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Polynomial and Parallelizable Preconditioning for Block Tridiagonal Positive Definite Matrices

Shaohui Yang, Toshiyuki Ohtsuka, Brian Plancher, Colin N. Jones

TL;DR

This work targets efficient GPU-enabled solutions for moderately large SPD block-tridiagonal KKT systems arising in optimal control problems. It introduces a parallelizable, parametric multi-splitting polynomial preconditioner, derives spectrum- and SPD-focused theoretical guarantees, and identifies an optimal parameter set that yields highly clustered eigenvalues. A polynomial extension with Chebyshev-like coefficients further accelerates convergence, supported by explicit closed-form preconditioners for small problem sizes and comprehensive complexity analysis. Numerical experiments on synthetic OCP-derived KKT systems demonstrate reduced PCG iterations and mat-vec counts, highlighting practical impact for real-time MPC and trajectory optimization on parallel hardware.

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.

Polynomial and Parallelizable Preconditioning for Block Tridiagonal Positive Definite Matrices

TL;DR

This work targets efficient GPU-enabled solutions for moderately large SPD block-tridiagonal KKT systems arising in optimal control problems. It introduces a parallelizable, parametric multi-splitting polynomial preconditioner, derives spectrum- and SPD-focused theoretical guarantees, and identifies an optimal parameter set that yields highly clustered eigenvalues. A polynomial extension with Chebyshev-like coefficients further accelerates convergence, supported by explicit closed-form preconditioners for small problem sizes and comprehensive complexity analysis. Numerical experiments on synthetic OCP-derived KKT systems demonstrate reduced PCG iterations and mat-vec counts, highlighting practical impact for real-time MPC and trajectory optimization on parallel hardware.

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.

Paper Structure

This paper contains 23 sections, 11 theorems, 19 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric positive definite block tridiagonal matrix
  4. Matrix splittings and polynomial preconditioning
  5. Splittings of s.p.d. block tridiagonal matrix
  6. Diagonal splitting
  7. Stair splittings
  8. m-step preconditioners from multi-splitting
  9. Parametric multi-splitting family
  10. Spectrum analysis
  11. Symmetric positive definiteness analysis
  12. Example cases of parametric multi-splitting
  13. Optimal case a=1, b=-1
  14. Extreme case a=0, b=1
  15. Extreme case a=0.5, b=0
  16. ...and 8 more sections

Key Result

Lemma 2.1

neumann1877untersuchungen If $A = B- C$ is a convergent splitting of the nonsingular matrix $A$, then with eq:def-GH-from-BC,

Figures (2)

  • Figure 1: Sparsity pattern of $G_{ab}$ and $H_{ab}$ for different $(a,b)$ with $N = 7, n = 10$. The counts of nonzero entries of matrices are labeled on the top-right. $G_{ab}$ is block tridiagonal with the exception at $a=0$ (block Jacobi). $H_{ab}$ is block pentadiagonal with the exceptions at $a=0$ and $a=1$ (optimal).
  • Figure 2: Statistics for different $(a,b)$ with randomly generated s.p.d. block tridiagonal matrices $N = 30, n = 20$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1: Neumann series
  • Definition 2.4: Truncated Neumann series
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 16 more