论文标题
随机革兰氏 - schmidt过程，并应用于转基因
Randomized Gram-Schmidt process with application to GMRES
论文作者
论文摘要
为高维矢量或QR分解而开发了随机革兰氏阴性算法。所提出的过程在计算上可能比经典的革兰氏schmidt过程贵，而至少与修改后的革兰氏schmidt过程在数值上稳定。我们的方法基于随机素描，这是一种缩小尺寸的技术，由其小小的有效竞争的随机图像（所谓的草图）的内部产物估算高维矢量的内部产物。通过这种方式，可以通过其草图的正交化获得完整向量的大约正交性。拟议的革兰氏阴性算法可以在任何架构中提供降低计算成本。可以通过更高的精度执行非主导操作来扩大随机素描的好处。在这种情况下，可以通过与问题的维度无关的工作单元圆形循环来保证数值稳定性。提出的革兰氏schmidt过程可以应用于Arnoldi迭代，并导致新的Krylov子空间方法来求解方程或特征值问题的高维系统。其中，我们选择了随机的转基因方法作为方法的实际应用。
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.
