发表机构
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); Research Institute of Intelligent Complex Systems, Fudan University（上海数学与交叉学科研究院; 复旦大学智能复杂系统研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种保结构的LOBPCG方法，利用Jordan-Wielandt铅笔的谱对称性分别构造左右奇异向量搜索空间，通过小规模SVD和IHL技巧实现高效计算，数值实验验证其优于非结构化和Gram矩阵方法。
AI 中文摘要
我们开发了一种保结构的局部最优块预条件共轭梯度（LOBPCG）方法，用于计算矩阵对 \((A, B)\) 的若干最大正广义奇异值及其相关的广义奇异向量，其中 \(B\) 具有满列秩。该方法源自 Hermitian 正定 Jordan--Wielandt 铅笔，因此避免了显式构造正规方程矩阵。通过利用该铅笔的块结构和谱对称性，我们为左、右广义奇异向量构造了独立的搜索子空间。由此产生的 Rayleigh--Ritz 过程仅需要对一个小型投影交叉矩阵进行奇异值分解。我们改进了 Hetmaniuk--Lehoucq（IHL）技巧，以在两个分量空间中构造稳定的共轭搜索方向。我们还建立了分量式与增广式 Rayleigh--Ritz 过程产生的系数矩阵之间的显式对应关系。这种对应关系允许在增广的保结构 LOBPCG 公式中使用分量式 IHL 构造，而无需显式组装增广基。数值实验证明了所提方法相对于非结构化和基于 Gram 矩阵的 LOBPCG 实现的计算优势。
英文摘要
We develop a structure-preserving locally optimal block preconditioned conjugate gradient(LOBPCG) method for computing several of the largest positive generalized singular values of a matrix pair \((A, B)\), where \(B\) has full column rank, together with the associated generalized singular vectors. The method is derived from the Hermitian-definite Jordan--Wielandt pencil and therefore avoids explicitly forming the normal-equation matrices. By exploiting the block structure and spectral symmetry of this pencil, we construct separate search subspaces for the left and right generalized singular vectors. The resulting Rayleigh--Ritz procedure requires only a singular value decomposition of a small projected cross matrix. We adapt the improved Hetmaniuk--Lehoucq (IHL) trick to construct stable conjugate search directions in the two component spaces. We also establish an explicit correspondence between the coefficient matrices produced by the componentwise and augmented Rayleigh--Ritz procedures. This correspondence permits the componentwise IHL construction to be used in an augmented, structure-preserving LOBPCG formulation without explicitly assembling the augmented basis. Numerical experiments demonstrate the computational advantages of the proposed method over unstructured and Gram-matrix-based LOBPCG implementations.
