parametrized multi-splitting polynomial preconditioner framework. We first construct and prove the optimal parametrization theoretically in terms of the least amount of distinct eigenvalues and the narrowest spectrum range. We then compare the theoretical time complexity of solving the linear system directly or iteratively. We finally show through numerical experiments how much the preconditioning improves the convergence of OCP linear systems solves.
Cite
@article{arxiv.2503.15269,
title = {Polynomial and Parallelizable Preconditioning for Block Tridiagonal Positive Definite Matrices},
author = {Shaohui Yang and Toshiyuki Ohtsuka and Brian Plancher and Colin N. Jones},
journal= {arXiv preprint arXiv:2503.15269},
year = {2025}
}
Comments
The initial submission of this work was reviewed in IEEE Control Systems Letters (L-CSS) with an option for presentation at the 2025 Conference on Decision and Control (CDC). This revised version incorporates all feedback from reviewers and editors, addressing theoretical clarifications, structural improvements, and methodological refinements suggested during the review process