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We propose a level-set approach to characterize the region occupied by the solid in Stefan problems with and without surface tension, based on their recent probabilistic reformulation. The level-set function is parameterized by a feed-forward neural network, whose parameters are trained using the probabilistic formulation of the Stefan growth condition. The algorithm can handle Stefan problems where the liquid is supercooled and can capture surface tension effects through the simulation of particles along the moving boundary together with an efficient approximation of the mean curvature. We demonstrate the effectiveness of the method on a variety of examples with and without radial symmetry.
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On median filters for motion by mean curvature
The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.
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- Award ID(s):
- 2012015
- PAR ID:
- 10573243
- Publisher / Repository:
- American Mathematical Society
- Date Published:
- Journal Name:
- Mathematics of Computation
- ISSN:
- 0025-5718
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/mcom/3940
