解极值本征值问题的三角化无网格化方法
Triangularized Orthogonalization-free Method for Solving Extreme Eigenvalue Problems
摘要：
提出了一种新的无正交化方法和两种求解极值本征值问题的具体算法。在基于梯度的算法之上，所提出的算法修改多列梯度，使得较早的列与较晚的列解耦。几乎可以保证算法整体收敛到特征向量而不是特征空间。局部地，算法以取决于特征间隙的收敛速度线性地收敛。动量加速，精确的lineswitch，和列锁定，以进一步加快这两种算法，并减少其计算成本。我们证明了这两种算法的效率在几个随机矩阵具有不同的光谱分布和矩阵计算化学。
英文摘要：
A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier columns are decoupled from later ones. Global convergence to eigenvectors instead of eigenspace is guaranteed almost surely. Locally, algorithms converge linearly with convergence rate depending on eigengaps. Momentum acceleration, exact linesearch, and column locking are incorporated to further accelerate both algorithms and reduce their computational costs. We demonstrate the efficiency of both algorithms on several random matrices with different spectrum distribution and matrices from computational chemistry.
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