A gradient method in a Hilbert space with an optimized inner product: achieving a Newton-like convergence

In this paper we introduce a new gradient method which attains quadratic convergence in a certain sense. Applicable to infinite-dimensional unconstrained minimization problems posed in a Hilbert space $H$, the approach consists in finding the energy gradient $g(\lambda)$ defined with respect to an optimal inner product selected from an infinite family of equivalent inner products $(\cdot,\cdot)_\lambda$ in the space $H$. The inner products are parameterized by a space-dependent weight function $\lambda$. At each iteration of the method, where an approximation to the minimizer is given by an element $u\in H$, an optimal weight $\hlambda$ is found as a solution of a nonlinear minimization problem in the space of weights $\Lambda$. It turns out that the projection of $\kappa g(\hlambda)$, where $0<\kappa \ll 1$ is a fixed step size, onto a certain finite-dimensional subspace generated by the method is consistent with Newton's step $h$, in the sense that $P_u(\kappa g(\hlambda))=P_u(h)$, where $P_u$ is an operator describing the projection onto the subspace. As demonstrated by rigorous analysis, this property ensures that thus constructed gradient method attains quadratic convergence for error components contained in these subspaces, in addition to the linear convergence typical of the standard gradient method. We propose a numerical implementation of this new approach and analyze its complexity. Computational results obtained based on a simple model problem confirm the theoretically established convergence properties, demonstrating that the proposed approach performs much better than the standard steepest-descent method based on Sobolev gradients. The presented results offer an explanation of a number of earlier empirical observations concerning the convergence of Sobolev-gradient methods.