We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but also with consideration of its use within the conjugate gradient (CG) method. We propose three different strategies for selecting a scaling parameter, which aims to position the eigenvalues of the preconditioned matrix in a way that reduces the energy norm of the error, the quantity that CG monotonically decreases at each iteration. Our focus is on accelerating convergence especially in the early iterations, which is particularly important when CG is truncated due to computational cost constraints. Numerical experiments provide in data assimilation confirm that the scaled spectral preconditioner can significantly improve early CG convergence with negligible computational cost.

翻译：本文研究一种缩放谱预条件子，用于高效求解对称正定线性系统序列。我们设计的缩放预条件子不仅作为线性系统逆的近似，还充分考虑了其在共轭梯度法中的应用。我们提出了三种不同的缩放参数选择策略，旨在通过调整预条件矩阵的特征值分布来降低误差的能量范数——该量值在共轭梯度法的每次迭代中单调递减。研究重点在于加速收敛过程，特别是在早期迭代阶段，这对于因计算成本限制而需要截断共轭梯度法的情况尤为重要。数据同化领域的数值实验证实，缩放谱预条件子能以可忽略的计算成本显著提升共轭梯度法的早期收敛速度。