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<p>In this paper, we introduce an enhanced variant of the PRESB preconditioning technique in Liang et al. (Numer. Algorithms <b>79</b>(2), 575-596, 2018) within the framework of Krylov subspace acceleration techniques. Unlike conventional applications where it is primarily employed to accelerate nonsymmetric Krylov subspace methods such as GMRES, we explore its conjunction with the more economical conjugate gradient method in the context of restrictive preconditioning techniques developed in Bai and Li (IMA J. Numer. Anal. <b>23</b>(4), 561–580 <CitationRef CitationID="CR10">2003</CitationRef>). The proposed method is specifically designed to address complex-valued block two-by-two linear systems arising from time-harmonic eddy-current optimal control problems. Capitalizing on the computational efficiency and favorable spectral properties of the PRESB type preconditioners, it offers efficient implementation and exhibits a notably sharp constant convergence rate. Numerical experiments are conducted to validate the effectiveness of this approach and its advantages over the nonstandard inner product based <i>Bramble-Pasciak CG</i> (BPCG) method, the original PRESB preconditioned GMRES method and classical optimal <i>block diagonal</i> (BD) preconditioned MINRES, as well as <i>block triangular</i> (BT) preconditioned GMRES methods in existing literature.</p>

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On PRESB preconditioning within restrictively preconditioned conjugate gradient method for time-harmonic eddy-current optimal control problems

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Zhao-Zheng Liang

摘要

In this paper, we introduce an enhanced variant of the PRESB preconditioning technique in Liang et al. (Numer. Algorithms 79(2), 575-596, 2018) within the framework of Krylov subspace acceleration techniques. Unlike conventional applications where it is primarily employed to accelerate nonsymmetric Krylov subspace methods such as GMRES, we explore its conjunction with the more economical conjugate gradient method in the context of restrictive preconditioning techniques developed in Bai and Li (IMA J. Numer. Anal. 23(4), 561–580 2003). The proposed method is specifically designed to address complex-valued block two-by-two linear systems arising from time-harmonic eddy-current optimal control problems. Capitalizing on the computational efficiency and favorable spectral properties of the PRESB type preconditioners, it offers efficient implementation and exhibits a notably sharp constant convergence rate. Numerical experiments are conducted to validate the effectiveness of this approach and its advantages over the nonstandard inner product based Bramble-Pasciak CG (BPCG) method, the original PRESB preconditioned GMRES method and classical optimal block diagonal (BD) preconditioned MINRES, as well as block triangular (BT) preconditioned GMRES methods in existing literature.