A model reduction for highly non-linear problems using wavelets and the Gauss-Newton method

Abstract

A global regularized Gauss-Newton method is proposed to obtain a zero residual for square nonlinear problems on an affine subspace. The affine subspace is characterized by using wavelets which enable us to solve the problem without making simulations before solving it. We pose the problem as a zero-overdetermined nonlinear composite function where the inside function provided the solution we are seeking. A Gauss-Newton method is presented together with its standard Newton's assumptions that guarantee to retain the q-quadratic rate of convergence. To avoid the singularity and the high-nonlinearity a regularized strategy is presented which preserves the fast rate of convergence. A line-search method is included for global convergence. We rediscover that the Petrov-Galerkin (PG) inexact directions for the Newton method are the Gauss-Newton (GN) directions for the composite function. The results obtained in a set of large-scale problems show the capability of the method for reproducing their essential features while reducing the computational cost associated with high-dimensional problems by a substantial order of magnitude.