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The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput. 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput. 24(3), 865--885 (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 the above work and playing 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 named as the normalized Wolfe-Powell-type LMM (NWP-LMM) are introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm 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 LMM algorithms. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMMs family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better among its LMM companions.
In this paper, combining normalized nonmonotone search strategies with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is globally convergent, is proposed to find multiple (unstable) saddle points of nonconvex function
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $Omega$. One of the requirements leading to approximation robustness is to place the interpola
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This work is a follow-up to our previous contribution (Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs), SIAM J. Numer. Anal., 2018), and contains further insights on some
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