Stabilization of unstable procedures: the recursive projection method
Stabilization of unstable procedures: the recursive projection method
复制标题
不稳定过程的稳定：递归投影法
DOI：
10.1137/0730057
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发表时间：
1993
影响因子：
2.9
通讯作者：
H. Keller
中科院分区：
文献类型：
--
作者：
Gautam M. Shroff;H. Keller
求解非线性参数相关问题的不动点迭代法可以在一定的参数值区间内收敛，并随着参数的变化而发散。递归投影法（RPM），它稳定这样的程序通过计算投影到不稳定的子空间。在这个子空间上执行牛顿或特殊牛顿迭代，并且在补上使用不动点迭代。随着参数的继续进行，投影被有效地更新，可能增加或减少不稳定子空间的维度。当不稳定子空间的维数小于系统的维数时，该方法是非常有效的。给出了收敛性证明，并在不稳定子空间上引入伪弧长延拓，使延拓能越过折叠。RPM中的一个重要的应用程序，其中的“黑箱”时间积分方案是稳定的，使其能够计算不稳定的稳态的例子。RPM也可以用来加速迭代程序时，缓慢收敛是由于一些缓慢衰减的模式。
Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a “black-box” time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes.