Nonmonotone local minimax methods for finding multiple saddle points

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.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics on January 3, 202

Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well

Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study

A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles

In order to find the first few unconstrained saddles of functionals with different types of variational structures, a new local minimax method (LMM), based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc., is developed. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark examples commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE of both M-type (focusing) and W-type (defocusing). The Newton’s method will also be investigated and used to accelerate the local convergence and increase the accuracy. The Nehari manifold is used in the algorithm to satisfy a crucial condition for convergence. The numerical computation is also accelerated and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by the LMM with the Nehari manifold. To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail, and a generalized dynamic system of points on it is given. The corresponding LMM with a correction technique is also justified and a convergence analysis is presented, then it is tested on an indefinite M-type case. A numerical investigation of bifurcation for an indefinite problem will be given to provide numerical evidence for PDE analysts for future stud

A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles

In order to find the first few unconstrained saddles of functionals with different types of variational structures, a new local minimax method (LMM), based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc., is developed. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark examples commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE of both M-type (focusing) and W-type (defocusing). The Newton’s method will also be investigated and used to accelerate the local convergence and increase the accuracy. The Nehari manifold is used in the algorithm to satisfy a crucial condition for convergence. The numerical computation is also accelerated and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by the LMM with the Nehari manifold. To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail, and a generalized dynamic system of points on it is given. The corresponding LMM with a correction technique is also justified and a convergence analysis is presented, then it is tested on an indefinite M-type case. A numerical investigation of bifurcation for an indefinite problem will be given to provide numerical evidence for PDE analysts for future stud

Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well

Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study

On Computing Multiple Solutions of Nonlinear PDEs Without Variational Structure

Variational structure plays an important role in critical point theory and methods. However many differential problems are non-variational i.e. they are not the Euler- Lagrange equations of any variational functionals, which makes traditional critical point approach not applicable. In this thesis, two types of non-variational problems, a nonlinear eigen solution problem and a non-variational semi-linear elliptic system, are studied. By considering nonlinear eigen problems on their variational energy profiles and using the implicit function theorem, an implicit minimax method is developed for numerically finding eigen solutions of focusing nonlinear Schrodinger equations subject to zero Dirichlet/Neumann boundary condition in the order of their eigenvalues. Its mathematical justification and some related properties, such as solution intensity preserving, bifurcation identification, etc., are established, which show some significant advantages of the new method over the usual ones in the literature. A new orthogonal subspace minimization method is also developed for finding multiple (eigen) solutions to defocusing nonlinear Schrodinger equations with certain symmetries. Numerical results are presented to illustrate these methods. A new joint local min orthogonal method is developed for finding multiple solutions of a non-variational semi-linear elliptic system. Mathematical justification and convergence of the method are discussed. A modified non-variational Gross-Pitaevskii system is used in numerical experiment to test the method