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A New Matrix Truncation Method for Improving Approximate Factorisation Preconditioners

Andreas A. Bock, Martin S. Andersen

TL;DR

This experimental work presents a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems, and proposes a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly.

Abstract

In this experimental work, we present a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems. We explore this divergence as a measure of discrepancy between a preconditioner and a matrix. Given an approximate factorisation of a given matrix, the proposed framework informs the construction of a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition (TSVD). This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values, and we prove that such truncations are minimisers of the aforementioned divergence. We present several numerical examples showing that the proposed preconditioner can reduce the number of PCG iterations compared to a preconditioner constructed using a TSVD for the same rank. We also propose a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly (e.g. in a large-scale setting) and demonstrate its effectiveness over TSVD-based approaches.

A New Matrix Truncation Method for Improving Approximate Factorisation Preconditioners

TL;DR

This experimental work presents a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems, and proposes a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly.

Abstract

In this experimental work, we present a general framework based on the Bregman log determinant divergence for preconditioning Hermitian positive definite linear systems. We explore this divergence as a measure of discrepancy between a preconditioner and a matrix. Given an approximate factorisation of a given matrix, the proposed framework informs the construction of a low-rank approximation of the typically indefinite factorisation error. The resulting preconditioner is therefore a sum of a Hermitian positive definite matrix given by an approximate factorisation plus a low-rank matrix. Notably, the low-rank term is not generally obtained as a truncated singular value decomposition (TSVD). This framework leads to a new truncation where principal directions are not based on the magnitude of the singular values, and we prove that such truncations are minimisers of the aforementioned divergence. We present several numerical examples showing that the proposed preconditioner can reduce the number of PCG iterations compared to a preconditioner constructed using a TSVD for the same rank. We also propose a heuristic to approximate the proposed preconditioner in the case where exact truncations cannot be computed explicitly (e.g. in a large-scale setting) and demonstrate its effectiveness over TSVD-based approaches.
Paper Structure (15 sections, 4 theorems, 73 equations, 6 figures, 9 tables)

This paper contains 15 sections, 4 theorems, 73 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Approximation problem
  4. On the asymmetry of the Bregman divergence
  5. A preconditioner with Bregman log determinant truncation
  6. Approximating the Bregman log determinant truncation
  7. Estimating the largest eigenvalues
  8. Smallest eigenvalues
  9. Approximating the proposed preconditioner
  10. Randomised methods
  11. Numerical experiments with PCG
  12. Improving incomplete Cholesky using low-rank compensation
  13. Results for larger instances
  14. Summary & outlook
  15. Acknowledgements

Key Result

Theorem 1

$\mathopen{\hbox{$\m@th{\langle}$}\mkern2mu\hbox{$\m@th{\langle}$}}\tilde{E} \mathclose{\hbox{$\m@th{\rangle}$}\mkern2mu\hbox{$\m@th{\rangle}$}}_{ r }$ is a minimiser of eq:Bregman_truncation:problem.

Figures (6)

  • Figure 1: Bregman curve $\gamma$ in \ref{['eq:bregman_curve']}.
  • Figure 2: This depicts the situation in \ref{['ex:breg_vs_svd:diagonal']} where we compare a TSVD and BLD truncations of a diagonal matrix $I + \tilde{E}$. The continuous curve traced on the left is \ref{['eq:bregman_curve']}.
  • Figure 3: Reverse Bregman curve $\nu$ defined in \ref{['eq:bregman_curve:inv']}. The dashed line is the Bregman curve $\gamma$ from \ref{['eq:bregman_curve']}.
  • Figure 4: Left: spectrum of $\tilde{E} = Q^{-1} S Q^{-*} - I$, where $S$ is the matrix HB/1138_bus taken from SuiteSparse, and $Q$ is its zero fill incomplete Cholesky factorisation. Right: values of $\tilde{E}$ under the image of $\gamma$ and $\nu$ (defined in \ref{['eq:bregman_curve']} and \ref{['eq:bregman_curve:inv']}, respectively).
  • Figure 5: Experiment with HB/lund_a ($n=147$ and $r=7$). Top: SVD and Bregman curve. Bottom: eigenvalues of $\tilde{E}$ and PCG convergence.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2: Bregman log determinant truncation
  • Example 1: Diagonal matrix
  • Theorem 1
  • proof
  • Definition 3: Reverse Bregman log determinant truncation
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • ...and 3 more