U.S.-Federal Republic of Germany Cooperative Research on Pointwise Quasi-Newton Methods (Applied Mathematics)

该奖项支持北卡罗来纳州立大学C.T.Kelley博士对德意志联邦共和国特里尔大学的几次短期访问，以与E.W.Sachs博士合作研究逐点准牛顿方法。他们的合作已经产生了解决积分方程组、微分方程组和控制中的非线性问题的新算法。他们将继续合作，重点放在高度非线性问题、全局收敛和离散化策略分析上。凯利博士在非线性方程方面的强项和萨克斯博士在最优控制和优化方面的优势，以及泛函分析方面的联合优势，对于这项研究的持续成功非常重要。拟牛顿方法是求解非线性方程组和优化问题的迭代格式。它们已经成为在计算机上进行快速收敛计算的强大工具。当牛顿方法不可能或成本太高而无法实现时，例如，当雅可比矩阵不能以合理的成本计算时，这种方法是有用的。如果考虑到问题的结构，这些方法的性能可以得到提高。萨克斯博士和凯利博士一直在研究一类新的方法，当问题是无限维问题的离散化时，例如微分方程或最优控制问题，这种方法考虑结构。所考虑的方法比应用标准方法对原始连续问题的有限维近似效果更好。

This award supports several short visits by Dr. C. T. Kelley of North Carolina State University to the University of Trier, Federal Republic of Germany, for collaborative research on pointwise quasi-Newton methods with Dr. E. W. Sachs. Their joint work has already produced new algorithms for solution of nonlinear problems in integral equations, differential equations, and control. They will continue their collaboration with emphasis on highly nonlinear problems, global convergence, and analysis of discretization strategies. The strengths of Dr. Kelley in nonlinear equations and of Dr. Sachs in optimal control and optimization, together with a joint strength in functional analysis, are important to the continued success of this research. Quasi-Newton methods are iterative schemes for solution of nonlinear equations and optimization problems. They have become a powerful tool for rapidly convergent calculations on computers. Such methods are useful when Newton's method is impossible or too costly to implement, for example, when the Jacobian matrix can not be computed at reasonable cost. Performance of these methods can be enhanced if the structure of the problem is taken into account. Drs. Sachs and Kelley have been studying a new class of methods that consider structure when the problem is a discretization of an infinite dimensional problem, such as a differential equation or optimal control problem. The methods considered perform better than application of standard methods to a finite dimensional approximation of the original continuous problem.