Discretization and index-robust error analysis for constrained high-index saddle dynamics on the high-dimensional sphere  

作　　者：Lei Zhang Pingwen Zhang Xiangcheng Zheng 

机构地区：[1]Beijing International Center for Mathematical Research,Center for Machine Learning Research,Center for Quantitative Biology,Peking University,Beijing 100871,China [2]School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China [3]School of Mathematical Sciences,Laboratory of Mathematics and Applied Mathematics,Peking University,Beijing 100871,China [4]School of Mathematics,Shandong University,Jinan 250100,China

出　　处：《Science China Mathematics》2023年第10期2347-2360,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.12225102,12050002 and 12288101);the National Key Research and Development Program of China(Grant No.2021YFF1200500).

摘　　要：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 constraints,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 the high-dimensional sphere and then improve it by providing index-robust error analysis in an averaged norm by adjusting the relaxation parameters.The developed results provide mathematical support for the accuracy of numerical computations.

关 键 词：saddle dynamics saddle point solution landscape error estimate index-robust 

分 类 号：O241.8[理学—计算数学]