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Symmetric Stair Preconditioning of Linear Systems for Parallel Trajectory Optimization

Xueyi Bu, Brian Plancher

TL;DR

It is proved that the new parallel-friendly symmetric stair preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum.

Abstract

There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems. Iterative methods are particularly well-suited for parallel solves of such systems. However, fast and stable convergence of iterative methods is reliant on the application of a high-quality preconditioner that reduces the spread and increase the clustering of the eigenvalues of the target matrix. To improve the performance of these approaches, we present a new parallel-friendly symmetric stair preconditioner. We prove that our preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum. Numerical experiments with typical trajectory optimization problems reveal that as compared to the best alternative parallel preconditioner from the literature, our symmetric stair preconditioner provides up to a 34% reduction in condition number and up to a 25% reduction in the number of resulting linear system solver iterations.

Symmetric Stair Preconditioning of Linear Systems for Parallel Trajectory Optimization

TL;DR

It is proved that the new parallel-friendly symmetric stair preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum.

Abstract

There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems. Iterative methods are particularly well-suited for parallel solves of such systems. However, fast and stable convergence of iterative methods is reliant on the application of a high-quality preconditioner that reduces the spread and increase the clustering of the eigenvalues of the target matrix. To improve the performance of these approaches, we present a new parallel-friendly symmetric stair preconditioner. We prove that our preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum. Numerical experiments with typical trajectory optimization problems reveal that as compared to the best alternative parallel preconditioner from the literature, our symmetric stair preconditioner provides up to a 34% reduction in condition number and up to a 25% reduction in the number of resulting linear system solver iterations.
Paper Structure (12 sections, 8 theorems, 48 equations, 3 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 48 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Direct Trajectory Optimization
  4. Parallel Iterative Solvers
  5. Parallel Preconditioners
  6. Stair Preconditioners
  7. Eigenpairs of $\Psi_l$ and $\Psi_l$
  8. Eigenpairs of $\Phi_{add}$
  9. The Symmetric Stair Precontioner
  10. Numerical Results
  11. Conclusion and Future Work
  12. Appendix

Key Result

Lemma III.1

Given the stair-splittings of a symmetric block tridiagonal matrix $S=\Psi_l - P_l = \Psi_r - P_r$ for a $n\times n$ block $S$, where each block is $m\times m$, $\Psi_l^{-1}P_l$ and $\Psi_r^{-1}P_r$ have the same spectrum.

Figures (3)

  • Figure 1: Distribution of the Eigenvalues of the additive and symmetric stair preconditioners matching the theoretical results in Equation \ref{['eq:finalVals']}.
  • Figure 2: The relative condition number resulting from different preconditioners, normalized to the value of the Jacobi preconditioner, showing the improved performance of the symmetric stair preconditioner.
  • Figure 3: The number of PCG iterations required for convergence to the same exit tolerance across different preconditioners and problems, again showing the improved performance of the symmetric stair preconditioner.

Theorems & Definitions (16)

  • Lemma III.1
  • proof
  • Lemma III.2
  • proof
  • Theorem III.3
  • proof
  • Theorem III.4
  • proof
  • Theorem III.5
  • proof
  • ...and 6 more