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The stability of split-preconditioned FGMRES in four precisions

Erin Carson, Ieva Daužickaitė

TL;DR

The paper develops a four-precision framework for split-preconditioned FGMRES, enabling the use of different precisions for A, the left preconditioner, the right preconditioner, and the working computations. It derives normwise backward- and forward-error bounds under realistic perturbation models and provides concrete guidance for selecting u_A, u_L, and u_R to achieve backward error on the order of the working precision, with results applicable to general preconditioners. Through dense and SuiteSparse experiments, it demonstrates how split-preconditioning can improve conditioning and how precision choices influence convergence and accuracy, highlighting the central role of the left preconditioner in forward error control. Overall, the work offers a practical, theory-backed strategy for robust mixed-precision FGMRES in four precisions, applicable to a broad class of preconditioners and problems.

Abstract

We consider the split-preconditioned FGMRES method in a mixed precision framework, in which four potentially different precisions can be used for computations with the coefficient matrix, application of the left preconditioner, application of the right preconditioner, and the working precision. Our analysis is applicable to general preconditioners. We obtain bounds on the backward and forward errors in split-preconditioned FGMRES. Our analysis further provides insight into how the various precisions should be chosen; under certain assumptions, a suitable selection guarantees a backward error on the order of the working precision.

The stability of split-preconditioned FGMRES in four precisions

TL;DR

The paper develops a four-precision framework for split-preconditioned FGMRES, enabling the use of different precisions for A, the left preconditioner, the right preconditioner, and the working computations. It derives normwise backward- and forward-error bounds under realistic perturbation models and provides concrete guidance for selecting u_A, u_L, and u_R to achieve backward error on the order of the working precision, with results applicable to general preconditioners. Through dense and SuiteSparse experiments, it demonstrates how split-preconditioning can improve conditioning and how precision choices influence convergence and accuracy, highlighting the central role of the left preconditioner in forward error control. Overall, the work offers a practical, theory-backed strategy for robust mixed-precision FGMRES in four precisions, applicable to a broad class of preconditioners and problems.

Abstract

We consider the split-preconditioned FGMRES method in a mixed precision framework, in which four potentially different precisions can be used for computations with the coefficient matrix, application of the left preconditioner, application of the right preconditioner, and the working precision. Our analysis is applicable to general preconditioners. We obtain bounds on the backward and forward errors in split-preconditioned FGMRES. Our analysis further provides insight into how the various precisions should be chosen; under certain assumptions, a suitable selection guarantees a backward error on the order of the working precision.
Paper Structure (17 sections, 2 theorems, 41 equations, 4 figures, 12 tables, 1 algorithm)

This paper contains 17 sections, 2 theorems, 41 equations, 4 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Finite precision analysis of FGMRES in four precisions
  3. Comparison with existing bounds
  4. Choosing the precisions
  5. Forward error
  6. Left-, right-, or split-preconditioning
  7. Example: LU preconditioner
  8. Numerical example: synthetic dense systems with split-preconditioning
  9. Numerical example: left- and right-preconditioning
  10. Numerical example: application problems with split-preconditioning
  11. Concluding remarks
  12. Proof of Theorem \ref{['th:backward_error']}
  13. Left preconditioner
  14. Stage 1: MGS
  15. Stage 2: Least squares
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\bar{x}_k$ be the approximate solution to eq:Ax=b_split_precond computed by Algorithm alg:fgmres. Under the assumptions eq:Ml_assump, eq:Mr_assump, where $c_0(n) = 18.53 n^{3/2}$ and $\bar{s}_k$ are the sines computed for the Givens rotations, and where $c_{13}(n,k)$ is defined in eq:yk_norm, the residual for the left-preconditioned system is bounded by where and the normwise relative bac

Figures (4)

  • Figure 3.1: Synthetic problem, $c=5$. BE is the relative backward error and FE is the relative forward error, $\zeta$ is as defined in \ref{['eq:zeta']}, and $\rho$ is as defined in \ref{['eq:rho_def_assump']}.
  • Figure 3.2: As in Figure \ref{['fig:dense_c5']}, but for full left- and right-preconditioning. The left panel shows results for different choices of $u_L$ and the right panel shows results for different choices of $u_R$. IC is the iteration count.
  • Figure 3.3: SuiteSparse problems rajat14 and arc130. BE is the relative backward error and FE is the relative forward error.
  • Figure 3.4: SuiteSparse problems west0132 and fs_183_3. BE is the relative backward error and FE is the relative forward error.

Theorems & Definitions (2)

  • Theorem 2.1
  • Corollary 2.2