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

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

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

Nondifferentiable Optimization: Motivations and Applications

IIASA has been involved in research on nondifferentiable optimization since 1976. The Institute's research in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimization has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition and to review developments in this field IIASA held this Workshop in Sopron (Hungary) in September 1984. This volume contains selected papers presented at the Workshop. It is divided into four sections dealing with the following topics: (I) Concepts in Nonsmooth Analysis; (II) Multicriteria Optimization and Control Theory; (III) Algorithms and Optimization Methods; (IV) Stochastic Programming and Applications

Shape optimization for a sharp interface model of distortion compensation

We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface

Bounding extreme events in nonlinear dynamics using convex optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.Comment: Revised according to comments by reviewers. Added references and rearranged introduction, conclusions, and proofs. 38 pages, 7 figures, 4 tables, 4 appendices, 87 reference