A Partial Differential Equation Obstacle Problem for the Level Set Approach to Visibility
Journal of Scientific Computing(2020)
摘要
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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关键词
Visibility, Level set method, Viscosity solutions, Fast sweeping method, 35J15, 35J60, 65N06, 65N12
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