Journal Article10.1007/S00791-006-0020-2
Robust Multigrid Methods for Vector-valued Allen–Cahn Equations with Logarithmic Free Energy
Ralf Kornhuber,Rolf Krause +1 more
TL;DR: The algorithms are shown to be robust in the sense that convergence is preserved for arbitrary values of temperature, including the deep quench limit, and the convergence speed as well is independent of temperature.
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Abstract: We present efficient and robust multigrid methods for the solution of large, nonlinear, non-smooth systems as resulting from implicit time discretization of vector-valued Allen-Cahn equations with isotropic interfacial energy and logarithmic potential. The algorithms are shown to be robust in the sense that convergence is preserved for arbitrary values of temperature, including the deep quench limit. Numerical experiments indicate that the convergence speed as well is independent of temperature.
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Citations
A Nonsmooth Multiscale Method for Solving Frictional Two-Body Contact Problems in 2D and 3D with Multigrid Efficiency
TL;DR: The nonsmooth multiscale approach is general in the sense that it can be used in the context of geometric as well as algebraic multigrid methods and can be applied to contact problems with Tresca friction and Coulomb friction.
61
Color Image Segmentation by the Vector-Valued Allen–Cahn Phase-Field Model: A Multigrid Solution
David Kay,Alessandro Tomasi +1 more
TL;DR: An efficient numerical solution of a PDE-driven model for color image segmentation using a multigrid splitting of a finite element space is presented, thereby producing an efficient and robust method for the segmentation of large images.
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Interacting diffusions approximating the porous medium equation and propagation of chaos
TL;DR: In this paper, a system of interacting diffusions was studied and it was shown that for a large number of particles its empirical measure approximates the solution of the porous medium equation.
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Semi-supervised learning with sparse grids
Jochen Garcke,Michael Griebel +1 more
- 01 Jan 2005
TL;DR: This article formulate the semi-supervised learning problem by a regularization approach and discretize the resulting problem in function space by the sparse grid method and solve the arising equations using the so-called combination technique.
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A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms
Hyun Geun Lee,June-Yub Lee +1 more
TL;DR: In this paper, the authors proposed a second-order operator splitting method for AC type equations with nonlinear source terms, which decomposes the original AC equation into three sub-equations with the free-energy evolution term, the heat evolution term and a non-linear source term, respectively.
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