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H-CMRH: a novel inner product free hybrid Krylov method for large-scale inverse problems

Ariana N. Brown, Malena Sabaté Landman, James G. Nagy

TL;DR

Investigation of the iterative regularization properties of two Krylov methods for solving large-scale ill-posed problems suggests that H-CMRH exhibits comparable performance to the established hybrid GMRES method in terms of stabilizing semiconvergence, but H-CMRH has does not require any inner products, and requires less work and storage per iteration.

Abstract

This study investigates the iterative regularization properties of two Krylov methods for solving large-scale ill-posed problems: the changing minimal residual Hessenberg method (CMRH) and a novel hybrid variant called the hybrid changing minimal residual Hessenberg method (H-CMRH). Both methods share the advantages of avoiding inner products, making them efficient and highly parallelizable, and particularly suited for implementations that exploit randomization and mixed precision arithmetic. Theoretical results and extensive numerical experiments suggest that H-CMRH exhibits comparable performance to the established hybrid GMRES method in terms of stabilizing semiconvergence, but H-CMRH has does not require any inner products, and requires less work and storage per iteration.

H-CMRH: a novel inner product free hybrid Krylov method for large-scale inverse problems

TL;DR

Investigation of the iterative regularization properties of two Krylov methods for solving large-scale ill-posed problems suggests that H-CMRH exhibits comparable performance to the established hybrid GMRES method in terms of stabilizing semiconvergence, but H-CMRH has does not require any inner products, and requires less work and storage per iteration.

Abstract

This study investigates the iterative regularization properties of two Krylov methods for solving large-scale ill-posed problems: the changing minimal residual Hessenberg method (CMRH) and a novel hybrid variant called the hybrid changing minimal residual Hessenberg method (H-CMRH). Both methods share the advantages of avoiding inner products, making them efficient and highly parallelizable, and particularly suited for implementations that exploit randomization and mixed precision arithmetic. Theoretical results and extensive numerical experiments suggest that H-CMRH exhibits comparable performance to the established hybrid GMRES method in terms of stabilizing semiconvergence, but H-CMRH has does not require any inner products, and requires less work and storage per iteration.
Paper Structure (17 sections, 3 theorems, 63 equations, 13 figures, 2 tables, 3 algorithms)

This paper contains 17 sections, 3 theorems, 63 equations, 13 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Changing Minimal Residual Hessenberg Method
  3. The Hessenberg Process and CMRH
  4. The Relationship Between CMRH and GMRES
  5. Regularizing Properties of CMRH
  6. Singular Value Decomposition Analysis
  7. Spectral Filtering Properties
  8. Hybrid Changing Minimal Residual Hessenberg Method
  9. Residual Norms for Hybrid CMRH and Hybrid GMRES
  10. Regularization Parameters
  11. Stopping Criterion
  12. Numerical Results
  13. Comparison with other inner product free methods
  14. Results on low precision arithmetic
  15. Results on deblurring test problems
  16. ...and 2 more sections

Key Result

Proposition 1

\newlabelprop:10 Let $H_{k+1,k}$ and $H^{A}_{k+1,k}$ be the Hessenberg matrices associated to the Hessenberg and Arnoldi processes, respectively, at iteration $k$, then or, equivalently, $H^{A}_{k+1,k} = R_{k+1}H_{k+1,k}R_{k}^{-1}.$

Figures (13)

  • Figure 1: Singular values for Deriv2-example 2 and Heat.
  • Figure 1: Relative error norm histories for problems with different spectral properties.
  • Figure 2: Singular values for Shaw and Spectra.
  • Figure 2: Relative error norm histories for Deriv2 and Shaw in precisions 'q52' and 'q43' (respectively). Early termination of GMRES is due to underflow (left) and overflow (right) computing vector norms.
  • Figure 3: Singular values for modified Spectra.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Theorem 2
  • Proof 1
  • Theorem 1
  • Proof 2