论文标题
高维球体上受约束的高索引动力学的离散化和索引射击误差分析
Discretization and index-robust error analysis for constrained high-index saddle dynamics on high-dimensional sphere
论文作者
论文摘要
我们将数值离散化开发并分析为受约束的高索引动力学，搜索限制在高维单元球体上的高索引鞍点的动力学。与没有约束的马鞍动力学相比，受约束的高索引动力学具有更复杂的动力学形式，并且由于约束而需要其他操作，例如缩回和向量传输，这显着使数值方案和相应的数值分析变得复杂。此外，由于现有的数值分析结果通常取决于鞍点的索引，因此，如果在许多应用中该指数很高，则可以降低证明的数值准确性，这表明相对于索引缺乏稳健性。为了解决这些问题，我们得出了高维球体上约束高索引鞍动力学数字离散化的误差估计，然后通过调整放宽参数在平均规范中提供索引 - 固定错误分析来改进它。开发的结果为数值计算的准确性提供了数学支持。
We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.
