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Fourth order partial differential equations on general geometries
 UNIVERSITY OF CALIFORNIA LOS ANGELES
, 2005
"... We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the ..."
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Cited by 24 (4 self)
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We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces (Bertalmío, Cheng, Osher, and Sapiro 2001) to fourth order PDEs including the CahnHilliard equation and a lubrication model for curved surfaces. By representing a surface in ¡ N as the level set of a smooth function, φ ¢ we compute the PDE using only finite differences on a standard Cartesian mesh in ¡ N. The higher order equations introduce a number of challenges that are of small concern when applying this method to first and second order PDEs. Many of these problems, such as timestepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the CahnHilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.
DIFFUSE INTERFACE MODELS ON GRAPHS FOR CLASSIFICATION OF HIGH DIMENSIONAL DATA
, 2012
"... There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDEba ..."
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Cited by 22 (10 self)
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There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDEbased imaging community. The algorithms are based on the GinzburgLandau functional which has classical PDE connections to total variation minimization. Convexsplitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graphtheoretic problems. We present diverse computational examples involving both basic clustering and semisupervised learning for different applications. Case studies include feature identification in images, segmentation in social networks, and segmentation of shapes in high dimensional datasets.
CahnHilliard inpainting and a generalization for grayvalue images
 SIAM J. Imaging Sci
"... Abstract. The CahnHilliard equation is a fourth order reaction diffusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images using the CahnHilliard equation is a new approach in image processing. In this pape ..."
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Cited by 20 (8 self)
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Abstract. The CahnHilliard equation is a fourth order reaction diffusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images using the CahnHilliard equation is a new approach in image processing. In this paper we discuss the stationary state of the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smooth curvature of level sets. Key words. CahnHilliard equation, TV minimization, image inpainting AMS subject classifications. 49J40 1. Introduction. An important task in image processing is the process of filling in missing parts of damaged images based on the information obtained from the surrounding areas. It is essentially a type of interpolation and is referred to as inpainting. Given an image f in a suitable Banach space of functions defined on Ω ⊂ R2, an open and bounded domain, the problem is to reconstruct the original image u in the damaged domain D ⊂ Ω, called inpainting domain. In
2D phase diagram for minimizers of a Cahn–Hilliard functional with longrange interactions
, 2011
"... This paper presents a twodimensional investigation of the phase diagram for global minimizers to a Cahn–Hilliard functional with longrange interactions. Based upon the H−1 gradient flow, we introduce a hybrid numerical method to navigate through the complex energy landscape and access an accurate ..."
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Cited by 9 (1 self)
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This paper presents a twodimensional investigation of the phase diagram for global minimizers to a Cahn–Hilliard functional with longrange interactions. Based upon the H−1 gradient flow, we introduce a hybrid numerical method to navigate through the complex energy landscape and access an accurate depiction of the ground state of the functional. We use this method to numerically compute the phase diagram in a (finite) neighborhood of the orderdisorder transition. We demonstrate a remarkably strong agreement with the standard asymptotic estimates for stability regions based upon a small parameter measuring perturbation from the orderdisorder transition curve.
UNCONDITIONALLY STABLE SCHEMES FOR HIGHER ORDER INPAINTING
"... Abstract. Inpainting methods with third and fourth order equations have certain advantages in comparison with equations of second order such as the smooth interpolation of image information even over large distances. Because of this such methods became very popular in the last couple of years. Solvi ..."
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Cited by 9 (7 self)
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Abstract. Inpainting methods with third and fourth order equations have certain advantages in comparison with equations of second order such as the smooth interpolation of image information even over large distances. Because of this such methods became very popular in the last couple of years. Solving higher order equations numerically can be a computational demanding task though. Discretizing a fourth order evolution equation with a brute force method may restrict the time steps to a size up to order ∆x 4 where ∆x denotes the step size of the spatial grid. In this work we will present a more educated way of discretization, namely efficient semiimplicit schemes that are guaranteed to be unconditionally stable. We will explain the main idea of these schemes and present applications in image processing for inpainting with the CahnHilliard equation, TVH −1 inpainting, and inpainting with LCIS (low curvature image simplifiers). 1.
Efficient preconditioners for large scale binary CahnHilliard models
 COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
"... In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the CahnHilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element d ..."
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Cited by 8 (1 self)
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In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the CahnHilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the thetamethod in time. The preconditioner utilizes to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Large scale umerical experiments are included as well as performance comparisons with other solution methods.
A biharmonicmodified forward time stepping method for fourth order . . .
, 2009
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Fast Multiclass Segmentation using Diffuse Interface Methods on Graphs
"... We present two graphbased algorithms for multiclass segmentation of highdimensional data. The algorithms use a diffuse interface model based on the GinzburgLandau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs ..."
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Cited by 5 (2 self)
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We present two graphbased algorithms for multiclass segmentation of highdimensional data. The algorithms use a diffuse interface model based on the GinzburgLandau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional’s doublewell potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical MerrimanBenceOsher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current stateoftheart multiclass segmentation algorithms.
Image Inpainting Using a FourthOrder Total Variation Flow
, 2009
"... We introduce a fourthorder total variation flow for image inpainting proposed in [5]. The wellposedness of this new inpainting model is discussed and its efficient numerical realization via an unconditionally stable solver developed in [15] is presented. 1 ..."
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Cited by 4 (2 self)
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We introduce a fourthorder total variation flow for image inpainting proposed in [5]. The wellposedness of this new inpainting model is discussed and its efficient numerical realization via an unconditionally stable solver developed in [15] is presented. 1
A Method for Finding Structured Sparse Solutions to Nonnegative Least Squares Problems with Applications
 SIAM J. Imaging Sciences
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