Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension

Authors:
Thomas C. Cecil;Stanley J. Osher;Jianliang Qian
Affiliations:
ICES, University of Texas, Austin, 1 University Station, C0200, Austin, TX 78712, United States;Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, United States;Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, United States
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

We introduce simplex free adaptive tree numerical methods for solving static and time-dependent Hamilton-Jacobi equations arising in level set problems in arbitrary dimension. The data structure upon which our method is built in a generalized n-dimensional binary tree, but it does not require the complicated splitting of cubes into simplices (aka generalized n-dimensional triangles or hypertetrahedrons) that current tree-based methods require. It has enough simplicity that minor variants of standard numerical Hamiltonians developed for uniform grids can be applied, yielding consistent, monotone, convergent schemes. Combined with the fast sweeping strategy, the resulting tree-based methods are highly efficient and accurate. Thus, without changing more than a few lines of code when changing dimension, we have obtained results for calculations in up to n=7 dimensions.