103 research outputs found
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
Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics
In this paper, we propose in a Hilbertian setting a second-order
time-continuous dynamic system with fast convergence guarantees to solve
structured convex minimization problems with an affine constraint. The system
is associated with the augmented Lagrangian formulation of the minimization
problem. The corresponding dynamics brings into play three general time-varying
parameters, each with specific properties, and which are respectively
associated with viscous damping, extrapolation and temporal scaling. By
appropriately adjusting these parameters, we develop a Lyapunov analysis which
provides fast convergence properties of the values and of the feasibility gap.
These results will naturally pave the way for developing corresponding
accelerated ADMM algorithms, obtained by temporal discretization
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
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
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
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