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

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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.