The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The stee...The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12171148 and 11771138)the Construct Program of the Key Discipline in Hunan Province.Wei Liu was supported by National Natural Science Foundation of China(Grant Nos.12101252 and 11971007)+2 种基金supported by National Natural Science Foundation of China(Grant No.11901185)National Key Research and Development Program of China(Grant No.2021YFA1001300)the Fundamental Research Funds for the Central Universities(Grant No.531118010207).
文摘The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple solutions.The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional LMMs.In this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search rules.The corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent directions.As a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also verified.Consequently,it extends the framework of traditional LMMs.In addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the NWP-LMM.Finally,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms.In practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.
