Probabilistic validation of homology computations for nodal domains
Probabilistic validation of homology computations for nodal domains
复制标题
节点域同源性计算的概率验证
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Thomas Wanner
中科院分区：
文献类型：
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作者：
K. Mischaikow;Thomas Wanner
同源性一直被认为是一个重要的计算工具，用于量化复杂的结构。在许多应用中，这些结构作为实值函数的节点域出现，因此仅适用于基于适当离散化的数值研究。这种方法立即引起了一个问题，即由此产生的同源性计算有多准确。在本文中，我们提出了一个概率的方法来量化的有效性的同源计算的节点域的随机场在一个和两个空间的维，它crosshes明确的概率先验界的适用性，某些离散化的大小。我们说明了我们的结果的特殊情况下的随机周期场和随机三角多项式。
Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random fields in one and two space dimensions, which furnishes explicit probabilistic a priori bounds for the suitability of certain discretization sizes. We illustrate our results for the special cases of random periodic fields and random trigonometric polynomials.