平面Morse-Smale复形的证明计算
Certified Computation of planar Morse-Smale Complexes
摘要：
Morse-Smale复形是计算几何和拓扑学中各种问题的全局拓扑分析的重要工具。Morse-Smale复形的算法已经在分段线性流形的情况下提出。然而，以前在这一领域的研究是不完整的光滑函数的情况下。在目前的文件中，我们解决以下问题：给定一个任意复杂的Morse-Smale系统的平面域，是否有可能计算其认证（拓扑正确）Morse-Smale复杂？为此，我们开发了一个算法，使用区间算术计算证明临界点和分离线形成的Morse-Smale复形的光滑函数在有界平面区域上。我们的算法也可以计算几何上接近的Morse-Smale复形。
英文摘要：
The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse-Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse-Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse-Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse-Smale complexes.
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