A curvilinear search method for \(p\)-harmonic flows on spheres. (English) Zbl 1193.49030
Summary: 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.
MSC:
49M15 | Newton-type methods |
65N06 | Finite difference methods for boundary value problems involving PDEs |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
68U10 | Computing methodologies for image processing |
90C26 | Nonconvex programming, global optimization |
90C30 | Nonlinear programming |
94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |