140 research outputs found
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
Adaptive local minimax Galerkin methods for variational problems
In many applications of practical interest, solutions of partial differential
equation models arise as critical points of an underlying (energy) functional.
If such solutions are saddle points, rather than being maxima or minima, then
the theoretical framework is non-standard, and the development of suitable
numerical approximation procedures turns out to be highly challenging. In this
paper, our aim is to present an iterative discretization methodology for the
numerical solution of nonlinear variational problems with multiple (saddle
point) solutions. In contrast to traditional numerical approximation schemes,
which typically fail in such situations, the key idea of the current work is to
employ a simultaneous interplay of a previously developed local minimax
approach and adaptive Galerkin discretizations. We thereby derive an adaptive
local minimax Galerkin (LMMG) method, which combines the search for saddle
point solutions and their approximation in finite-dimensional spaces in a
highly effective way. Under certain assumptions, we will prove that the
generated sequence of approximate solutions converges to the solution set of
the variational problem. This general framework will be applied to the specific
context of finite element discretizations of (singularly perturbed) semilinear
elliptic boundary value problems, and a series of numerical experiments will be
presented
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
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