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

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

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