论文标题
通过语法引导的基于多移民的概括性基于多移民的Helmholtz预处理
Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming
论文作者
论文摘要
求解无限的Helmholtz方程不仅对于理解许多物理现象至关重要，而且代表了成功应用数值方法的缺陷基准问题。在这里，我们介绍了一种新的方法，用于将有效的预处理迭代求解器与多目标语法引导的基因编程有关Helmholtz问题。我们的方法基于一种新颖的无上下文语法，该语法能够构建多方面的预定器，该杂种预处理在每个离散水平上采用量身定制的操作顺序。为了找到对给定领域概括的求解器，我们提出了一种连续的问题难度适应的自定义方法，在该方法中，我们评估了预审人员对越来越多的条件问题实例的效率。我们通过将基于多机构的预处理进化为二维无限性Helmholtz问题来证明我们的方法的有效性，该问题的表现优于几种人为设计的方法，用于不同的波数，直到有超过一百万个未知数的线性方程系统。
Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find solvers that generalize well over the given domain, we propose a custom method of successive problem difficulty adaption, in which we evaluate a preconditioner's efficiency on increasingly ill-conditioned problem instances. We demonstrate our approach's effectiveness by evolving multigrid-based preconditioners for a two-dimensional indefinite Helmholtz problem that outperform several human-designed methods for different wavenumbers up to systems of linear equations with more than a million unknowns.
