Stepsize control for path tracking
Stepsize control for path tracking
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路径跟踪的步长控制
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
C. Wampler
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作者：
Daniel J. Bates;J. Hauenstein;A. Sommese;C. Wampler
当数值跟踪隐式定义的路径时，例如同伦连续方法所需要的，通过使用自适应步长和自适应多精度方法来提高效率和可靠性。通过同时适应精度和步长，可以进一步提高效率和可靠性。本文提出了一种同时调整精度和步长的策略，以消除单独调整这两个量时可能发生的某些类型的路径故障，同时还减少了在路径上每单位前进所花费的计算工作量。本文研究由n+1个未知数的n个方程隐含定义的一维路径的路径跟踪算法。特别是，当多精度计算可用时，即当计算的精度可以在计算期间改变时，我们考虑这种算法。我们处理一种常见类型的路径跟踪器，它使用欧拉预测器沿着路径的切线前进，并使用牛顿校正器使预测点更接近路径。本文的目的是描述一种启发式算法，用于同时调整精度和步长，以减少跟踪路径的计算成本，同时保持高可靠性。在固定精度跟踪中，设置步长的试错法是有效的：失败时缩短步长，重复成功时延长步长。如果精度不够高，那么无论迈出多小的一步，这一步都可能失败，因此试错法反复地缩短步长，直到由于缺乏进展而宣布失败。2000年数学学科分类。主65H10；辅助65H20、65G50、14Q99。贝茨得到了科罗拉多州立大学和数学及其应用研究所(IMA)的支持。豪恩斯坦得到了圣母大学邓肯教席、圣母大学应用数学中心和美国国家科学基金会资助的DMS0410047和DMS0712910的支持。Sommese得到了圣母大学邓肯教席的支持；国家科学基金会提供了0410047丹麦马克和0712910丹麦马克的赠款。Wanter得到了美国国家科学基金会拨款dms-0410047和dms-0712910的支持。
When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy for adjusting precision and stepsize together to eliminate certain types of path failures that can occur when adapting the two quantities independently, while also reducing the computational effort expended per unit advance along the path. This paper concerns path tracking algorithms for tracing out a one dimensional path defined implicitly by n equations in n+1 unknowns. In particular, we consider such algorithms when multiprecision calculations are available, that is, when the precision of the computations can be changed during the computation. We treat a common type of path tracker that uses an Euler predictor to step ahead along the tangent to the path and a Newton corrector to bring the predicted point closer to the path. The objective of this paper is to describe a heurstic for adjusting precision and stepsize together to reduce the computational cost of tracking the path while maintaining high reliability. In fixed precision tracking, a trial-and-error approach to setting the stepsize is effective: shorten the step upon failure, and lengthen it upon repeated successes. If the level of precision is inadequate, the step may fail no matter how small the step is made, so the trial-and-error approach repeatedly shortens the stepsize until failure is declared due to lack of progress. 2000 Mathematics Subject Classification. Primary 65H10; Secondary 65H20, 65G50, 14Q99. Bates was supported by Colorado State University and the Institute for Mathematics and Its Applications (IMA). Hauenstein was supported by the Duncan Chair of the University of Notre Dame; the University of Notre Dame Center for Applied Mathematics; and NSF grants DMS-0410047 and DMS0712910. Sommese was supported by the Duncan Chair of the University of Notre Dame; and NSF grants DMS-0410047 and DMS-0712910. Wampler was supported by NSF grants DMS-0410047 and DMS-0712910.