Construction of the Discrete Approximation Sequence

Since the fundamental paper of Moser (cf. [105]), it has been understood analytically that regularization is necessary as a postconditioning step in the application of approximate Newton methods, based upon the system differential map. A development of these ideas in terms of current numerical methods and complexity estimates was given by the author in [65]. The approach of Moser is often termed Nash-Moser iteration, because of the fundamental link to generalized implicit function theorems (cf. [108]), specifically, the Nash implicit function theorem. It was proposed by the author in [70], and analyzed further in [71], to use the fixed point map as a basis for the linearization, and thereby avoid the loss of derivatives phenomenon identified by Moser, and termed a numerical loss of derivatives in [65]. In the context of numerical analysis, this loss occurs because the approximation of the identity condition, involved in approximate Newton methods, is not robust with respect to differentiation up to the order of the nonlinear differential system (see (7.8) below). In this chapter, we shall discuss the implications of this fact. It is at the core of preferring the fixed point formulation to the differential formulation. We begin with the former. Our discussion of this case is brief, because of the developments of the preceding chapters.