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An Efficient Scaled spectral preconditioner for sequences of symmetric positive definite linear systems

Youssef Diouane, Selime Gürol, Oussama Mouhtal, Dominique Orban

TL;DR

This work targets efficient solution of sequences of SPD systems that arise in data assimilation by introducing a scaled spectral preconditioner F_θ that clusters the top k eigenvalues at a parameter θ while leaving the remainder of the spectrum unchanged. The authors develop three strategies for selecting θ and provide rigorous results showing that, when θ lies in the interval [λ_{k+1}, λ_k], PCG improves or at least preserves the energy-norm reduction compared to unpreconditioned CG, with an explicit optimal θ_r for minimizing the initial residual. They further connect the approach to deflated CG by analyzing mid-range clustering, deriving bounds that relate to deflation performance, and demonstrating that suitable θ choices can emulate deflation-like behavior at low cost. Numerical experiments in a data-assimilation setting using Lorenz-96 validate that the scaled LMP substantially accelerates early CG iterations with minimal overhead, making it competitive with deflated CG while avoiding its higher computational burden. Overall, the paper provides both theoretical guarantees and practical guidance for leveraging eigenvalue placement in preconditioners to boost the efficiency of CG on SPD sequences.

Abstract

We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but also with consideration of its use within the conjugate gradient (CG) method. We propose three different strategies for selecting a scaling parameter, which aims to position the eigenvalues of the preconditioned matrix in a way that reduces the energy norm of the error, the quantity that CG monotonically decreases at each iteration. Our focus is on accelerating convergence especially in the early iterations, which is particularly important when CG is truncated due to computational cost constraints. Numerical experiments provide in data assimilation confirm that the scaled spectral preconditioner can significantly improve early CG convergence with negligible computational cost.

An Efficient Scaled spectral preconditioner for sequences of symmetric positive definite linear systems

TL;DR

This work targets efficient solution of sequences of SPD systems that arise in data assimilation by introducing a scaled spectral preconditioner F_θ that clusters the top k eigenvalues at a parameter θ while leaving the remainder of the spectrum unchanged. The authors develop three strategies for selecting θ and provide rigorous results showing that, when θ lies in the interval [λ_{k+1}, λ_k], PCG improves or at least preserves the energy-norm reduction compared to unpreconditioned CG, with an explicit optimal θ_r for minimizing the initial residual. They further connect the approach to deflated CG by analyzing mid-range clustering, deriving bounds that relate to deflation performance, and demonstrating that suitable θ choices can emulate deflation-like behavior at low cost. Numerical experiments in a data-assimilation setting using Lorenz-96 validate that the scaled LMP substantially accelerates early CG iterations with minimal overhead, making it competitive with deflated CG while avoiding its higher computational burden. Overall, the paper provides both theoretical guarantees and practical guidance for leveraging eigenvalue placement in preconditioners to boost the efficiency of CG on SPD sequences.

Abstract

We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but also with consideration of its use within the conjugate gradient (CG) method. We propose three different strategies for selecting a scaling parameter, which aims to position the eigenvalues of the preconditioned matrix in a way that reduces the energy norm of the error, the quantity that CG monotonically decreases at each iteration. Our focus is on accelerating convergence especially in the early iterations, which is particularly important when CG is truncated due to computational cost constraints. Numerical experiments provide in data assimilation confirm that the scaled spectral preconditioner can significantly improve early CG convergence with negligible computational cost.
Paper Structure (19 sections, 10 theorems, 70 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 19 sections, 10 theorems, 70 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. CG algorithm
  5. Convergence properties of CG
  6. Properties of a good preconditioner
  7. Preconditioning for a sequence of linear systems
  8. A scaled spectral preconditioner
  9. On the choice of the scaling parameter
  10. theta providing lower error in energy norm
  11. Optimal choice for theta with respect to the initial residual
  12. theta as the mid-range between lambdak and lambdan
  13. Discussion
  14. Numerical Experiments
  15. Setup
  16. ...and 4 more sections

Key Result

Theorem 4.1

\newlabeltheorem:ErrorPrecSys0 Let $\hat{x}_\ell(\theta)$ be generated at iteration $\ell$ of Algo:PCG applied to $A x = b$ with preconditioner eq:ftheta. Then, where $\eta_i = s_i^\top r_0$ is the $i$-th component of the initial residual in the basis $S$.

Figures (4)

  • Figure 1: Quadratic cost function values along all CG iterates (left) and with respect to the number of matrix-vector product with the matrix $A^{(1)}$ and $A^{(2)}$ (right).
  • Figure 1: Quadratic cost function values along all CG iterates and with respect to the number of matrix-vector product for the HighObs scenario.
  • Figure 2: Spectrum of $U_{\theta_1}^{(1)} A^{(2)} U_{\theta_1}^{(1)}$ for different values of $\theta_1$ on a logarithmic scale. LowObs scenario $(k = 45)$.
  • Figure 2: Spectrum of $U_{\theta_1}^{(1)} A^{(2)} U_{\theta_1}^{(1)}$ with different $\theta_1$ for the HighObs scenario $(k = 26)$.

Theorems & Definitions (19)

  • Theorem 4.1
  • Proof 1
  • Proposition 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • Theorem 4.4
  • Proof 4
  • Corollary 4.5
  • Theorem 4.6
  • ...and 9 more