论文标题
基于分裂的随机迭代方法来解决不确定的最小二乘问题
Splitting-based randomized iterative methods for solving indefinite least squares problem
论文作者
论文摘要
不确定的最小二乘问题（ILS）问题是著名的线性最小二乘问题的概括。相对于签名矩阵，它最大程度地减少了不确定的二次形式。对于这个问题，我们首先根据其自身的结构提出了一种令人印象深刻的简单有效分裂方法（SP）方法，并证明其对任何初始值都“无条件地”收敛。此外，为了避免实施一些矩阵乘法并计算大型矩阵的倒数并考虑随机策略的加速度和效率，我们根据SP方法以及随机的Kaczmarz，Ghuss-seeidel和coativeer discent方法开发了两种随机迭代方法，并描述了它们的转化属性。数值结果表明，与ILS问题的最新迭代方法相比，我们的三种方法在计算时间和迭代数字中均具有相当不错的性能，并且还证明了这两种随机方法在计算时间期间确实产生了显着的加速度。
The indefinite least squares (ILS) problem is a generalization of the famous linear least squares problem. It minimizes an indefinite quadratic form with respect to a signature matrix. For this problem, we first propose an impressively simple and effective splitting (SP) method according to its own structure and prove that it converges 'unconditionally' for any initial value. Further, to avoid implementing some matrix multiplications and calculating the inverse of large matrix and considering the acceleration and efficiency of the randomized strategy, we develop two randomized iterative methods on the basis of the SP method as well as the randomized Kaczmarz, Gauss-Seidel and coordinate descent methods, and describe their convergence properties. Numerical results show that our three methods all have quite decent performance in both computing time and iteration numbers compared with the latest iterative method of the ILS problem, and also demonstrate that the two randomized methods indeed yield significant acceleration in term of computing time.
