Refine
Year of publication
Language
- English (1103)
- German (10)
- Multiple languages (1)
Keywords
- optimal control (27)
- stability (14)
- integer programming (11)
- Stochastic programming (9)
- finite elements (9)
- mixed integer programming (9)
- Hamiltonian matrix (8)
- finite element method (8)
- model reduction (8)
- state constraints (8)
We review some known results for POD model reduction applied to ODEs. Then, these
results are generalized to several types of DAEs. We provide algorithms for the
model reduction and error bounds for the reduced order models. Some limits of
the approach are pointed out and alternative methods for reduced order subspace approximation
are presented. The POD approach is tested and evaluated for a medium sized DAE example
from multibody dynamics.
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-calles Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Lojasiewicz-Simon-type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms $f$, $g$ are real analytic. Moreover, we provide an estimate for the convergence rate.
A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity
(2012)
In this paper, we prove the existence
and global boundedness from above for a solution to an
integrodifferential model for nonisothermal multi-phase
transitions under nonhomogeneous third type boundary conditions.
The system couples a quasilinear internal energy balance
ruling the evolution of the absolute temperature with a vectorial
integro-differential inclusion governing the vectorial
phase-parameter dynamics. The specific heat and the heat
conductivity k are allowed to depend both on the order parameter
$\chi$ and on the absolute temperature $\teta$ of the system, and
the convex component of the free energy may or may not be
singular. Uniqueness and continuous data dependence are
also proved under additional assumptions.
We prove the existence, uniqueness, thermodynamic consistency,
global boundedness from both above and below, and continuous data
dependence for a strong solution to an
integrodifferential model for nonisothermal phase transitions
under nonhomogeneous mixed boundary conditions.
The specific heat is allowed to depend on the order parameter,
and the convex component of the free energy may or may not
be singular.
For a nice Markov process such as Brownian motion on a bounded domain, we introduce a non-linear potential operator defined in terms of running
suprema, and we prove a non-linear Riesz representation of a given function as
the sum of a harmonic function and a non-linear potential. The proof involves
a family of optimal stopping problems in analogy to the general construction
of Bank and El Karoui [3], but here the analysis is carried out in terms of
probabilistic potential theory.
The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster near the zero eigenvalues. For
the solution of the eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which uses a hybrid method that combines the Jacobi-Davidson, the shift-invert residual Arnoldi and nonlinear Arnoldi methods to compute the clustered eigenvalues. Numerical results illustrate that the method is robust even for the case of computing many eigenvalues in very large problems.
In this paper we propose a new finite element realization of the Perfectly Matched
Layer method (PML-method). Our approach allows to deal with a wide class of
polygonal domains and with certain types of inhomogeneous exterior domains.
Among the covered inhomogeneities are open waveguide structures playing an essential
role in integrated optics. We give a detailed insight into implementation
aspects. Numerical examples show exponential convergence behavior to the exact
solution with the thickness of the PML sponge layer.