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GPBiLQ and GPQMR: Two iterative methods for unsymmetric partitioned linear systems

Kui Du, Jia-Jun Fan, Fang Wang

TL;DR

The paper introduces GPBiLQ and GPQMR for solving unsymmetric partitioned linear systems by employing a simultaneous biorthogonal tridiagonal reduction, yielding short-recurrence iterations with fixed storage. GPBiLQ and GPQMR are connected to GPBiCG and GPMR, respectively, with GPBiCG obtainable cheaply from GPBiLQ under mild non-breakdown conditions. Through numerical experiments on large sparse matrices, the authors demonstrate tradeoffs: GPBiLQ/GPBiCQ/GPQMR offer storage and computational advantages over GMRES-type methods but can exhibit slower and less monotone convergence compared to GPMR, with exact-equivalence to TriCG/TriMR in the B = A^T case. The methods extend to SPD-multiplied variants and offer practical alternatives for block-structured, unsymmetric systems with controllable memory usage, supported by open-source MATLAB code and public data.

Abstract

We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.

GPBiLQ and GPQMR: Two iterative methods for unsymmetric partitioned linear systems

TL;DR

The paper introduces GPBiLQ and GPQMR for solving unsymmetric partitioned linear systems by employing a simultaneous biorthogonal tridiagonal reduction, yielding short-recurrence iterations with fixed storage. GPBiLQ and GPQMR are connected to GPBiCG and GPMR, respectively, with GPBiCG obtainable cheaply from GPBiLQ under mild non-breakdown conditions. Through numerical experiments on large sparse matrices, the authors demonstrate tradeoffs: GPBiLQ/GPBiCQ/GPQMR offer storage and computational advantages over GMRES-type methods but can exhibit slower and less monotone convergence compared to GPMR, with exact-equivalence to TriCG/TriMR in the B = A^T case. The methods extend to SPD-multiplied variants and offer practical alternatives for block-structured, unsymmetric systems with controllable memory usage, supported by open-source MATLAB code and public data.

Abstract

We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.
Paper Structure (19 sections, 3 theorems, 157 equations, 4 figures, 4 tables)

This paper contains 19 sections, 3 theorems, 157 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A simultaneous biorthogonal tridiagonal reduction process
  3. Derivations of GPBiLQ and GPBiCG
  4. From GPBiLQ to GPBiCG
  5. Residual estimates
  6. Computational expense and storage
  7. Derivation of GPQMR
  8. Residual estimates
  9. Computational expense and storage
  10. Numerical experiments
  11. Extensions and concluding remarks
  12. GPBiLQ and GPBiCG details
  13. Computations for the LQ factorization of $\mathbf H_k$
  14. Computations for $\varpi_1$, $\ldots$, $\varpi_{2k-2}$, $\widetilde{\varpi}_{2k-1}$, and $\widetilde{\varpi}_{2k}$
  15. Computations for $\vartheta_k$, $\varrho_k$, $\chi_k$, and $\varsigma_k$
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathbf A\in\mathbb R^{m\times n}$ and $\mathbf B\in\mathbb R^{n\times m}$, and $p:=\min\{m,n\}$. There exist $\mathbf V\in\mathbb R^{m\times p}$ and $\mathbf U\in\mathbb R^{n\times p}$ with orthonormal columns, and upper Hessenberg $\mathbf H\in\mathbb R^{p\times p}$ and $\mathbf F\in\mathbb R

Figures (4)

  • Figure 1: Convergence curves on the unsymmetric system \ref{['lin']} with matrices well1033 as $\mathbf A^\top$ and illc1033 as $\mathbf B$, $\lambda=1$, and $\mu=-0.1$.
  • Figure 2: Convergence curves on the unsymmetric system \ref{['lin']} with matrices well1850 as $\mathbf A^\top$ and illc1850 as $\mathbf B$, $\lambda=1$, and $\mu=-0.05$.
  • Figure 3: Convergence curves on the symmetric system \ref{['lin']} with matrix lp_osa_07 as $\mathbf A=\mathbf B^\top$, $\lambda=1$, and $\mu=-1$.
  • Figure 4: Convergence curves on the symmetric system \ref{['lin']} with matrix lpi_klein3 as $\mathbf A=\mathbf B^\top$, $\lambda=1$, and $\mu=-1$.

Theorems & Definitions (4)

  • Theorem 1: montoison2023gpmr
  • Remark 2
  • Proposition 3
  • Proposition 4