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NEWTON'S METHOD IN THE CONTEXT OF GRADIENTS

This paper gives a common theoretical treatment for gradient and Newton type methods for general classes of problems. First, for Euler-Lagrange equations Newton's method is characterized as an (asymptotically) optimal variable steepest descent method. Second, Sobolev gradient type minimization is developed for general problems using a continuous Newton method which takes into account a 'boundary condition' operator. equations may be placed into an optimization setting. Sometimes it is possible to show that a given system of PDEs are Euler-Lagrange equations for some functional . In the more general case one looks for the critical points of a least-squares functional associated with the given system. Furthermore, one can approach Newton type methods also in two dierent ways: from numerical aspect it is the study of the discrete (i.e. iterative) solution method that is mostly relevant, whereas continuous Newton methods can lead to attractive theoretical results. The first part of this paper characterizes Newton's method in the Euler-Lagrange case as an (asymptotically) optimal variable steepest descent method for the itera- tive minimization of the corresponding functional. The second part treats the more