Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides
Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides
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DOI：
10.1137/21m1436774
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发表时间：
2021-06
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影响因子：
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通讯作者：
Kensuke Aihara;A. Imakura;K. Morikuni
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文献类型：
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作者：
Kensuke Aihara;A. Imakura;K. Morikuni
整体和块Krylov子空间方法是求解具有多个右端的大型稀疏线性方程组的有效迭代方法。然而，全球或块Lanczos型求解器往往表现出大的振荡的残差范数，并可能有一个大的残差差距有关的损失可达到的精度的近似。传统的残差平滑方案抑制这些振荡，但不能提高可达到的精度，而最近的残差平滑方案，使单右手侧Lanczos型求解器的可达到的精度的提高。该方案的基本概念是近似和残差的原始序列和平滑序列相互影响，从而避免了舍入误差的严重传播。在本研究中，我们将这种交叉交互式残差平滑扩展到求解具有多个右侧的线性方程组的情况。所得到的平滑方法可以减少剩余的差距与低的额外成本相比，他们原来的同行。我们通过舍入误差分析和数值实验证明了所提出的方法的有效性。
Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms and may have a large residual gap relating to the loss of attainable accuracy of the approximations. Conventional residual smoothing schemes suppress these oscillations but cannot improve the attainable accuracy, whereas a recent residual smoothing scheme enables the improvement of the attainable accuracy for single right-hand side Lanczos-type solvers. The underlying concept of this scheme is that the primary and smoothed sequences of the approximations and residuals influence one another, thereby avoiding the severe propagation of rounding errors. In the present study, we extend this cross-interactive residual smoothing to the case of solving linear systems with multiple right-hand sides. The resulting smoothed methods can reduce the residual gap with a low additional cost compared to their original counterparts. We demonstrate the effectiveness of the proposed approach through rounding error analysis and numerical experiments.