In this paper, we describe and analyze the spectral properties of a symmetric positive definite inexact block preconditioner for a class of symmetric, double saddle-point linear systems. We develop a spectral analysis of the preconditioned matrix, showing that its eigenvalues can be described in terms of the roots of a cubic polynomial with real coefficients. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds, in solving large-scale PDE-constrained optimization problems.

翻译：本文针对一类对称双鞍点线性系统，描述并分析了一种对称正定非精确块预条件子的谱性质。我们对预条件化矩阵进行了谱分析，证明其特征值可由实系数三次多项式的根刻画。通过求解大规模偏微分方程约束优化问题，我们验证了所提预条件子的有效性，并证实了理论界的确立。