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On-line visualization in parallel computationsPester, M. 30 October 1998 (has links)
The investigation of new parallel algorithms for MIMD computers
requires some postprocessing facilities for quickly evaluating
the behavior of those algorithms We present two kinds of
visualization tool implementations for 2D and 3D finite element
applications to be used on a parallel computer and a host
workstation.
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Local theory of a collocation method for Cauchy singular integral equations on an intervalJunghanns, P., Weber, U. 30 October 1998 (has links)
We consider a collocation method for Cauchy singular integral equations on the interval
based on weighted Chebyshev polynomials , where the coefficients of the operator are
piecewise continuous. Stability conditions are derived using Banach algebra methods,
and numerical results are given.
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On a SQP-multigrid technique for nonlinear parabolic boundary control problemsGoldberg, H., Tröltzsch, F. 30 October 1998 (has links)
An optimal control problem governed by the heat equation with nonlinear boundary
conditions is considered. The objective functional consists of a quadratic terminal
part and a quadratic regularization term. It is known, that an SQP method converges
quadratically to the optimal solution of the problem. To handle the quadratic optimal
control subproblems with high precision, very large scale mathematical programming
problems have to be treated. The constrained problem is reduced to an unconstrained
one by a method due to Bertsekas. A multigrid approach developed by Hackbusch is
applied to solve the unconstrained problems. Some numerical examples illustrate the
behaviour of the method.
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Estimates for the condition numbers of large semi-definite Toeplitz matricesBöttcher, A., Grudsky, S. M. 30 October 1998 (has links)
This paper is devoted to asymptotic estimates for the condition numbers
$\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$
of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where
$\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes
of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ ,
or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues,
and the finite section method are discussed. We also consider Wiener-Hopf integral
operators and multidimensional Toeplitz operators.
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Asymptotic Expansions for Second-Order Moments of Integral Functionals of Weakly Correlated Random FunctionsScheidt, Jrgen vom, Starkloff, Hans-Jrg, Wunderlich, Ralf 30 October 1998 (has links)
In the paper asymptotic expansions for
second-order moments of integral functionals
of a class of random functions are considered.
The random functions are assumed to be
$\epsilon$-correlated, i.e. the values are not
correlated excluding a $\epsilon$-neighbourhood
of each point. The asymptotic expansions are
derived for $\epsilon \to 0$. With the help of
a special weak assumption there are found
easier expansions as in the case of general
weakly correlated functions.
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Local theory of projection methods for Cauchy singular integral equations on an intervalJunghanns, P., U.Weber 30 October 1998 (has links)
We consider a finite section (Galerkin) and a collocation method for Cauchy singular
integral equations on the interval based on weighted Chebyshev polymoninals, where
the coefficients of the operator are piecewise continuous.
Stability conditions are derived using Banach algebra techniques, where
also the system case is mentioned. With the help of
appropriate Sobolev spaces a result on convergence rates is proved.
Computational aspects are discussed in order to develop
an effective algorithm. Numerical results, also
for a class of nonlinear singular integral equations,
are presented.
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Error Estimation for Anisotropic Tetrahedral and Triangular Finite Element MeshesKunert, G. 30 October 1998 (has links)
Some boundary value problems yield anisotropic solutions, e.g. solutions
with boundary layers. If such problems are to be solved with the finite
element method (FEM), anisotropically refined meshes can be
advantageous.
In order to construct these meshes or to control the error
one aims at reliable error estimators.
For \emph{isotropic} meshes many estimators are known, but they either fail
when used on \emph{anisotropic} meshes, or they were not applied yet.
For rectangular (or cuboidal) anisotropic meshes a modified
error estimator had already been found.
We are investigating error estimators on anisotropic tetrahedral or
triangular meshes because such grids offer greater geometrical flexibility.
For the Poisson equation a residual error estimator, a local Dirichlet problem
error estimator, and an $L_2$ error estimator are derived, respectively.
Additionally a residual error estimator is presented for a singularly
perturbed reaction diffusion equation.
It is important that the anisotropic mesh corresponds to the anisotropic
solution. Provided that a certain condition is satisfied, we have proven
that all estimators bound the error reliably.
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Solving Large-Scale Generalized Algebraic Bernoulli Equations via the Matrix Sign FunctionBarrachina, Sergio, Benner, Peter, Quintana-Ortí, Enrique S. 11 September 2006 (has links)
We investigate the solution of large-scale generalized algebraic Bernoulli equations as those arising in control and systems theory in the context of stabilization of linear dynamical systems, coprime factorization of rational matrix-valued functions, and model reduction. The algorithms we propose, based on a generalization of the Newton iteration for the matrix sign function, are easy to parallelize, yielding an efficient numerical tool to solve large-scale problems. Both the accuracy and the parallel performance of our implementations on a cluster of Intel Xeon processors are reported.
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Anisotropic mesh construction and error estimation in the finite element methodKunert, Gerd 13 January 2000 (has links)
In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh.
However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators.
Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error.
The focus of this paper is on error estimation on anisotropic meshes.
It is known that such error estimation is reliable and efficient only
if the anisotropic mesh is aligned with the anisotropic solution.
The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution.
The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form.
Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.
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Anisotropic mesh construction and error estimation in the finite element methodKunert, Gerd 27 July 2000 (has links)
In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error.
The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution.
The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.
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