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Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

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Science China Mathematics 2023年第10期66卷 2361-2384页

作者： Wei Liu Ziqing Xie Wenfan Yi Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) Hunan Normal UniversityChangsha 410081China School of Mathematics Hunan UniversityChangsha 410082China

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 *** 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 *** 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 *** 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 *** a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also ***,it extends the framework of traditional *** addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the ***,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 *** practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.

关键词： semilinear elliptic PDE multiple unstable solution local minimax method normalized strong Wolfe-Powell-type search rule conjugate-gradient-type descent direction general descent direction global convergence

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NONMONOTONE local minimax methodS FOR FINDING MULTIPLE SADDLE POINTS

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Journal of Computational Mathematics 2024年第3期42卷 851-884页

作者： Wei Liu Ziqing Xie Wenfan Yi South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) Hunan Normal UniversityChangshaHunan 410081China Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics School of MathematicsHunan UniversityChangshaHunan 410082China

In this paper,by designing a normalized nonmonotone search strategy with the BarzilaiBorwein-type step-size,a novel local minimax method(LMM),which is a globally convergent iterative method,is proposed and analyzed to find multiple(unstable)saddle points of nonconvex functionals in Hilbert *** to traditional LMMs with monotone search strategies,this approach,which does not require strict decrease of the objective functional value at each iterative step,is observed to converge faster with less ***,based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold,by generalizing the Zhang-Hager(ZH)search strategy in the optimization theory to the LMM framework,a kind of normalized ZH-type nonmonotone step-size search strategy is introduced,and then a novel nonmonotone LMM is *** feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative ***,in order to speed up the convergence of the nonmonotone LMM,a globally convergent Barzilai-Borwein-type LMM(GBBLMM)is presented by explicitly constructing the Barzilai-Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each ***,the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures:one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary *** numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

关键词： Multiple saddle points local minimax method Barzilai-Borwein gradient method Normalized nonmonotone search strategy Global convergence

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