Convolution Kernels and Stability of Threshold Dynamics Methods

Esedoḡlu, Selim; Jacobs, Matt

doi:10.1137/16m1087552

Cited by 13 publications

(21 citation statements)

References 24 publications

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“…Chapters 1-3 in [4], to this localized setting. As for any minimizing movements scheme, the comparison of χ n to the previous time step χ n−1 in the minimization problem (14) yields an energy-dissipation inequality which serves well as an a priori estimate, but which fails to be sharp by a factor of 2. To obtain a sharp inequality we follow the ideas of De Giorgi.…”

Section: De Giorgi's Variational Interpolation and Idea Of Proofmentioning

confidence: 99%

“…Dividing by √ h, recalling the definitions (15) and (16) of the localized distance and energy, and using the semi-group and symmetry properties of the kernel and the symmetry of σ yield (14). and furthermore we recall the (not necessarily unique) variational interpolation u h (t) of χ and χ 1 := u h (h), cf.…”

Section: Proofsmentioning

confidence: 99%

“…The recent work [10] of Elsey and Esedoglu is inspired by the variational viewpoint [15] and shows that not all anisotropies can be obtained when structural features such as positivity of the kernel are required. However, variants of the scheme developed by Esedoglu and Jacobs [14] share the same stability conditions even for more general kernels.…”

Section: Introductionmentioning

confidence: 99%

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Brakke’s inequality for the thresholding scheme

Laux

Otto

2020

Calc. Var.

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We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke's notion of mean curvature flow. Our proof is based on a localized version of the minimizing movements interpretation of Esedoglu and the second author. We apply De Giorgi's variational interpolation to the thresholding scheme and pass to the limit in the resulting energy-dissipation inequality. The result is conditional in the sense that we assume the time-integrated energies of the approximations to converge to those of the limit.

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Section: De Giorgi's Variational Interpolation and Idea Of Proofmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Brakke’s inequality for the thresholding scheme

Laux

Otto

2020

Calc. Var.

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“…A stronger, Gamma convergence version of Proposition 2 is given in [14] for a class of kernels that include sign changing ones. In polar coordinates, the expression for the surface tension σ K that corresponds to a given convolution kernel K is:…”

Section: Previous Workmentioning

confidence: 99%

“…The present paper is devoted to providing a decisive, constructive answer to this question, by showing how to choose the kernel K given a desired possibly anisotropic surface tension and possibly anisotropic mobility for the interface. Combined with new multiphase versions of threshold dynamics recenty proposed in [14], the kernel constructions of this paper yield unconditionally stable schemes for the weighted mean curvature flow of a general n-phase network by allowing n-choose-2 anistropic surface tensions and n-choose-2 anisotropic mobilities (one pair for each interface in the network) to be individually specified. This full level of generality is also a first for threshold dynamics schemes.…”

Section: Introductionmentioning

confidence: 99%

Kernels with prescribed surface tension & mobility for threshold dynamics schemes

Esedoḡlu

Jacobs

2017

Journal of Computational Physics

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We show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces, including networks of them, in both two and three dimensions. Moreover, we discuss necessary and sufficient conditions for the positivity of the kernel which, in the case of two-phase flow, ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow. In particular, we show, in a barrier-type statement, that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions, and give a complete characterization. Among other results is a threshold dynamics scheme that is guaranteed to dissipate a nonlocal approximation to the interfacial energy in the fully anisotropic, multiphase setting, using the new kernel construction.

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