论文标题
PLSS：一个投影的线性系统求解器
PLSS: A Projected Linear Systems Solver
论文作者
论文摘要
我们提出了解决方形或矩形一致的线性系统AX = b的迭代投影方法。现有的投影方法使用素描矩阵（可能是随机的）来生成一系列小型投影子问题，但即使较小的系统也可能成本高昂。我们开发一个过程，该过程将一列附加到素描矩阵中的每条迭代中，并在迭代数量的有限数量中收敛，无论该草图是随机的还是确定性的。通常，我们的过程生成了近似解决方案XK的正交更新。通过选择草图为所有以前的残差集，我们可以在最多级别（a）迭代（精确算术）中获得一个简单的递归更新和收敛。通过为草图选择一系列身份列，我们开发了Kaczmarz方法的概括。在大型稀疏系统的实验中，我们的方法（PLSS）具有残留草图与LSQR和LSMR具有竞争力，并且与最先进的随机方法相比，残留和身份草图对其进行了比较。
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but even the smaller systems can be costly. We develop a process that appends one column to the sketching matrix each iteration and converges in a finite number of iterations whether the sketch is random or deterministic. In general, our process generates orthogonal updates to the approximate solution xk. By choosing the sketch to be the set of all previous residuals, we obtain a simple recursive update and convergence in at most rank(A) iterations (in exact arithmetic). By choosing a sequence of identity columns for the sketch, we develop a generalization of the Kaczmarz method. In experiments on large sparse systems, our method (PLSS) with residual sketches is competitive with LSQR and LSMR, and with residual and identity sketches compares favorably with state-of-the-art randomized methods.
