On the entropy of parabolic Allen–Cahn equation

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

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A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions

Advances in Difference Equations ◽

10.1186/s13662-020-02616-x ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 3

Author(s):

Nauman Khalid ◽

Muhammad Abbas ◽

Muhammad Kashif Iqbal ◽

Dumitru Baleanu

Keyword(s):

Numerical Investigation ◽

Spline Functions ◽

B Spline ◽

Cahn Equation

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BSDEs with polynomial growth generators

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953300000216 ◽

2000 ◽

Vol 13 (3) ◽

pp. 207-238 ◽

Cited By ~ 41

Author(s):

Philippe Briand ◽

René Carmona

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Existence And Uniqueness ◽

Polynomial Growth ◽

Probabilistic Representation ◽

Existence And Uniqueness Results ◽

State Variable ◽

Cahn Equation ◽

Uniqueness Results ◽

Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

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