Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points
Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points
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DOI：
10.1007/s10957-008-9469-0
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发表时间：
2009-02
影响因子：
1.9
通讯作者：
Xin-long Luo;C. Kelley;L. Liao;H. Tam
中科院分区：
文献类型：
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作者：
Xin-long Luo;C. Kelley;L. Liao;H. Tam
Rosenbrock方法是求解常微分方程刚性初值问题的常用方法。与其他隐式方法如后向差分公式或隐式Runge-Kutta方法相比，它的一个优点是不需要在每次迭代时求解非线性方程。在这篇文章中，我们介绍了一个信赖域技术来选择一个特殊的初始值问题，即梯度系统从一个无约束优化问题的二阶Rosenbrock方法的时间步长。该技术不同于局部误差方法。讨论了求解梯度方程组平衡点的新方法的局部收敛性和全局收敛性。最后给出了一些有意义的数值结果。
Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented.