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- primaldual (1)
- prime implicand (1)
- probabilistic schemes (1)
- probability metrics (1)
- progressive mesh coding (1)
- projected Lyapunov equations (1)
- projector chain (1)
- protein folding (1)
- protein-ligand binding (1)
- pseudospectra (1)
- public transit (1)
- public transport optimization (1)
- purely imaginary (1)
- purely imaginary eigenvalue (1)
- purely imaginary eigenvalues (1)
- pyrometer control (1)
- quadratic cost functional (1)
- quadratic variation (1)
- quadrature formula (1)
- quai-Newton (1)
- quantum dots (1)
- quasi-Newton (1)
- quasi-Newton methods (1)
- quasi-static problems (1)
- quasilinear equation (1)
- quasilinear integrodifferential vectorial equation (1)
- quasilinear parabolic systems (1)
- radiative heat transfer (1)
- railway capacity (1)
- railway models (1)
- random demand (1)
- random discrete structures (1)
- random graphs (1)
- random graphs and hypergraphs (1)
- random matrix (1)
- random recourse (1)
- rank $k$ (1)
- rare transitions (1)
- rarefaction fans (1)
- rate independent materials (1)
- rate matrix (1)
- rate-dependent damage (1)
- rate-independent crack propagation (1)
- rate-independent systems (1)
- rational matrix (1)
- reachability (1)
- real distance to instability (1)
- real structured distance to instability (1)
- real time optimation (1)
- reciprocal (1)
- recourse (1)
- reflection argument (1)
- reflexive Banach spaces (1)
- regression (1)
- regularisation (1)
- regularity estimates (1)
- relaxation oscillations (1)
- required supply (1)
- resonance problems (1)
- resonances (1)
- resource-constrained project Schedulin (1)
- resource-constrained project scheduling (1)
- restricted distance to instability (1)
- restricted real distance to instability (1)
- resuspension (1)
- reverse engineering (1)
- reversible mass action type reactions (1)
- reversible reactions (1)
- robust hedging (1)
- robust optimization (1)
- robust stability (1)
- rogue waves (1)
- rolling-stock-planning (1)
- rough potential energy function (1)
- routing (1)
- saddle-point method (1)
- saddlepoint method (1)
- saturation (1)
- scattering theory (1)
- scenario (1)
- scheduling (1)
- scip (1)
- second order system (1)
- second-order systems (1)
- secular equation (1)
- self-adjoint dilation (1)
- self-adjoint eigenvalue problem (1)
- self-assembly of quantum dots (1)
- self-contact (1)
- self-heating (1)
- self-similar solutions (1)
- selfadjoint matrices (1)
- semi-infinite optimization (1)
- semi-insulating GaAs (1)
- semi-smooth Newton method (1)
- semi-smooth Newton methods (1)
- semiconductor design (1)
- semiconductor laser (1)
- semiconductor ring laser (1)
- semilinear elliptic equation (1)
- semilinear parabolic PDEs (1)
- semismooth Newton (1)
- sensitivities (1)
- sensitivity (1)
- set covering (1)
- set-union knapsack (1)
- sets with Lipschitz boundary (1)
- shape analysis (1)
- shape operator (1)
- shape signatures (1)
- shifted Laplacian preconditioner (1)
- shortest paths (1)
- sign symmetric (1)
- simple invariant subspace (1)
- simple recourse (1)
- simulation (1)
- single-molecule spectroscopy (1)
- singular matrix polynomial (1)
- singular pencil (1)
- singular pencils (1)
- singular perturbation balanced truncation (1)
- singular values (1)
- skew QR factorization (1)
- skew QRQ$^T$ factorization (1)
- skew Takagi factorization (1)
- skew-Hermitian matrix polynomial (1)
- skew-symmetric (1)
- skew-symmetric/symmetric pencil (1)
- sliding mode simulation (1)
- slow time scale (1)
- slow-fast systems (1)
- small denominators (1)
- small signal response (1)
- small strain (1)
- small-slope approximation (1)
- smooth $QR$ factorization (1)
- smooth QR factorization (1)
- smooth operators (1)
- smooth singular value decomposition (1)
- software (1)
- solitons (1)
- solvability (1)
- solver (1)
- sparse grids (1)
- sparse optimal control (1)
- sparse representations (1)
- sparse solution (1)
- sparse solution of linear systems (1)
- spectral approximation (1)
- spectral asymptotics (1)
- spectral clustering (1)
- spectral deferred correction (1)
- spectral factorization (1)
- spectral shift (1)
- spin-stabilized discretization (1)
- split graphs (1)
- split-step method (1)
- spurious solutions (1)
- stability analysis (1)
- staircase form (1)
- state constrained optimization (1)
- stationary distribution (1)
- stationary lasing modes (1)
- stationary point (1)
- stationary solutions (1)
- stationary thermistor model (1)
- statistical error (1)
- stellar subdivisions (1)
- stereographic projection (1)
- stochastic (1)
- stochastic control (1)
- stochastic dominance (1)
- stochastic optimization (1)
- stochastic ordering (1)
- stochastic semi-infinite programming (1)
- stopping time (1)
- storage function (1)
- strained multi quantum-wells (1)
- strangeness-free system (1)
- strong spectrum (1)
- structrured distance to instability (1)
- structure preserving method (1)
- structure-preserving doubling algorithm (1)
- structured (1)
- structured Kronecker canonical form (1)
- structured SVD (1)
- structured backward error (1)
- structured condition number (1)
- structured deflation method (1)
- structured doubling algorithm (1)
- structured eigenvalue backward error (1)
- structured mapping problem (1)
- structured matrices (1)
- structured model reduction (1)
- structured perturbation (1)
- structured perturbations (1)
- structured staircase form, structured Kronecker canonical form, symmetric pencil, Hermitian pencil, alternating pencil, palindromic pencil, linear quadratic control, $H_\infty$ control (1)
- stylized facts (1)
- sub-Riemannian geometry with drift (1)
- sub-quadratic growth (1)
- subdivision surfaces (1)
- subgrid (1)
- subspace recycling (1)
- sufficient optimality condition (1)
- sum of weighted completion times (1)
- super-cell method (1)
- surface modes (1)
- survivability (1)
- switched positive systems (1)
- switched system (1)
- switching (1)
- symmetric pencils (1)
- symmetry detection (1)
- symplectic pencil (1)
- systems theory (1)
- telecommunication networks (1)
- temperature reweighting (1)
- temperature-dependent dynamics (1)
- test equation (1)
- texture mapping (1)
- thermionic emission (1)
- thermistor model (1)
- thermistor system (1)
- thermodynamic constraints (1)
- thermomechanics (1)
- thickness of a domain (1)
- thin flow (1)
- time (1)
- time and space discretization (1)
- time integration (1)
- time-consistency (1)
- time-dependent Dirichlet conditions (1)
- time-discretization (1)
- time-optimal control (1)
- timed automata (1)
- timestep control (1)
- total weighted completion time (1)
- trace formula (1)
- traces (1)
- track allocation framework (1)
- tractability index (1)
- transfer operator (1)
- transition probabilities (1)
- transition state theory (1)
- transmission at material interfaces (1)
- transmission problems (1)
- transparent boundary (1)
- transparent boundary conditions (1)
- transport theory (1)
- traveling magnetic fields (1)
- traveling wave equations (1)
- traveling wave model (1)
- triangular Toeplitz systems (1)
- triangularization (1)
- trimmed linearization (1)
- trip timetable (1)
- truncation (1)
- trust region (1)
- two way canonical form (1)
- two-dimensional Schrödinger equation (1)
- two-stage linear-quadratic problems (1)
- two-stage models (1)
- two-stage stochastic program (1)
- unbounded domain (1)
- undercompressive waves (1)
- unidirectional (1)
- unidirectional coupling (1)
- uniform Caccioppoli-like inequality (1)
- unit sphere (1)
- unitary congruence (1)
- updating factorization (1)
- upscaling (1)
- utility based pricing (1)
- utility indifference hedging and pricing (1)
- utility maximization (1)
- value function (1)
- vanishing viscosity (1)
- vanishing viscosity approach (1)
- variable fixing (1)
- variational analysis and optimization (1)
- variational evolution (1)
- variational formulations (1)
- vehicle dispatching (1)
- vehicle rotation planning (1)
- vehicle scheduling (1)
- velocity method (1)
- von Mises model (1)
- water reservoir management (1)
- weak convergence (1)
- weather derivative (1)
- weather risk (1)
- weighted regularity (1)
- well-posedness (1)
- zero curvature (1)
Zonotopes With Large 2D Cuts
(2009)
We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
The timetable is the essence of the service offered by any provider
of public transport'' (Jonathan Tyler, CASPT 2006). Indeed, the
timetable has a major impact on both operating costs and on passenger
comfort. Most European agglomerations and railways use periodic timetables in which operation repeats in regular intervals. In contrast, many North and South American municipalities use trip timetables in which the vehicle trips are scheduled individually subject to frequency constraints. We compare these two
strategies with respect to vehicle operation costs. It turns out that
for short time horizons, periodic timetabling can be suboptimal; for
sufficiently long time horizons, however, periodic timetabling can
always be done in an optimal way'.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
We prove the existence, uniqueness, regularity and smooth dependence
of the weak solution on the initial data for a certain class of semilinear first order
dissipative hyperbolic systems with spacially discontinuous coefficients. Such
kind of hyperbolic problems have succesfully been used to describe the dynamics
of distributed feedback multisection semiconductor lasers in recent years. We
show that in a suitable function space of continuous functions the weak solutions
generate a smooth semiflow.
Den kürzesten Weg in einem Graphen zu finden ist ein klassisches Problem der Graphentheorie. Über einen Vortrag zu diesem Thema beim Tag der Mathematik 2007 von R. Borndörfer kam ich in Kontakt mit dem Konrad-Zuse-Zentrum (ZIB), das sich u.a. mit Wegeoptimierung beschäftigt. Ein Forschungsschwerpunkt dort ist im Rahmen eines Projekts zur Chipverifikation das Zählen von Lösungen, das, wie wir sehen werden, eng mit dem Zählen von Wegen zusammenhängt.
Anhand von zwei Fragen aus der Graphentheorie soll diese Facharbeit unterschiedliche Lösungsmethoden untersuchen. Wie bestimmt man den kürzesten Weg zwischen zwei Knoten in einem Graphen und wie findet man alle möglichen Wege?
Nach einer Einführung in die Graphentheorie und einer Konkretisierung der Probleme wird zunächst für beide eine Lösung mit auf Graphen basierenden Algorithmen vorgestellt. Während der Algorithmus von Dijkstra sehr bekannt ist, habe ich für das Zählen von Wegen einen eigenen Algorithmus auf der Basis der Tiefensuche entwickelt.
Im zweiten Teil der Arbeit wird das Konzept der ganzzahligen Programmierung vorgestellt und die Lösungsmöglichkeiten für Wegeprobleme, die sich darüber ergeben.
In den letzten Jahren ist die Bedeutung computerunterstützter Darstellungen von Mathematik
im Lern- als auch im Forschungsbereich stark gestiegen. Obwohl bereits massive Anstrengungen
unternommen werden, die Mathematikausbildung im Ingenieursbereich durch Neue
Medien zu unterstützen und auszubauen[Mum], gibt es dennoch nur eine geringe Zahl von
Projekten, die auf die Studenten der Mathematik und Physik abzielen. Ferner beschränkten
sich viele Projekte nur auf die Verwaltung von Dokumenten, die den Lernenden — obgleich
teilweise aufgelockert durch eingestreute aktive Inhalte—zu einem passiven Konsumenten des
Lerninhaltes machen und die darum selten geeignet sind, die eigentständige, selbstgesteuerte
Auseinandersetzung mit der Materie zu fördern.
Wir präsentieren in diesem Artikel das Konzept des „Virtuellen Labors“, welches die Metapher
eines Laborpraktikums innerhalb eines Computernetzwerkes nachbildet und damit das zweite
Standbein der universitären Ausbildung in die Neuen Medien abbildet. Das Labor „Cinderella“
[KRG04] zur Untersuchung euklidischer und nicht-euklidischer Geometrie mag hier als ein
Beispiel dienen.
Wir werden im ersten Teil ein didaktisches Anforderungsprofil an derartige Labore definieren,
dann in einem zweiten Teil die sich daraus untersuchenden Konsequenzen für die Software-
Architektur darstellen und im dritten Teil ein am DFG-ForschungzentrumMATHEON der Berliner
Universitäten entwickeltes Labor für Elemente der statistischen Mechanik präsentieren.
This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.
The classical approach to investigating polynomial eigenvalue problems is linearization, where the
polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely
many linearizations with widely varying properties, but in practice the companion forms are typically used. However,
these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes
be required.
In this paper we develop a systematic approach to generating large classes of linearizations for matrix polynomials.
Given a polynomial P, we show how to simply construct two vector spaces of pencils that generalize the companion
forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors of these pencils are
shown to be closely related to those of P. A distinguished subspace is then isolated, and the special properties of
these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured
linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations, issues to be
addressed in further work.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod
to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have
to connect a one-dimensional model with director variables to a three-dimensional
model without directors.
Two alternative coupling conditions are proposed,
which correspond to two different configuration trace spaces.
For both we show existence of solutions of the coupled problems, using the direct
method of the calculus of variations. From the first-order optimality conditions
we also derive the corresponding conditions for the dual variables. These are
then interpreted in mechanical terms.
We consider the problem of utility maximization for small traders on incomplete
financial markets. As opposed to most of the papers dealing with this
subject, the investors’ trading strategies we allow underly constraints described
by closed, but not necessarily convex, sets. The final wealths obtained by trading
under these constraints are identified as stochastic processes which usually are
supermartingales, and even martingales for particular strategies. These strategies
are seen to be optimal, and the corresponding value functions determined
simply by the initial values of the supermartingales. We separately treat the
cases of exponential, power and logarithmic utility.
Under market frictions like illiquidity or transaction costs, contingent claims
can incorporate some inevitable intrinsic risk that cannot be completely hedged
away but remains with the holder. In general, they cannot be synthesized by
dynamical trading in liquid assets and hence not be priced by no-arbitrage arguments alone. Still, an agent can determine a valuation with respect to her
preferences towards risk. The utility indifference value for a variation in the
quantity of illiquid assets held by the agent is defined as the compensating variation
of wealth, under which her maximal expected utility remains unchanged.
In this work numerical methods for the solution of two classes of structured generalized eigenvalue problems, $Ax=\lambda Bx$, are developed. Those classes are the palindromic ($B=A^T$) and the even ($A=A^T$, $B=-B^T$) eigenvalue problems.
The spectrum of these problems is not arbitrary, rather do eigenvalues occur in pairs.
We will construct methods for palindromic and even eigenvalue problems that are of cubic complexity and that are guaranteed to produce eigenvalues that are paired to working precision.
At the heart of both methods is a new URV-type matrix decomposition, that simultaneously transforms three matrices to skew triangular form, i.e., to a form that is triangular with respect to the Northeast-Southwest diagonal.
The algorithm to compute this URV decomposition uses several other methods to reduce a single square matrix to skew triangular form: the skew QR factorization and the skew QRQ$^T$ decomposition. Moreover, a method to compute the singular value decomposition of a complex, skew symmetric matrix is presented and used.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem.
We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.
We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.
We shortly review the uncoupling-coupling method, a Markov chain
Monte Carlo based approach to compute statistical properties of systems like
medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the
decomposition of reversible nearly uncoupled Markov chains into rapidly mixing
subchains. We show how the underlying scheme of uncoupling-coupling can also be
applied to stochastic differential equations where it can be translated into a domain
decomposition technique for partial differential equations.
This paper deals with MIP-based primal heuristics to be used within a branch-and-cut approach for solving multi-layer telecommunication network design problems. Based on a mixed-integer programming formulation for two network layers, we present three heuristics for solving important subproblems, two of which solve a sub-MIP. On multi-layer planning instances with many parallel logical links, we show the effectiveness of our heuristics in finding good solutions early in the branch-and-cut search tree.
For a system of globally pulse-coupled phase-oscillators,
we derive conditions for stability of the completely synchronous
state and all possible two-cluster states and explain
how the different states are naturally connected via
bifurcations. The coupling is modeled using the phaseresponse-
curve (PRC), which measures the sensitivity of
each oscillator’s phase to perturbations. For large systems
with a PRC, which turns to zero at the spiking threshold,
we are able to find the parameter regions where multiple
stable two-cluster states coexist and illustrate this by an
example. In addition, we explain how a locally unstable
one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon
of intermittent, asymptotic synchronization with abating
beats away from the perfect synchrony.
We prove that the Random-Edge simplex algorithm requires an
expected number of at most 13n/pd pivot steps on any simple d-polytope with
n vertices. This is the first nontrivial upper bound for general polytopes. We
also describe a refined analysis that potentially yields much better bounds for
specific classes of polytopes. As one application, we show that for combinatorial
d-cubes, the trivial upper bound of 2d on the performance of Random-Edge
can asymptotically be improved by any desired polynomial factor in d.
We present a new inexact nonsmooth Newton method for the solution
of convex minimization problems with piecewise smooth, pointwise
nonlinearities. The algorithm consists of a nonlinear smoothing
step on the fine level and a linear coarse correction.
Suitable postprocessing guarantees global convergence
even in the case of a single multigrid step for each linear subproblem.
Numerical examples show that the overall efficiency
is comparable to multigrid for similar linear problems.
We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.
We use the traveling wave model for simulating and analyzing
nonlinear dynamics of complex semiconductor ring laser devices.
This modeling allows to consider temporal-spatial distributions
of the coun\-ter-pro\-pa\-ga\-ting slowly varying optical fields
and the carriers, what can be important when studying
non-homogeneous ring cavities, propagation of short pulses or fast switching.
By performing numerical integration of the model
equations we observe several dynamic regimes as well as transitions
between them. The computation of ring cavity modes explains some
peculiarities of these regimes.
A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions.
Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some
problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are
sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schroedinger and the drift-diffusion equation
but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.
In this article, we use numerical simulation to investigate transient temperature
phenomena during sublimation growth of SiC single crytals via physical
vapor transport (also called the modified Lely method). We consider the evolution
of temperatures at the SiC source and at the SiC seed crystal, which
are highly relevant to the quality of the grown crystals, but inaccessible to
direct measurements. The simulations are based on a transient mathematical
model for the heat transport, including heat conduction, radiation, and radio
frequency (RF) induction heating. Varying the position of the induction coil
as well as the heating power, it is shown that the measurable temperature difference
between the bottom and the top of the growth apparatus can usually
not be used as a simple indicator for the respective temperature difference
between SiC source and seed. Moreover, it is shown that there can be a time
lack of 1.5 hours between the heating of the temperature measuring points
and the heating of the interior of the SiC source.
Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting group-velocity horizon can be exploited for adiabatic compression of the soliton down into the few-cycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.
Building on a transformation formula that we previously found in connection with Landau-Lifshitz-Gilbert equations, we present a strategy for utilizing compensated compactness methods in the context of Ginzburg-Landau approximation for harmonic maps and related problems in magnetism and superconductivity. Its applicability is illustrated by a number of new examples.
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum- well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.
We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by one-dimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatio--temporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy--momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet--Neumann method. The large algebraic systems of the bone--cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method.
Several qualitative properties of equilibria in electrical circuits are analyzed in this paper. Specifically, non-singularity, hyperbolicity, and asymptotic stability are addressed in terms of the circuit topology, which is captured through the use of Modified Nodal Analysis (MNA) models. The differential-algebraic or semistate nature of these models drives the analysis of the spectrum to a matrix pencil setting, and puts the results beyond the ones already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The analysis combines results coming from graph theory, matrix analysis, matrix pencil theory, and Lyapunov theory for DAEs. The study is restricted to problems with independent sources; qualitative properties of circuits including controlled sources are the focus of future research.
Time-lag in Derivative Convergence Time-lag in Derivative Convergence for Fixed Point Iterations
(2004)
In an earlier study it was proven and experimentally confirmed on a 2D Euler code
that fixed point iterations can be differentiated to yield first and second order derivatives of
implicit functions that are defined by state equations. It was also asserted that the resulting
approximations for reduced gradients and Hessians converge with the same R-factor as the
underlying fixed point iteration.
A closer look reveals now that nevertheless these derivative values lag behind the functions
values in that the ratios of the corresponding errors grow proportional to the iteration counter
or its square towards infinity. This rather subtle effect is caused mathematically by the
occurrence of nontrivial Jordan blocks associated with degenerate eigenvalues. We elaborate
the theory and report its confirmation through numerical experiments.
We consider anisotropic Allen--Cahn equations with interfacial energy
induced by an anisotropic surface energy density $\gamma$.
Assuming that $\gamma$
is positive, positively homogeneous of degree one,
strictly convex in tangential directions to the unit sphere,
and sufficiently smooth, we show stability of
various time discretizations. In particular,
we consider a fully implicit and a linearized time discretization
of the interfacial energy combined with implicit
and semi-implicit time discretizations
of the double-well potential. In the semi-implicit variant,
concave terms are taken explicitly.
The arising discrete spatial problems are solved by
globally convergent truncated nonsmooth Newton multigrid methods.
Numerical experiments show the accuracy of the different
discretizations.
We also illustrate that pinch-off under anisotropic
mean curvature flow is no longer frame invariant,
but depends on the orientation of the initial configuration.
In this paper we lay the foundation for a numerical algorithm to
simulate high-dimensional coupled FBSDEs under weak coupling or
monotonicity conditions. In particular we prove convergence of a
time discretization and a Markovian iteration. The iteration
differs from standard Picard iterations for FBSDEs in that the
dimension of the underlying Markovian process does not increase
with the number of iterations. This feature seems to be
indispensable for an efficient iterative scheme from a numerical
point of view. We finally suggest a fully explicit numerical
algorithm and present some numerical examples with up to
10-dimensional state space.
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of $p(x)$-Laplacian-type with discontinuous exponent $p$, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori $L^1$ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two diffculties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
This paper analyzes a model for phase
transformation in shape-memory alloys
induced by temperature changes and by
mechanical loading. We assume that the temperature is prescribed and
formulate the problem within the framework
of the energetic theory of
rate-independent processes. Existence and uniqueness results are proved.
Tree spanner problems have important applications in network design, e.g. in the telecommunications industry. Mathematically, there have been considered quite a number of maxstretch tree spanner problems and of average stretch tree spanner problems. We propose a unified notation for 20 tree spanner problems, which we investigate for graphs with general positive weights, with metric weights, and with unit weights. This covers several prominent problems of combinatorial optimization. Having this notation at hand, we can clearly identify which problems coincide. In the case of unweighted graphs, the formally 20 problems collapse to only five different problems. Moreover, our systematic notation for tree spanner problems enables us to identify a tree spanner problem whose complexity status has not been solved so far. We are able to provide an NP-hardness proof. Furthermore, due to our new notation of tree spanner problems, we are able to detect that an inapproximability result that is due to Galbiati (2001, 2003) in fact applies to the classical max-stretch tree spanner problem. We conclude that the inapproximability factor for this problem thus is 2-ε, instead of only (1+sqrt(5))/2 ~ 1.618 according to Peleg and Reshef (1999).
The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6
(2010)
Let $B$ be a fractional Brownian motion with Hurst parameter
$H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums
for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability.
We show, however, that they do converge in law in the Skorohod space of
c\`adl\`ag functions. Moreover, we show that the resulting stochastic
integral satisfies a change of variable formula with a correction term
that is an ordinary It\^o integral with respect to a Brownian motion
that is independent of $B$.
The Steiner connectivity problem is a generalization of
the Steiner tree problem. It consists in finding a minimum cost set of
simple paths to connect a subset of nodes in an undirected graph.
We show that important polyhedral and algorithmic results on the
Steiner tree problem carry over to the Steiner connectivity problem,
namely, the Steiner cut and the Steiner partition inequalities, as
well as the associated polynomial time separation algorithms, can be
generalized. Similar to the Steiner tree case, a certain directed
formulation, which is stronger than the natural undirected one,
plays a central role.
We show that the spectrum of linear delay differential equations with
large delay splits into two different parts. One part, called the
strong spectrum, converges to isolated points when the delay parameter
tends to infinity. The other part, called the pseudocontinuous spectrum,
accumulates near criticality and converges after rescaling to a set
of spectral curves, called the asymptotic continuous spectrum. We
show that the spectral curves and strong spectral points provide a
complete description of the spectrum for sufficiently large delay
and can be comparatively easily calculated by approximating expressions.