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In this paper we introduce a new gradient method which attains quadraticconvergence in a certain sense. Applicable to infinite-dimensionalunconstrained minimization problems posed in a Hilbert space $H$, the approachconsists in finding the energy gradient $g(\lambda)$ defined with respect to anoptimal inner product selected from an infinite family of equivalent innerproducts $(\cdot,\cdot)_\lambda$ in the space $H$. The inner products areparameterized by a space-dependent weight function $\lambda$. At each iterationof 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 nonlinearminimization problem in the space of weights $\Lambda$. It turns out that theprojection of $\kappa g(\hlambda)$, where $0<\kappa \ll 1$ is a fixed stepsize, onto a certain finite-dimensional subspace generated by the method isconsistent with Newton's step $h$, in the sense that $P_u(\kappag(\hlambda))=P_u(h)$, where $P_u$ is an operator describing the projection ontothe subspace. As demonstrated by rigorous analysis, this property ensures thatthus constructed gradient method attains quadratic convergence for errorcomponents contained in these subspaces, in addition to the linear convergencetypical of the standard gradient method. We propose a numerical implementationof this new approach and analyze its complexity. Computational results obtainedbased on a simple model problem confirm the theoretically establishedconvergence properties, demonstrating that the proposed approach performs muchbetter than the standard steepest-descent method based on Sobolev gradients.The presented results offer an explanation of a number of earlier empiricalobservations concerning the convergence of Sobolev-gradient methods.

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