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A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems

Sk. Safique Ahmad, Pinki Khatun

Abstract

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.

A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems

Abstract

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.
Paper Structure (8 sections, 9 theorems, 98 equations, 8 figures, 9 tables, 1 algorithm)

This paper contains 8 sections, 9 theorems, 98 equations, 8 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The parameterized enhanced shift-splitting (PESS) iterative process and the preconditioner
  3. Convergence analysis of the PESS iterative process
  4. Spectral distribution of the PESS preconditioned matrix
  5. Local PESS (LPESS) preconditioner
  6. The strategy of parameter selection
  7. Numerical experiments
  8. Conclusion

Key Result

Lemma 3.1

CAOSS19 Let $A\in{\mathbb R}^{n\times n}$ be a SPD matrix and let $B\in {\mathbb R}^{m\times n}$ and $C\in {\mathbb R}^{p\times m}$ be full row rank matrices. Then, the saddle point matrix $\mathcal{A}$ is positive stable.

Figures (8)

  • Figure 1: Convergence curves for IT versus RES of PGMRES processes employing BD, IBD, MAPSS, SL, SS, RSS, EGSS, RPGSS, PESS and LPESS$(s=12)$ preconditioners in Case II for Example \ref{['ex1']}.
  • Figure 2: Spectral distributions of $\mathcal{A},{\mathscr{P}}_{BD}^{-1}\mathcal{A}, {\mathscr{P}}_{IBD}^{-1}\mathcal{A}, {{\mathscr{P}}_{MAPSS}^{-1}\mathcal{A}, {\mathscr{P}}_{SL}^{-1}\mathcal{A},} {\mathscr{P}}_{SS}^{-1}\mathcal{A}, {\mathscr{P}}_{EGSS}^{-1}\mathcal{A}, {{\mathscr{P}}_{RPGSS}^{-1}\mathcal{A},} {\mathscr{P}}_{PESS}^{-1}\mathcal{A}$ and ${\mathscr{P}}_{LPESS}^{-1}\mathcal{A}$ for Case II with $l=16$ for Example \ref{['ex1']}.
  • Figure 3: Spectral bounds for ${\mathscr{P}}_{PESS}^{-1}\mathcal{A}$ and ${\mathscr{P}}_{LPESS}^{-1}\mathcal{A}$ for Case II with $l=16$ for Example \ref{['ex1']}.
  • Figure 5: Relationship of norm error of solution with increasing noise percentage, employing proposed ${\mathscr{P}}_{PESS}$ for Case II with $s=12$ for $l=16,32,48,64$ and $80$ for Example \ref{['ex1']}.
  • Figure 6: Convergence curves for IT versus RES of the PGMRES processes by employing IBD, MAPSS, SL, SS, RSS, EGSS, RPGSS, PESS and LPESS preconditioners in Case II for Example \ref{['ex2']}.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.1
  • Remark 3.1
  • Corollary 3.2
  • Remark 4.1
  • Proposition 4.1
  • Remark 4.2
  • proof
  • ...and 5 more