随机Gram-Schmidt过程及其在GMRES中的应用
Randomized Gram-Schmidt process with application to GMRES
摘要：
提出了一种随机化的Gram-Schmidt算法，用于高维向量的正交归一化或QR分解。所提出的过程可以比经典的Gram-Schmidt过程更少的计算开销，同时至少与修改的Gram-Schmidt过程一样数值稳定。我们的方法是基于随机素描，这是一种降维技术，包括在估计的内积的高维向量的内积的小有效计算的随机图像，所谓的草图。通过这种方式，全矢量的近似正交性可以通过它们的草图的正交化来获得。提出的Gram-Schmidt算法可以在任何架构中降低计算成本。通过以更高的精度执行非优势操作，可以放大随机草图的好处。在这种情况下，可以保证数值稳定性与工作单位的循环无关的问题的维数。本文提出的Gram-Schmidt过程可应用于Arnoldi迭代，从而得到求解高维方程组或本征值问题的新Krylov子空间方法。其中，我们选择了随机GMRES方法作为该方法的实际应用。
英文摘要：
A randomized Gram-Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram-Schmidt process while being at least as numerically stable as the modified Gram-Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently-computable random images, so-called sketches. In this way, an approximate orthogonality of the full vectors can be obtained by orthogonalization of their sketches. The proposed Gram-Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by performing the non-dominant operations in higher precision. In this case the numerical stability can be guaranteed with a working unit roundoff independent of the dimension of the problem. The proposed Gram-Schmidt process can be applied to Arnoldi iteration and result in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose randomized GMRES method as a practical application of the methodology.
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