A Curvilinear Search Method for p-Harmonic Flows on Spheres

The problem of finding p-harmonic flows arises in a wide range of applications including color image (chromaticity) denoising, micromagnetics, liquid crystal theory, and directional diffusion. In this paper, we propose an innovative curvilinear search method for minimizing p-harmonic energies over spheres. Starting from a flow (map) on the unit sphere, our method searches along a curve that lies on the sphere in a manner similar to that of a standard inexact line search descent method. We show that our method is globally convergent if the step length satisfies the Armijo–Wolfe conditions. Computational tests are presented to demonstrate the efficiency of the proposed method and a variant of it that uses Barzilai–Borwein steps.