论文标题
非线性方程的结构化系统
Structured Systems of Nonlinear Equations
论文作者
论文摘要
在方程的“结构化系统”中，每个方程都取决于变量的指定子集。在本文中，我们探讨了“几乎每个”具有固定结构的“几乎每个”系统的属性，以及如何从相应的连接图中读取属性。 如果$ f $变化很小，则解决系统$ f（p）= c $的解决方案$ p $被称为健壮。我们建立了确定取决于结构的鲁棒性的方法，如结构化系统的相应有向图的性能所示。了解线性和非线性结构化系统的关键是我们向前和向后瓶颈的变量的子集。特别是，当鲁棒性在结构化系统中失败时，这是由于存在独特的“向后瓶颈”，我们称之为“ minimax瓶颈”。我们提出了一种定位最小瓶颈的数值方法。我们通过将边缘添加到图表中显示如何将其删除。
In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution $p$ of a system $F(p)=c$ is called robust if it persists despite small changes in $F$. We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck", that we call a "minimax bottleneck". We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph.
