用于求解大规模非线性方程组的序列残差法的正割加速
Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations
AI总结：
针对序列残差法在临界情形收敛慢的问题，本文提出用序列正割法的一种变体进行加速，并在偏微分方程离散化产生的大规模非线性方程组上验证了性能。
摘要：
序列残差法（Sequential Residual Methods）试图通过沿一个与残差相关的方向迭代更新当前近似解，来求解非线性方程组 $F(x)=0$。因此，其内存需求极小，这些方法对于求解大规模非线性系统颇具吸引力。然而，在临界情形下，这些算法的收敛可能较慢，因此加速过程受到欢迎。本文建议采用序列正割法（Sequential Secant Method）的一种变体来加速序列残差法。通过将所得算法应用于求解由偏微分方程离散化产生的超大规模问题，说明了其性能。
英文摘要：
Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. In this paper, we suggest to employ a variation of the Sequential Secant Method in order to accelerate Sequential Residual Methods. The performance of the resulting algorithm is illustrated by applying it to the solution of very large problems coming from the discretization of partial differential equations.
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