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Searching the solution landscape by generalized high-index saddle dynamics

Science China Mathematics 2021年第8期64卷 1801-1816页

作者： Jianyuan Yin Bing Yu Lei Zhang School of Mathematical Sciences Peking UniversityBeijing 100871China Beijing International Center for Mathematical Research Center for Quantitative BiologyPeking UniversityBeijing 100871China

We introduce a generalized numerical algorithm to construct the solution landscape,which is a pathway map consisting of all the stationary points and their connections.Based on the high-index optimizationbased shrinking dimer(Hi OSD)method for gradient systems,a generalized high-index saddle dynamics(GHi SD)is proposed to compute any-index saddles of dynamical systems.Linear stability of the index-k saddle point can be proved for the GHi SD system.A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape,which not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses,but also reveals the relationships between different solutions.Numerical examples,including a three-dimensional example and the phase field model,demonstrate the novel concept of the solution landscape by showing the connected pathway maps.

关键词： saddle point energy landscape solution landscape pathway map dynamical system phase field

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

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Science China Mathematics 2023年第10期66卷 2347-2360页

作者： Lei Zhang Pingwen Zhang Xiangcheng Zheng Beijing International Center for Mathematical Research Center for Machine Learning ResearchCenter for Quantitative BiologyPeking UniversityBeijing 100871China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China School of Mathematical Sciences Laboratory of Mathematics and Applied MathematicsPeking UniversityBeijing 100871China School of Mathematics Shandong UniversityJinan 250100China

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

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Error Estimate for Semi-implicit Method of Sphere-Constrained High-Index Saddle Dynamics

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Chinese Annals of Mathematics,Series B 2023年第5期44卷 765-780页

作者： Lei ZHANG Pingwen ZHANG Xiangcheng ZHENG Beijing International Center for Mathematical Research Center for Machine Learning ResearchCenterfor Quantitative BiologyPeking UniversityBeijing 100871China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China School of MathematicalSciences Laboratory of Mathematics and Applied MathematicsPeking UniversityBeijing 100871China School of Mathematics Shandong UniversityJinan 250100China

The authors prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics,which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere.Due to the semi-implicit treatment and the novel computational procedure,the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations,and the computations of the state variable and directional vectors are coupled with the retraction,the vector transport and the orthonormalization procedure,which significantly complicates the analysis.They address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

关键词： Saddle point Constrained saddle dynamics solution landscape Semiimplicit Numerical analysis

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