A conservative Allen–Cahn equation with a curvature-dependent Lagrange multiplier

2022 ◽  
Vol 126 ◽  
pp. 107838
Author(s):  
Soobin Kwak ◽  
Junxiang Yang ◽  
Junseok Kim
Keyword(s):  
Lagrange Multiplier ◽  
Cahn Equation

Lagrange multiplier and singular limit of double obstacle problems for the Allen-Cahn equation with constraint

2016 ◽  
Vol 40 (1) ◽  
pp. 5-21 ◽  
Author(s):  
Mohammad Hassan Farshbaf-Shaker ◽  
Takeshi Fukao ◽  
Noriaki Yamazaki
Keyword(s):  
Lagrange Multiplier ◽  
Singular Limit ◽  
Obstacle Problems ◽  
Cahn Equation

scholarly journals Convergence of a mass conserving Allen–Cahn equation whose Lagrange multiplier is nonlocal and local

2014 ◽  
Vol 16 (2) ◽  
pp. 243-268 ◽  
Author(s):  
Matthieu Alfaro ◽  
Pierre Alifrangis
Keyword(s):  
Lagrange Multiplier ◽  
Cahn Equation

scholarly journals Mass-conserving diffusion-based dynamics on graphs

2021 ◽  
pp. 1-49
Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP
Keyword(s):  
Multiscale Modeling ◽  
Classification Problems ◽  
Theoretical Justification ◽  
Finite Graphs ◽  
Discrete Scheme ◽  
Cahn Equation ◽  
Separating Flows ◽  
Special Case ◽  
Appl Math ◽  
Convergence Of The Scheme

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.

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

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nauman Khalid ◽  
Muhammad Abbas ◽  
Muhammad Kashif Iqbal ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Investigation ◽  
Spline Functions ◽  
B Spline ◽  
Cahn Equation

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
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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