Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems
Abstract
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.
Subject
VariationalCritical Point
Multiple Saddles
G-differntiable
G-Saddles
Peak Selection
L-orthogonal Selection
Locally Lipschitz Continuity
Local Minmax Method
Local Min-orthogonal Method
Local Minmaxmin Method
Stepsize Rule
Characterization
Convergence
Numerical Examples
PS Condition
Compact
Strong-weak Topology
Gradient
Citation
Li, Meiqin (2017). Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /173210.