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The parallel complexity of embedding algorithms for the solution of systems of nonlinear equations

机译：非线性方程组求解的嵌入算法的并行复杂度

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摘要

Embedding algorithms used to solve nonlinear systems of equations do so by constructing a continuous family of systems and solving the given system by tracking the continuous curve of solutions to the family. Solving nonlinear equations by a globally convergent embedding algorithm requires the evaluation and factoring of a Jacobian matrix at many points along the embedding curve. Ways to optimize the Jacobian matrix on a hypercube are described. Several static and dynamical strategies for assigning components of the Jacobian to processors on the hypercube are investigated. It is found that a static rectangular grid mapping is the preferred choice for inclusion in a robust parallel mathematical software package. The static linear mapping is a viable alternative when there are many common subexpressions in the component evaluation, and the dynamic assignment strategy should only be considered when there is large variation in the evaluation times for the components, leading to a load imbalance on the processors.
机译：用来求解方程组非线性系统的嵌入算法是通过构造一个连续的系统族并通过跟踪该族的解的连续曲线来求解给定系统来实现的。用全局收敛嵌入算法求解非线性方程需要在沿着嵌入曲线的许多点上对雅可​​比矩阵进行评估和分解。描述了在超立方体上优化雅可比矩阵的方法。研究了在超立方体上将雅可比矩阵的分量分配给处理器的几种静态和动态策略。发现静态矩形网格映射是包含在健壮的并行数学软件包中的首选。当组件评估中有许多常见的子表达式时，静态线性映射是可行的选择，并且仅当组件的评估时间有较大差异时才应考虑动态分配策略，从而导致处理器上的负载不平衡。

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