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- eigenvalues (2)
- Becker-Doering equations (1)
- Cauchy matrix (1)
- Hamiltonian matrix (1)
- Jacobi algorithm (1)
- Kronecker product (1)
- Schur form (1)
- Spark (1)
- anti-triangular form (1)
- a~priori error analysis (1)
- coagulation and fragmentation (1)
- compressed sensing (1)
- convergence to equilibrium (1)
- discontinuous Galerkin (1)
- elastoplasticity (1)
- existence of equilibrium (1)
- fully discrete problem (1)
- gyroscopic system (1)
- invariant subspace (1)
- matrix polynomial (1)
- minimal positive solution (1)
- mutual incoherence (1)
- non-symmetric Riccati equation (1)
- nonconvex Lyapunov function (1)
- nonlinear eigenvalue problem (1)
- palindromic (1)
- palindromic QR-algorithm (1)
- passive system (1)
- purely imaginary (1)
- quadrature formula (1)
- rate independent materials (1)
- restricted isometry property (1)
- robust control (1)
- secular equation (1)
- skew-Hamiltonian matrix (1)
- sparse solution of linear systems (1)
- structured deflation method (1)
- structured perturbation (1)
- symplectic matrix (1)
- transport theory (1)
Bei der Produktions- und Handelsplanung treffen Energieversorgungsunternehmen eine Reihe von Entscheidungen unter unsicheren Randbedingungen. Ein Optimierungsmodell für einen mittelfristigen Planungshorizont muss diese Unsicherheiten berücksichtigen, etwa durch Einbeziehung von statistischen Modellen für die zufallsbehafteten Eingangsdaten. Dadurch ist es prinzipiell möglich, Risikobetrachtungen direkt in die Optimierung zu integrieren. Wir demonstrieren in dieser Arbeit die Möglichkeit, spezielle dynamische Risikomaße, so genannte polyedrische Risikomaße, in die Zielfunktion der Optimierung mit aufzunehmen. Im Gegensatz zu vielen anderen Ansätzen wird dadurch die Komplexität des Problems nicht wesentlich erhöht. Das vorgestellte Modell stellt ein Werkzeug zur Entscheidungsunterstützung für kleinere Marktteilnehmer hinsichtlich der Beschaffungsplanung dar. Dabei werden insbesondere konkrete mittelfristig bindende Bezugsverträge mit der Möglichkeit verglichen, die Versorgung in erster Linie auf der Basis von Spot- und Futuremarkt zu planen.
The discontinuous Galerkin (dG) method provides a hierarchy of time discretization schemes for evolutionary problems. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in [Alberty, Carstensen: Discontinuous {G}alerkin Time Discretization in Elastoplasticity: Motivation, Numerical Algorithms and Applications, Comp. Meth. Appl. Mech. Engrg. \textbf{191} (2002)] with the help of duality in convex analysis to justify certain jump terms. Convincing numerical experiments have already been displayed in the literature.\This paper establishes a mathematical a~priori error analysis for the dG($1$) scheme with discontinuous piecewise linear polynomials in the temporal and first-order finite elements for the spacial discretization. One novel key idea in the a~priori convergence analysis is an optimal trace estimate under convex constraints. The numerical investigation of the empirical convergence rate in a benchmark concludes the paper.
Three properties of matrices: the spark, the mutual incoherence and the restricted isometry property have recently been introduced in the context of compressed sensing. We study these properties for matrices that are Kronecker products and show how these properties relate to those of the factors. For the mutual incoherence we also
discuss results for sums of Kronecker products.
Stewart's recently introduced Krylov-Schur algorithm
is a modification of the implicitly restarted Arnoldi algorithm which
employs reordered Schur decompositions to perform restarts and de-
ations in a numerically reliable manner. This paper describes a variant
of the Krylov-Schur algorithm suitable for addressing eigenvalue
problems associated with products of large and sparse matrices. It
performs restarts and de
ations via reordered periodic Schur decompositions
and, by taking the product structure into account, it is
capable to achieve qualitatively better approximations to the eigenvalues
of small magnitude.
This article describes Fortran 77 subroutines for computing eigenvalues and invariant subspaces
of Hamiltonian and skew-Hamiltonian matrices. The implemented algorithms are based on orthogonal
symplectic decompositions, implying numerical backward stability as well as symmetry
preservation for the computed eigenvalues. These algorithms are supplemented with balancing and
block algorithms, which can lead to considerable accuracy and performance improvements. As a
by-product, an efficient implementation for computing symplectic QR decompositions is provided.
We demonstrate the usefulness of the subroutines for several, practically relevant examples.
Recently, Dreyer and Duderstadt have proposed a modification of the
Becker-Doering cluster equations which now have a nonconvex
Lyapunov function. We start with existence and uniqueness results
for the modified equations. Next we derive an explicit criterion for
the existence of equilibrium states and solve the minimization
problem for the Lyapunov function. Finally, we discuss the long time
behavior in the case that equilibrium solutions do exist.
Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form
(2007)
We present structure-preserving numerical methods
for complex palindromic polynomial eigenvalue problems
via corresponding palindromic linearizations.
A key ingredient is the development of an appropriate condensed form ---
the anti-triangular Schur form.
Ill-conditioned problems which have eigenvalues near the unit circle,
in particular near +/-1, are discussed.
We show how a combination of unstructured methods
followed by a structured refinement can be used
to solve such problems very accurately.
A framework for the reduction of scenario trees as inputs of (linear) multistage stochastic programs is provided such that optimal values and approximate solution sets remain close to each other. The argument is based on upper bounds of the Lr-distance and the filtration distance, and on quantitative stability results for multistage stochastic programs. The important difference from scenario reduction in two-stage models consists in incorporating the filtration distance. An algorithm is presented for selecting and removing nodes of a scenario tree such that a prescribed error tolerance is met. Some numerical experience is reported.
Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations
(2007)
We discuss the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations. We
show that there is a substantial difference in the behavior under these perturbations. We also discuss the perturbation of real eigenvalues of real
skew-Hamiltonian matrices under structured perturbations and use these results to analyze the properties of the URV method of computing the
eigenvalues of Hamiltonian matrices.
We derive formulas for the minimal positive solution of a
particular non-symmetric Riccati
equation arising in transport theory. The formulas are based
on the eigenvalues of an
associated matrix. We use the formulas to explore some new
properties of the minimal positive solution and to derive
fast and highly accurate numerical methods. Some numerical tests
demonstrate the properties of the new methods.