An adaptive Newton algorithm based on numerical inversion: Regularization as postconditioner

Abstract

We examine a class of approximate inversion processes, satisfying estimates similar to those defined by finite element or truncated spectral approximations; these are to be used as approximate right inverses for Newton iteration methods. When viewed at the operator level, these approximations introduce a defect, or "loss of derivatives", of order one or more. Regularization is introduced as a form of defect correction. A superlinearly convergent, approximate Newton iteration is thereby obtained by using the numerical inversion adaptively, i.e., with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect. This adaptive choice makes possible ascribing an order to the convergent process, and this is identified as essentially optimal for elliptic problems, relative to complexity. The design of the algorithm involves multi-parameter selection, thereby opening up interesting avenues for elliptic problems, relative to complexity. This applies also to the regularization which may be carried out in the Fourier transform space, and is band-limited in the language of Whittaker-Shannon sampling theory. The norms employed in the analysis are of Holder space type; the iteration is an adaptation of Nash-Moser interation; and, the complexity studies use Vituskin's theory of information processing. Computational experience is described in the final section.