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A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static
elasto-plasticity with kinematic or isotropic hardening is considered. The associated path is induced
by a combined Moreau-Yosida / Tichonov regularization of the dual problem.
The sequence of solutions to the regularized problems is shown to converge strongly to the optimal
displacement-stress-strain triple of the original elasto-plastic contact problem in the space-continuous setting.
This property relies on the density of the intersection of certain convex sets which is shown as well.
It is also argued that the mappings associated with the resulting problems are Newton- or slantly differentiable.
Consequently, each regularized subsystem can be solved mesh-independently at a local
superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and
a numerical validation of the theoretical results is given.
The inverse eigenvalue problem for $T$-alternating matrix polynomials over arbitrary
algebraically closed fields of characteristic different from two is considered.
The main result shows that the necessary conditions obtained in \cite{MacMMM10} for a matrix
polynomial to be the Smith form of a $T$-alternating matrix polynomial are under mild
conditions also sufficient to be the Smith form of a $T$-alternating matrix polynomial with
invertible leading coefficient which is additionally in anti-triangular form.. In particular, this result
implies that any $T$-alternating matrix polynomial with invertible leading coefficient is
equivalent to a $T$-alternating matrix polynomial in anti-triangular form that has the
same finite and infinite elementary divisors as the original matrix polynomial.
Finally, the inverse eigenvalue problem for $T$-palindromic matrix polynomials is
considered excluding the case that both $+1$ and $-1$ are eigenvalues.
In this paper, we introduce a high order finite element (FEM) implementation using perfectly matched layer (PML) for the scattering by plasmonic structures inside layered media. The PML is proven to be very accurate and efficient by a comparative analysis with a commercial FEM software and the Multiple Multipole Program (MMP). A convergence analysis using hp-adaptive refinement inside the PML layer
shows that adaptive mesh refinement inside the PML layer is most efficient. Based on this convergence analysis an hp-strategy is proposed, which shows a remarkable error reduction for small additional computational costs.
Shielding sheets are commonly used in the protection of electronic devices. With their large aspect ratios they become a serious issue for the direct application of boundary element methods due to the occurring almost singular integrals. Impedance transmission conditions allow for boundary element formulations where only sheet mid-line has to be triangulated. We propose and analyse boundary element methods of second kind for impedance transmission conditions of two different types for the time-harmonic eddy current problem in two dimensions. In numerical experiments we study the effect on the magnetic shielding efficiency of the discretisation error due the boundary element discretisation and the modelling error due to the
impedance transmission condition.
We study the exact recovery of signals from quantized frame coefficients. Here, the basis of the quantization
is hard thresholding, and we present a simple algorithm for the recovery of reconstructable signals. The set of
non-reconstructable signals is shown to be star-shaped and symmetric with respect to the origin. Moreover, we
provide a criterion on the frame for the boundedness of this set. In this case, we also give a priori bounds.
According to the Helmholtz decomposition,
the irrotational parts of the momentum balance equations of
the incompressible Navier-Stokes equations are balanced by the
pressure gradient.
Unfortunately, nearly all mixed methods for incompressible flows
violate this fundamental property, resulting in the well-known
numerical instability of poor mass conservation.
The origin of this problem is the
lack of $L^2$-orthogonality between discretely divergence-free velocities
and irrotational vector fields.
In order to cure this,
a new variational crime using divergence-free
velocity reconstructions is proposed.
Applying lowest order Raviart-Thomas velocity reconstructions
to the nonconforming Crouzeix-Raviart element
allows to construct
a cheap flow discretization for general 2d and 3d
simplex meshes that possesses the same
advantageous robustness properties like divergence-free flow
solvers.
In the Stokes case,
optimal a-priori error estimates for the velocity gradients
and the pressure are derived. Moreover, the discrete velocity
is independent of the continuous pressure.
Several detailed linear and nonlinear
numerical examples illustrate the theoretical findings.
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
Primal heuristics are an important component of state-of-the-art codes for mixed integer nonlinear programming (MINLP). In this article we give a compact overview of primal heuristics for MINLP that have been suggested in the literature of recent years. We sketch the fundamental concepts of different classes of heuristics and discuss specific implementations. A brief computational experiment shows that primal heuristics play a key role in achieving feasibility and finding good primal bounds within a global MINLP solver.
Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series
(2013)
The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models,
is solvable only for certain values of the matrices $A,B$.
We present a numerical method to modify $A,B$ in order to make the matrix equation solvable.
Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$,
our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory.
The method is heuristic but works in practice, as is supported by several compelling numerical examples.
These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).
We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.