由单纯复形计算离散Morse复形
Computing discrete Morse complexes from simplicial complexes
摘要：
我们认为，有效地计算一个离散的莫尔斯复杂的任意尺寸和非常大的尺寸的单纯复形的问题。基于一个共同的基于图形的形式主义，我们分析了现有的数据结构的单纯形复合物，我们定义了一个有效的编码离散莫尔斯梯度最紧凑的这种表示。我们从理论上比较方法的基础上减少和coreductions计算离散莫尔斯梯度，证明减少和coreductions的组合产生新的相互等价的方法。我们设计并实现了一个新的算法来计算一个离散的Morse复形的单纯复形。我们表明，我们的方法规模非常好的大小和维数的单纯复杂的也通过比较与现有的唯一的公共域算法的离散莫尔斯复杂的计算。我们讨论了应用程序的计算多参数持久的同源性和极值图的可视化随时间变化的三维标量场。
英文摘要：
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial complexes, and we define an efficient encoding for the discrete Morse gradient on the most compact of such representations. We theoretically compare methods based on reductions and coreductions for computing a discrete Morse gradient, proving that the combination of reductions and coreductions produces new mutually equivalent approaches. We design and implement a new algorithm for computing a discrete Morse complex on simplicial complexes. We show that our approach scales very well with the size and the dimension of the simplicial complex also through comparisons with the only existing public-domain algorithm for discrete Morse complex computation. We discuss applications to the computation of multi-parameter persistent homology and of extrema graphs for visualization of time-varying 3D scalar fields.
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