论文标题
否决问题的共轭梯度方法中的误差积累
The error accumulation in the conjugate gradient method for degenerate problem
论文作者
论文摘要
在本文中，我们考虑了解决梯度中使用加性噪声最小化二次函数的问题的共轭梯度方法。考虑了三个噪声概念：线性项中的拮抗噪声，线性项中的随机噪声和二次术语中的噪声，以及第一和第二的组合与最后一个。在实验上获得的是，任何考虑的概念都没有误差积累，这与民间传说的观点不同，即与加速方法一样，必须发生错误积累。本文为为什么错误可能不会累积而动机。还通过实验研究了溶液误差对噪声的大小（比例）和使用偶联梯度方法的溶液大小的依赖性。提出了关于解决方案误差依赖性对噪声量表的依赖性以及解决方案的大小（2-词）的假设，并针对所考虑的所有概念进行了测试。事实证明，解决方案（按函数）中的误差线性取决于噪声量表。该作品包含说明每项研究的图表，以及对数值实验的详细描述，其中包括对矢量和矩阵噪声的方法的描述。
In this paper, we consider the conjugate gradient method for solving the problem of minimizing a quadratic function with additive noise in the gradient. Three concepts of noise were considered: antagonistic noise in the linear term, stochastic noise in the linear term, and noise in the quadratic term, as well as combinations of the first and second with the last. It was experimentally obtained that error accumulation is absent for any of the considered concepts, which differs from the folklore opinion that, as in accelerated methods, error accumulation must take place. The paper gives motivation for why the error may not accumulate. The dependence of the solution error both on the magnitude (scale) of the noise and on the size of the solution using the conjugate gradient method was also experimentally investigated. Hypotheses about the dependence of the error in the solution on the noise scale and the size (2-norm) of the solution are proposed and tested for all the concepts considered. It turned out that the error in the solution (by function) linearly depends on the noise scale. The work contains graphs illustrating each individual study, as well as a detailed description of numerical experiments, which includes an account of the methods of the noise of both the vector and the matrix.
