Jessica Bosch

Fast Solvers for Cahn-Hilliard Inpainting

The solution of Cahn--Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau--Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau--Yosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn--Hilliard inpainting models shows that the latter achieves better results.

Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

We consider the efficient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an effective Schur complement approximation. Numerical results illustrate the competitiveness of this approach.

A Fractional Inpainting Model Based on the Vector-Valued Cahn--Hilliard Equation

The Cahn--Hilliard equation provides a simple and fast tool for binary image inpainting. By now, two generalizations to gray value images exist: bitwise binary inpainting and TV-H

Preconditioning for Vector-Valued Cahn--Hilliard Equations

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Generalizing Diffuse Interface Methods on Graphs: Nonsmooth Potentials and Hypergraphs

inpainting. This paper outlines a model based on the vector-valued Cahn--Hilliard equation. Additionally, we generalize our approach to a fractional-in-space version. Fourier spectral methods provide efficient solvers since they yield a fully diagonal scheme. Furthermore, their application to three spatial dimensions is straightforward. Numerical examples show the superiority of the proposed fractional Cahn--Hilliard inpainting approach over its nonfractional version. It improves the peak signal-to-noise ratio and structural similarity index. Likewise, the experiments confirm that the proposed model competes with previous inpainting methods, such as the total variation inpainting approach and its fourth-order variant.

Fast Solvers for Cahn-Hilliard Variational Inequalities

The solution of vector-valued Cahn--Hilliard systems is of interest in many applications. We discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities. Concerning the nonsmooth systems, the necessary bound constraints are incorporated via the Moreau--Yosida regularization technique. We develop effective preconditioners for the efficient solution of the linear systems in saddle point form. Numerical results illustrate the efficiency of our approach. In particular, we numerically show mesh and phase independence of the developed preconditioner in the smooth case. The results in the nonsmooth case are also satisfying, and the preconditioned version always outperforms the unpreconditioned one.

Preconditioning of a Coupled Cahn-Hilliard Navier-Stokes system

Diffuse interface methods have recently been introduced for the task of semisupervised learning. The underlying model is well known in materials science but was extended to graphs using a Ginzburg--Landau functional and the graph Laplacian. We here generalize the previously proposed model by a nonsmooth potential function. Additionally, we show that the diffuse interface method can be used for the segmentation of data coming from hypergraphs. For this we show that the graph Laplacian in almost all cases is derived from hypergraph information. Additionally, we show that the formerly introduced hypergraph Laplacian coming from a relaxed optimization problem is well suited to be used within the diffuse interface method. We present computational experiments for graph and hypergraph Laplacians.