Analysis of Iterative Methods for Solving a Ginzburg-Landau Equation

Abstract

Abstract. Very recently we have proposed to use a complex GinzburgLandau equation for high contrast inpainting, to restore higher dimensional (volumetric) data (which has applications in frame interpolation), improving sparsely sampled data and to fill in fragmentary surfaces. In this paper we review digital inpainting algorithms and compare their performance with a Ginzburg-Landau inpainting model. For the solution of the Ginzburg-Landau equation we compare the performance of several numerical algorithms. A stability and convergence analysis is given and the consequences for applications to digital inpainting are discussed.

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

“…The method by Bornemann and März also creates an almost perfect triangle, as this approach is good at connecting level lines. This explains why it works very (5)), biharmonic interpolation (see (6)), triharmonic interpolation (see (7)), AMLE (see (8)), Charbonnier interpolation (see (9), λ = 1), regularised Charbonnier interpolation (see (11), λ = 0.1, σ = 8), EED interpolation (see (12), λ = 0.01, σ = 4), the method from [11] (ε = 5, κ = 100, σ = 4, ρ = 8), and the method from [65] (30 global and local iterations, dt = 17.86, α = 2.81, σ = 0.27).…”

“…The interpolation qualities of PDEs have become evident by an axiomatic analysis [20], by applying them to image inpainting [51,10,22,12,66,11] and by utilising them for upsampling digital images [49,7,8,3,73,58]. Extending this to image compression drives inpainting to the extreme: Only a small set of specifically selected pixels is stored, while the remaining image is reconstructed using the filling-in effect of PDE-based interpolation.…”

Thermodynamics of Nanoparticles: Experimental Protocol Based on a Comprehensive Ginzburg-Landau Interpretation

“…The nontrivial solution in one dimension leading to the expression of the order parameter is η ( x ) = e ( 2 / ε ) x − 1 e ( 2 / ε ) x + 1 ε corresponds to the width of the transition region. This function is plotted in Figure .…”

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

“…It also describes the amplitude evolution of instability waves in a large variety of dissipative systems in fluid mechanics, which are close to criticality [7]. Furthermore, it is used to model some types of chemical reactions, like the famous Belousov-Zhabotinsky reaction, to model boundary layers in multiphase systems, to describe the development of patterns and shocks in non-equilibrium systems [8], and the theory of the origin of wind waves on a water surface [9].…”