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

Liu, Wei; Xie, Zhongliang; Yi, Wenfan

doi:10.1007/s11425-021-2093-1

Cited by 1 publication

(3 citation statements)

References 31 publications

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“…It is worthwhile to point out that step-size search rules used in traditional LMMs include the optimal/exact step-size search rule [26], the normalized Armijo-type step-size search rule [27,41,44], the normalized Goldstein-type step-size search rule [29] and the normalized Wolfe-Powelltype step-size search rules [30]. And up to now all step-size search rules in traditional LMMs are monotone in the sense that the sequence {E(w k )} is monotonically decreasing.…”

Section: Lemma 25 ([41]mentioning

confidence: 99%

“…And up to now all step-size search rules in traditional LMMs are monotone in the sense that the sequence {E(w k )} is monotonically decreasing. Actually this feature is vital for the convergence analysis in traditional LMMs, see [27,29,30,41,52]. Since the work of this paper is closely related to the normalized Armijo-type step-size search rule, let us describe it a little more.…”

Section: Lemma 25 ([41]mentioning

confidence: 99%

“…The feasibility and global convergence analysis of this approach was also provided. Recently, as a subsequent work [30], both normalized Wolfe-Powell-type and strong Wolfe-Powell-type step-size search rules for the LMM were introduced and global convergence of the corresponding algorithms for general decrease directions were verified. The LMM with general descent direction provides the possibility for accelerating the convergence by appropriately choosing the descent direction.…”

Section: Introductionmentioning

confidence: 99%

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Nonmonotone Local Minimax Methods for Finding Multiple Saddle Points

Liu¹,

Yi²

2024

JCM

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In this paper, by designing a normalized nonmonotone search strategy with the Barzilai-Borwein-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 spaces. Compared 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 computations. Firstly, 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 constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, 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 iteration. Finally, 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 conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

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