从有限向量场数据到Forman意义下的组合动力系统
From Finite Vector Field Data to Combinatorial Dynamical Systems in the Sense of Forman
摘要：
主要目标是从有限的向量场数据构造一个Forman意义下的组合动力系统。我们使用一个线性最小化问题与二进制变量和线性等式约束。最小化问题的解导出了组合动力系统的容许匹配。它们是该方法的三个主要步骤：构造单纯复形，计算每个单纯形的向量，解决最小化问题，并应用诱导匹配。通过对Lotka-Volterra模型和Lorenz吸引子模型的实验，验证了该方法的有效性.我们能够检索的Lotka-Volterra模型的循环行为和混沌行为的洛伦兹吸引子。算法的两个扩展。我们采用了重心重分的方法来获得更好的分辨率。我们添加条件的最小化问题，以获得一个解决方案，诱导梯度匹配。
英文摘要：
The main goal is to construct a combinatorial dynamical system in the sense of Forman from finite vector field data. We use a linear minimization problem with binary variables and linear equality constraints. The solution of the minimization problem induces an admissible matching for the combinatorial dynamical system. They are three main steps for the method: Construct a simplicial complex, compute a vector for each simplex, solve the minimization problem, and apply the induced matching. We argue the effectiveness of the method by testing it on the Lotka-Volterra model and the Lorenz attractor model. We are able to retrieve the cyclic behaviour in the Lotka-Volterra model and the chaotic behaviour of the Lorenz attractor. Two extensions to the algorithm are shown. We use the barycentric subdivision to obtain a better resolution. We add conditions on the minimization problem to obtain a solution which induce a gradient matching.
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