嵌套子空间的增广拉格朗日法求解稀疏广义特征值问题

Nested Subspace Augmented Lagrangian Method for Solving Sparse Generalized Eigenvalue Problems

摘要: 稀疏广义特征值问题作为计算数学领域的重要研究课题，其核心在于确定满足广义特征方程$Ax=\lambda Bx$ 的特征对$\left(\lambda ,x\right)$ ，其中$A\in {ℝ}^{n\times n},B\in {ℝ}^{n\times n}$ 为给定的系数矩阵。值得注意的是，在工程结构分析、计算金融学、量子物理模拟及航空航天设计等实际应用场景中，所涉及的矩阵往往具有大规模、稀疏性以及非对称性等复杂特性，这对传统数值求解方法提出了严峻挑战。针对对称广义特征值问题的特殊数学结构，本文研究创新性地提出一种基于子空间构造与优化算法协同的数值计算方法。该方法通过建立具有良好逼近特性的子空间，结合增广拉格朗日半光滑牛顿法(ALMSsn)实现特征系统的高效求解，为解决大规模稀疏矩阵的广义特征值问题提供了新的技术途径。

Abstract: The sparse generalized eigenvalue problem, as a pivotal research subject in computational mathematics, focuses on determining eigenpairs $\left(\lambda ,x\right)$ that satisfy the generalized eigenvalue equation $Ax=\lambda Bx$ , where $A\in {ℝ}^{n\times n},B\in {ℝ}^{n\times n}$ are given coefficient matrices. It is noteworthy that in practical applications such as structural engineering analysis, computational finance, quantum physics simulations, and aerospace design, the involved matrices often exhibit complex characteristics including large-scale dimensions, sparsity, and asymmetry, posing significant challenges to conventional numerical solvers. Targeting the specific mathematical structure of symmetric generalized eigenvalue problems, this study innovatively proposes a numerical computational methodology that integrates subspace construction with optimization algorithms. By establishing subspaces with superior approximation properties and combining them with an augmented Lagrangian semi-smooth Newton method, the proposed approach enables efficient resolution of eigen-systems, thereby providing a novel technical pathway for addressing large-scale sparse generalized eigenvalue problems.