## Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature (1996)

Venue: | J. Comput. Phys |

Citations: | 22 - 5 self |

### BibTeX

@TECHREPORT{Ruuth96efficientalgorithms,

author = {Steven J. Ruuth},

title = {Efficient Algorithms for Diffusion-Generated Motion by Mean Curvature},

institution = {J. Comput. Phys},

year = {1996}

}

### OpenURL

### Abstract

We accept this thesis as conforming to the required standard

### Citations

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Citation Context ...= (~x; t) + c(t): (1:2) We shall refer to this type of motion as affine velocity front motion. Curve evolution has also been studied extensively for image analysis and the enhancement of planar shape =-=[43, 50, 42, 67, 41, 72]-=-. Important application areas for this subject include noise suppression, image recognition and image interpretation. Mean curvature Chapter 1. Introduction 3 Figure 1.2: Evolution of Multiple Grains ... |

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Citation Context ...ing of the motion by mean curvature of surfaces, we seek a method for visualizing our results. This section describes a simple approach for generating movies of evolving shapes using Matlab [46]. See =-=[6, 44]-=- for discussions on more advanced visualization techniques. Our principal task is to construct each frame of the movie. A relatively straightforward approach for generating surfaces with diffuse, ambi... |

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A Practical Guide to Splines
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Citation Context ... + 1) + x m fi (m\Gamma1) ` x + 1 2 ' + 1 2 (m + 1) \Gamma x m fi (m\Gamma1) ` x \Gamma 1 2 ' where fi (0) (x) = 8 ? ! ? : 1 if x 2 ( 1 2 ; 1 2 ] 0 otherwise. Chapter 3. A New, Spectral Method 54 See =-=[7]-=- for an algorithm that evaluates each of the components of f in only O(m 2 ) operations. 2. Using a fast Fourier transform, we next evaluate F jj 0 = 2n\Gamma1 X k=0 2n\Gamma1 X k 0 =0 f kk 0 e \Gamma... |

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Citation Context ...if i 6= 0 and j 6= 0 2 otherwise (3:13) Thus, simple functions must be integrated over a complicated, non-rectangular region, R t . This may be accomplished by recursively subdividing the domain (cf. =-=[62, 61]-=-), as we illustrate for the region, R, given in Figure 3.1a. We begin by evaluating U at the corners of a number of equally-sized subregions, so as to capture the large-scale features of the shape. Ty... |

1090 |
Fronts propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations
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Citation Context ...dical technology and architecture. Other applications for affine motion occur in certain flame propagation problems (e.g., Chapter 1. Introduction 4 grassfire flow) and automatic grid generation. See =-=[54, 63, 66]-=- for further details. 1.2 Methods for Curvature-Dependent Motion To study the phenomena outlined in the previous section, several numerical methods have been developed. Most of these can be divided in... |

494 |
Solving ordinary differential equations
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Citation Context ...oximation of M(�� ) to M can be expressed as M = M(�� ) +K�� +O(�� 2 ) where K is independent of �� , we find that the extrapolation 2M(�� ) \Gamma M(2�� ) = M +O(�� 2 =-=) is second order accurate. See [32, 34, 37]-=- for treatments of extrapolation for initial value problems. The extrapolation process may be repeated, of course, as in Romberg's integration to obtain methods which we expect are even higher order a... |

394 | Chebyshev and fourier spectral methods
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Citation Context ...fl�� ). In practice, however, our implementations simply select an n satisfying nss j ln(ffl)j �� 2 �� (3:11) and verify the corresponding results by repeating the calculation with a diffe=-=rent n (cf. [8]-=-). Chapter 3. A New, Spectral Method 26 3.2 Calculation of the Fourier Coefficients The values of the Fourier coefficients, c ij , of equation (3.10) must still be determined at the beginning of each ... |

392 | Introduction to Implicit Surfaces
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Citation Context ...ing of the motion by mean curvature of surfaces, we seek a method for visualizing our results. This section describes a simple approach for generating movies of evolving shapes using Matlab [46]. See =-=[6, 44]-=- for discussions on more advanced visualization techniques. Our principal task is to construct each frame of the movie. A relatively straightforward approach for generating surfaces with diffuse, ambi... |

353 | Algorithms for minimization without derivatives
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Citation Context ...l set that preserves phase areas; i.e., determine the value c satisfying A(c; t) = A ` 1 2 ; 0 ' : (5:36) Solving (5.36) for c may be accomplished by a variety of line search algorithms. For example, =-=[11]-=- gives an efficient and reliable approach based on a combination Chapter 5. Numerical Experiments and Visualization 95 Figure 5.10: A Test Problem for the Nonlocal Curvature Algorithm 0 0.2 0.4 0.6 0.... |

345 | A fast level set method for propagating interfaces
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Citation Context ...rrying out local mesh refinement is rather involved for level set methods when curvature terms arise (see [15]). An alternative approach is to place a narrow band of grid points around the front (cf. =-=[1]-=-). Even this optimized, finite difference approach can lead to a prohibitive number of operations per step when an accurate solution is sought. For example, consider the motion by mean curvature of a ... |

342 |
Image selective smoothing and edge detection by nonlinear diffusion
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Citation Context ...ppropriate for the case of grain growth [14]. It has also been argued that these boundary conditions (2.4) are the most natural for image processing since they do not impose any value at the boundary =-=[2]-=-. Although our experiments will concentrate on the important zero flux case, other types of boundary conditions are sometimes considered. In particular, Dirichlet conditions have been used for computi... |

308 | Motion of level sets by mean curvature
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Citation Context ...ure, F (), we obtain a surface \Gamma(t) = fx : \Phi(x; t) = 0g which moves with a normal velocity F (). It has been rigorously proven that this method converges for the case of mean curvature motion =-=[27]-=-. Furthermore, by assigning a function, \Phi i , to each region and evolving according to the method an extension to problems with junctions is possible [48]. This coupled Osher-Sethian approach allow... |

300 |
Spectral Methods in Fluid Dynamics
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Citation Context ...le features of the shape. Typically, n \Theta n subregions are selected because the corresponding U-values can be evaluated in just O(n 2 log(n)) operations using a fast Fourier transform (see, e.g., =-=[17]-=-). If the phase at all four corners of any subregion corresponds to white, then we assume that the subregion does not intersect with R and hence no contribution to the Fourier coefficients is made. Th... |

265 | A variational level set approach to multiphase motion
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Citation Context ...ues [53] and the determination of self-similar solutions [21] under curvature flow. Computations of minimal surfaces have also been carried out by evolving surfaces according to mean curvature motion =-=[20, 74]-=-. Such surfaces find application in numerous areas [20] including soap film shapes, relativity theory, medical technology and architecture. Other applications for affine motion occur in certain flame ... |

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Citation Context ...otion causes the most highly curved portions of a curve to smooth most rapidly. Indeed, any simple, closed curve in the plane shrinks to a small circle and disappears, regardless of the initial shape =-=[31, 36]-=-. For higher dimensional shapes, the most curved portions also move most quickly; however, topological merging and breaking is also possible [20]. Certain idealized models for grain growth fit into th... |

247 |
Finite difference schemes and partial differential equations
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Citation Context ...n \Deltat = \Delta h ~ U n+1 give a strong decay of high frequency error modes. Other methods, such as CrankNicolson, ~ U n+1 \Gamma ~ U n \Deltat = \Delta h 0 @ ~ U n+1 + ~ U n 2 1 A and ADI methods =-=[69]-=- give a very weak decay of these error modes [3, 69]. After the sharpening step of the MBO-method, the solution is discontinuous. Because high frequency modes make a very important contribution to suc... |

244 | Constraints on deformable models: Recovering 3d shape and nonrigid motion - Terzopoulos, Witkin, et al. - 1988 |

219 |
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Citation Context ...= (~x; t) + c(t): (1:2) We shall refer to this type of motion as affine velocity front motion. Curve evolution has also been studied extensively for image analysis and the enhancement of planar shape =-=[43, 50, 42, 67, 41, 72]-=-. Important application areas for this subject include noise suppression, image recognition and image interpretation. Mean curvature Chapter 1. Introduction 3 Figure 1.2: Evolution of Multiple Grains ... |

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Citation Context ...valuates all n 2 Fourier coefficients of (3.22) in O ` N p log 2 ` 1 ffl ' + n 2 log(n) ' (3:26) operations, where ffl is the precision of the computation. Essentially the same bounds are obtained in =-=[24]-=- for an interpolation using Gaussian Bells and in [68] for an algorithm which uses Lagrange interpolation and Green's theorem. In practice, numerical experiments [24, 68, 5] suggest that Beylkin's alg... |

142 | Flow by mean curvature of convex surfaces into spheres
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Citation Context ...ed barbell using a step size, Chapter 5. Numerical Experiments and Visualization 87 �� = 0:0004. From these images, it is clear that the center handle pinches off to form two pieces. As expected f=-=rom [39]-=-, these convex shapes become nearly spherical as they disappear. Thick-Stemmed Barbell A wider stem can produce a qualitatively different motion. For example, Figure 5.6 displays the motion of a thick... |

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Solid shape
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Citation Context ...= (~x; t) + c(t): (1:2) We shall refer to this type of motion as affine velocity front motion. Curve evolution has also been studied extensively for image analysis and the enhancement of planar shape =-=[43, 50, 42, 67, 41, 72]-=-. Important application areas for this subject include noise suppression, image recognition and image interpretation. Mean curvature Chapter 1. Introduction 3 Figure 1.2: Evolution of Multiple Grains ... |

115 |
Computing minimal surfaces via level set curvature flow
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Citation Context ...le and disappears, regardless of the initial shape [31, 36]. For higher dimensional shapes, the most curved portions also move most quickly; however, topological merging and breaking is also possible =-=[20]-=-. Certain idealized models for grain growth fit into the framework of mean curvature motion. For example, when a liquid metal solidifies, crystallization begins in many locations with random orientati... |

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Citation Context ...of high frequency error modes. Other methods, such as CrankNicolson, ~ U n+1 \Gamma ~ U n \Deltat = \Delta h 0 @ ~ U n+1 + ~ U n 2 1 A and ADI methods [69] give a very weak decay of these error modes =-=[3, 69]-=-. After the sharpening step of the MBO-method, the solution is discontinuous. Because high frequency modes make a very important contribution to such a result, it is essential to use a time-stepping s... |

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The Surface Evolver
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Citation Context ...ter 1. Introduction 2 Figure 1.1: A Region Evolving According to Mean Curvature Motion Initial Motion Later Region (Solid) is isotropic or independent of the orientation of the boundary of each grain =-=[9]-=-. By annealing the metal, it becomes warmed so that boundary atoms can change their phase. This produces an interface motion proportional to mean curvature motion [48, 51, 9]. In the interesting case ... |

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Citation Context ...n leads to O(n 2 N p ) operations. In this section, we consider recent methods for the fast evaluation of (3.19) and (3.20). Specifically, we discuss an unequally spaced fast Fourier transform method =-=[5] that we hav-=-e applied to the new, spectral method. This transform leads to an algorithm that typically requires only O ` 1 �� log 2 (�� ) ' operations per ��-step for the basic MBO-method. For consist... |

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Citation Context ...l (e.g., f 0 (u) = u(u \Gamma 1)(u \Gamma 1 2 )) and ffl is a small parameter. For the two-phase problem, it has been proven that certain phase field models produce mean curvature motion when ffl ! 0 =-=[59, 58, 12, 26]-=-. (Indeed, an important motivation for studying mean curvature motion is that it arises as a limit of certain phase field models.) By letting ffl be proportional to ru jruj , these methods provide a m... |

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Citation Context ...ion of the boundary of each grain [9]. By annealing the metal, it becomes warmed so that boundary atoms can change their phase. This produces an interface motion proportional to mean curvature motion =-=[48, 51, 9]-=-. In the interesting case where three or more grain orientations are present, junctions of moving surfaces can occur. This is illustrated in Figure 1.2 for the case of three grains. Although these imp... |

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Citation Context ...dical technology and architecture. Other applications for affine motion occur in certain flame propagation problems (e.g., Chapter 1. Introduction 4 grassfire flow) and automatic grid generation. See =-=[54, 63, 66]-=- for further details. 1.2 Methods for Curvature-Dependent Motion To study the phenomena outlined in the previous section, several numerical methods have been developed. Most of these can be divided in... |

77 |
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Citation Context ...if i 6= 0 and j 6= 0 2 otherwise (3:13) Thus, simple functions must be integrated over a complicated, non-rectangular region, R t . This may be accomplished by recursively subdividing the domain (cf. =-=[62, 61]-=-), as we illustrate for the region, R, given in Figure 3.1a. We begin by evaluating U at the corners of a number of equally-sized subregions, so as to capture the large-scale features of the shape. Ty... |

70 |
Fast reaction, slow diffusion, and curve shortening
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- 1989
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Citation Context ...l (e.g., f 0 (u) = u(u \Gamma 1)(u \Gamma 1 2 )) and ffl is a small parameter. For the two-phase problem, it has been proven that certain phase field models produce mean curvature motion when ffl ! 0 =-=[59, 58, 12, 26]-=-. (Indeed, an important motivation for studying mean curvature motion is that it arises as a limit of certain phase field models.) By letting ffl be proportional to ru jruj , these methods provide a m... |

67 |
Shortening embedded curves
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Citation Context ...otion causes the most highly curved portions of a curve to smooth most rapidly. Indeed, any simple, closed curve in the plane shrinks to a small circle and disappears, regardless of the initial shape =-=[31, 36]-=-. For higher dimensional shapes, the most curved portions also move most quickly; however, topological merging and breaking is also possible [20]. Certain idealized models for grain growth fit into th... |

66 | shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space,” Intern
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57 |
Diffusion generated motion by mean curvature
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Citation Context ... small time steps must be used whenever a fine spatial mesh is applied. A method based on the model of diffusion-dependent motion of level sets has recently been proposed by Merriman, Bence and Osher =-=[47, 48]-=-. We shall refer to this method as the MBO-method (cf. [25]) although the name DGCDM-algorithm has also been used [48]. The specifics of this method are elaborated upon in subsequent chapters, so will... |

54 |
On three-phase boundary motion and the singular limit of a vector valued Ginzburg–Landau equation
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- 1993
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Citation Context ... junctions of moving surfaces can occur. This is illustrated in Figure 1.2 for the case of three grains. Although these important problems have been the subject of some study in two dimensions (e.g., =-=[13, 14]-=-), little is known about their numerical solution, especially in three dimensions. An extension of the mean curvature model that we shall consider arises when the thermodynamic driving force of the in... |

54 |
Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature
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- 1993
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Citation Context ... no proof that the method actually approximates motion by mean curvature [71], although it is known that the Allen-Cahn equation (see below) yields Brakke's motion in the limit for two-phase problems =-=[40]-=-. Phase field methods (see [16, 71] and references therein) give the interface as a level Chapter 1. Introduction 5 set of a reaction-diffusion equation such as the Allen-Cahn equation, u t = ffl\Delt... |

54 |
TwO-dimensional motion of idealized grain boundaries
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Citation Context ...ion of the boundary of each grain [9]. By annealing the metal, it becomes warmed so that boundary atoms can change their phase. This produces an interface motion proportional to mean curvature motion =-=[48, 51, 9]-=-. In the interesting case where three or more grain orientations are present, junctions of moving surfaces can occur. This is illustrated in Figure 1.2 for the case of three grains. Although these imp... |

53 |
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Citation Context ...evaluation of Equations (3.21) and (3.22) have been developed. Lagrange interpolation has been used to replace function values at arbitrary points by several function values on an equally spaced grid =-=[55, 10]-=-. Taylor expansions have also been used to correct for deviations from an equally spaced grid [70]. Although these methods produce a significant speed-up over a direct approximation of the summations,... |

49 |
A simple proof for the convergence of an approximation scheme for computing mean curvature motions
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- 1995
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Citation Context ... is often exceedingly slow for three dimensional and affine velocity motions. Similarly to all other methods for multiple-phase problems, no convergence results are known for the MBO-method. However, =-=[25, 4]-=- do give rigorous convergence proofs for two-phase mean curvature motion and [45] gives some further asymptotic results. Chapter 1. Introduction 8 In this thesis, we describe a fast, new algorithm whi... |

45 |
Motion by mean curvature as the singular limit of Ginzburg{Landau dynamics
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- 1991
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Citation Context ...l (e.g., f 0 (u) = u(u \Gamma 1)(u \Gamma 1 2 )) and ffl is a small parameter. For the two-phase problem, it has been proven that certain phase field models produce mean curvature motion when ffl ! 0 =-=[59, 58, 12, 26]-=-. (Indeed, an important motivation for studying mean curvature motion is that it arises as a limit of certain phase field models.) By letting ffl be proportional to ru jruj , these methods provide a m... |

44 |
Nonlocal reaction-diffusion equations and nucleation
- RUBINSTEIN, STERNBERG
- 1992
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Citation Context ...(t) wheresav (t) is the average mean curvature over the surface at time t. Such motion preserves phase volumes (or areas in two dimensions) and occurs as a limit of a nonlocal model for binary alloys =-=[57]-=-. In the context of image enhancement, this nonlocal motion has also been suggested as a possible smoothing which preserves the area of shapes [64]. Theoretical aspects of mean curvature motion have a... |

44 |
Area and length preserving geometric invariant scale-spaces
- Sapiro, Tannenbaum
- 1995
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Citation Context ...urs as a limit of a nonlocal model for binary alloys [57]. In the context of image enhancement, this nonlocal motion has also been suggested as a possible smoothing which preserves the area of shapes =-=[64]-=-. Theoretical aspects of mean curvature motion have also been studied extensively (see, e.g., [31, 36]). Some areas of computational interest include the approximation of bifurcation values [53] and t... |

42 |
Convergence of an algorithm for mean curvature motion
- Evans
- 1993
(Show Context)
Citation Context ...applied. A method based on the model of diffusion-dependent motion of level sets has recently been proposed by Merriman, Bence and Osher [47, 48]. We shall refer to this method as the MBO-method (cf. =-=[25]-=-) although the name DGCDM-algorithm has also been used [48]. The specifics of this method are elaborated upon in subsequent chapters, so will not be repeated here. This method naturally handles compli... |

42 |
Singular perturbation theory of traveling waves in excitable media
- Tyson
- 1988
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Citation Context ...nsider, for example, the motion by mean curvature of the boundary of the spiral region given in Figure 2.11. (The curvature-dependent motion of similar shapes has been considered in biological models =-=[73]-=-.) Since the local curvature of the boundary of this problem varies tremendously, it is impractical to satisfy (2.7) everywhere using a uniform mesh. To achieve a more efficient finite difference algo... |

41 |
Geometric models of crystal growth
- Taylor, Cahn, et al.
- 1992
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Citation Context ...f the mean curvature model that we shall consider arises when the thermodynamic driving force of the interface motion depends on the volume phase change (i.e. bulk effects) as well as surface effects =-=[71]-=-. In this case, the normal velocity of the surface is given by its mean curvature plus a constant, v n (~x; t) = (~x; t) + c(t): (1:2) We shall refer to this type of motion as affine velocity front mo... |

35 |
Theory, algorithms, and applications of level set methods for propagating interfaces, Acta Numer
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- 1996
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29 |
Finite Dierence Schemes and Partial Dierential Equations (2nd Edition
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- 2004
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Citation Context ...h asbackward Euler, ~U n+1 ; ~ U n t = ~U n+1 h give a strong decay of high frequency error modes. Other methods, such as CrankNicolson, ~U n+1 ; ~ U n t = h 0 @ ~ U n+1 + ~ U n 2 1 A and ADI methods =-=[69]-=- give avery weak decay of these error modes [3, 69]. After the sharpening step of the MBO-method, the solution is discontinuous. Because high frequency modes make avery important contribution to such ... |

28 |
The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits
- Caginalp
- 1990
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Citation Context ...lly approximates motion by mean curvature [71], although it is known that the Allen-Cahn equation (see below) yields Brakke's motion in the limit for two-phase problems [40]. Phase field methods (see =-=[16, 71]-=- and references therein) give the interface as a level Chapter 1. Introduction 5 set of a reaction-diffusion equation such as the Allen-Cahn equation, u t = ffl\Deltau \Gamma 1 ffl f 0 (u): Here f 0 i... |

24 |
Fast algorithm for spectral analysis of unevenly sampled data
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- 1989
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Citation Context ...evaluation of Equations (3.21) and (3.22) have been developed. Lagrange interpolation has been used to replace function values at arbitrary points by several function values on an equally spaced grid =-=[55, 10]-=-. Taylor expansions have also been used to correct for deviations from an equally spaced grid [70]. Although these methods produce a significant speed-up over a direct approximation of the summations,... |

23 |
A short note on the evolution of a surface by its mean curvature
- Grayson
- 1989
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Citation Context ...ated by a tetrahedron. For example, the shaded region in tetrahedron TABCD of Figure 3.9 would be approximated by T abcD . 3 Nonsmooth corners may arise from singularities in the solution (see, e.g., =-=[35]-=-). Each corner can produce an O(h 3 ) error in the phase volumes. However, because these corners are rapidly smoothed away, they typically do not affect the overall order of accuracy of the method. Ch... |

19 | Curvature Flow and Entropy Conditions Applied to Grid Generation
- Sethian
- 1994
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Citation Context ...dical technology and architecture. Other applications for affine motion occur in certain flame propagation problems (e.g., Chapter 1. Introduction 4 grassfire flow) and automatic grid generation. See =-=[54, 63, 66]-=- for further details. 1.2 Methods for Curvature-Dependent Motion To study the phenomena outlined in the previous section, several numerical methods have been developed. Most of these can be divided in... |