Abstract
In this article we consider the problem of finding the visibility set from a given point when the obstacles are represented as the level set of a given function. Although the visibility set can be computed efficiently by ray tracing, there are advantages to using a level set representation for the obstacles, and to characterizing the solution using a partial differential equation (PDE). A nonlocal PDE formulation was proposed in Tsai et al. (J Comput Phys 199(1):260–290. https://doi.org/10.1016/j.jcp.2004.02.015, 2004): in this article we propose a simpler PDE formulation, involving a nonlinear obstacle problem. We present a simple numerical scheme and show its convergence using the framework of Barles and Souganidis. Numerical examples in both two and three dimensions are presented.
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The second author thanks the hospitality of the Mathematics and Statistics department of McGill University during its visit where the work for this paper was carried out.
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This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA9550-18-1-0167.
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Oberman, A., Salvador, T. A Partial Differential Equation Obstacle Problem for the Level Set Approach to Visibility. J Sci Comput 82, 14 (2020). https://doi.org/10.1007/s10915-019-01106-x
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DOI: https://doi.org/10.1007/s10915-019-01106-x