[Paper Review] Robust and effective eSIF preconditioning for general SPD matrices

Jianlin Xia|arXiv (Cornell University)|Jul 7, 2020

Electromagnetic Scattering and Analysis20 references4 citations

TL;DR

This paper introduces an enhanced SIF (eSIF) preconditioner for general symmetric positive definite (SPD) matrices, using a two-sided block triangular preprocessing and improved scaling-and-compression strategies to achieve significantly higher approximation accuracy, better condition number reduction, and superior eigenvalue clustering around 1. The eSIF method is unconditionally robust, guaranteeing positive definiteness without stabilization, and outperforms the original SIF in convergence speed and effectiveness across diverse test matrices.

ABSTRACT

We propose an unconditionally robust and highly effective preconditioner for general symmetric positive definite (SPD) matrices based on structured incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular preprocessing. A practical multilevel eSIF scheme is then designed. We give rigorous analysis for both the enhanced scaling-and-compression strategy and the multilevel eSIF preconditioner. The new eSIF framework has some significant advantages and overcomes some major limitations of the SIF strategy. (i) With the same tolerance for compressing the off-diagonal blocks, the eSIF preconditioner can approximate the original matrix to a much higher accuracy. (ii) The new preconditioner leads to much more significant reductions of condition numbers due to an accelerated magnification effect for the decay in the singular values of the scaled off-diagonal blocks. (iii) With the new preconditioner, the eigenvalues of the preconditioned matrix are much better clustered around $1$. (iv) The multilevel eSIF preconditioner is further unconditionally robust or is guaranteed to be positive definite without the need of extra stabilization, while the multilevel SIF preconditioner has a strict requirement in order to preserve positive definiteness. Comprehensive numerical tests are used to show the advantages of the eSIF preconditioner in accelerating the convergence of iterative solutions.

Motivation & Objective

• To address the robustness and accuracy limitations of the original SIF preconditioner for general SPD matrices.
• To overcome the strict positive definiteness requirement in multilevel SIF schemes that can lead to breakdown.
• To enhance the decay magnification effect in singular values of off-diagonal blocks to accelerate condition number reduction.
• To design a practical multilevel eSIF scheme that maintains high effectiveness and stability.
• To provide rigorous analysis of the enhanced scaling-and-compression strategy and its impact on preconditioner quality.

Proposed method

• Apply two-sided block triangular preprocessing to the SPD matrix before compression, improving the conditioning of off-diagonal blocks.
• Implement an enhanced scaling-and-compression strategy that uses scaled off-diagonal blocks $ C = L_1^{-1} A_{12} L_2^{-T} $ for low-rank approximation.
• Retain only the largest $ r $ singular values of $ C $, with truncation rank $ r $ controlling approximation accuracy.
• Construct a structured incomplete factorization based on the compressed blocks, ensuring the preconditioner remains SPD.
• Extend the one-level eSIF to a multilevel scheme by recursively applying the same strategy to diagonal blocks.
• Use rigorous analysis to prove that the eSIF preconditioner achieves a relative error bound of $ ilde{ ho} $, with condition number $ rac{1+ ilde{ ho}}{1- ilde{ ho}} $, where $ ilde{ ho} o 0 $ faster than in SIF.

Experimental results

Research questions

• RQ1 Can the eSIF preconditioner achieve higher approximation accuracy than the original SIF for the same compression tolerance?
• RQ2 Does the enhanced scaling-and-compression strategy lead to a faster decay magnification effect in the singular values of off-diagonal blocks?
• RQ3 Can the multilevel eSIF preconditioner be unconditionally robust, i.e., guaranteed to be positive definite without stabilization?
• RQ4 How does the eigenvalue distribution of the preconditioned matrix under eSIF compare to that under SIF, particularly in clustering around 1?
• RQ5 To what extent does eSIF accelerate the convergence of the PCG method across diverse SPD matrices?

Key findings

• The eSIF preconditioner achieves a relative error bound of $ 1.23 imes 10^{-14} $ for $ r=8 $, compared to $ 9.09 imes 10^{-13} $ for SIF, indicating significantly higher approximation accuracy.
• The condition number of the preconditioned matrix is reduced to 1.00 for eSIF (with $ r=8 $), while SIF reaches up to $ 6.21 imes 10^1 $, demonstrating superior conditioning.
• PCG convergence with eSIF requires only 2–3 iterations for $ r=8 $, while SIF requires 10–75 iterations, showing a 10–75× acceleration in convergence.
• For the MHD3200B matrix, eSIF reduces PCG iterations to 1/11 of SIF’s count, and for LinProg, the reduction is 1/7, confirming consistent acceleration across test cases.
• The eigenvalues of the preconditioned matrix under eSIF are tightly clustered around 1, with minimal spread, unlike SIF which shows wider dispersion.
• The multilevel eSIF preconditioner is unconditionally robust and does not require additional stabilization to preserve positive definiteness, unlike multilevel SIF.