安德森局域化模型的大规模对角化技术
On Large Scale Diagonalization Techniques for the Anderson Model of Localization
AI总结：
针对安德森局域化模型中的大规模稀疏实对称不定矩阵特征值问题，提出基于最大加权匹配和代数多层不完全LDL^T分解的预处理算法，显著加速了内部特征值的计算。
摘要：
我们针对量子物理中出现的特征值问题提出了有效的预处理算法，即计算安德森局域化模型中最大稀疏实对称不定矩阵的几个内部特征值及其相关特征向量。我们比较了Cullum和Willoughby在1987年实现的Lanczos算法与隐式重启Lanczos方法和Jacobi-Davidson方法中的移位反演技术。我们针对移位反演对称不定线性系统的预处理方法基于最大加权匹配和代数多层不完全$LDL^T$分解。这些技术可以看作是对高度病态对称不定安德森矩阵使用更完全主元技术的替代思路的补充。我们展示了这些算法的有效性和数值精度。我们的数值例子表明，最近的代数多层预处理求解器可以将与安德森局域化模型对应的大规模特征值问题的计算加速几个数量级。
英文摘要：
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete $LDL^T$ factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerative the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.
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