AI 中文总结
针对PDE约束优化的双鞍点系统，开发保留块结构的随机不精确块三角预条件子，经谱分析与数值实验验证其有效性、鲁棒性及可扩展性。
AI 中文摘要
针对PDE约束优化中产生的双鞍点系统，我们开发了一类新型不精确块三角预条件子。所提预条件子通过矩阵分解技术构造，同时保留原系统的固有块结构。对预条件矩阵进行了全面谱分析，给出了实特征值与非实特征值的显式界。为实现不精确预条件子的高效构造，引入随机策略选取所需子块。我们建立了期望近似误差的高概率界，误差估计由相关矩阵的特征值明确表征。数值实验验证了所提预条件子的有效性、鲁棒性与可扩展性，也验证了随机构造策略的效率。
英文摘要
We develop a new class of inexact block triangular preconditioners for double saddle-point systems arising from PDE-constrained optimization. The proposed preconditioners are constructed through matrix factorization techniques while preserving the inherent block structure of the original systems. A comprehensive spectral analysis of the preconditioned matrices is provided, yielding explicit bounds for both real and nonreal eigenvalues. To enable efficient construction of the inexact preconditioners, randomized strategies are introduced to select the required subblocks. We establish high-probability bounds for the expected approximation error, with the error estimates explicitly characterized in terms of the eigenvalues of the associated matrices. Numerical experiments demonstrate the effectiveness, robustness, and scalability of the proposed preconditioners, and validate the efficiency of the randomized construction strategies.
