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In this paper, we study an optimal control problem for a singular system of partial differential equations that models a nonisothermal phase transition with a nonconserved order parameter. The control acts through a third boundary condition for the absolute temperature and plays the role of the outside temperature. It is shown that the corresponding control-to-state mapping is well defined, and the existence of an optimal control and the first-order optimality conditions for a quadratic cost functional of Bolza type are established.
In a strong constant electric field, a dielectric particle
immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude.
We derive the governing equations for this experiment, and
ascertain that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations: explicit estimates are derived for their period and amplitude,
effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
In a strong constant electric field, a dielectric particle immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude. We derive the governing equations for this experiment, and ascertain
that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations:
explicit estimates are derived for their period and amplitude, effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
We develop and experimentally compare policies for the control of a system
of k elevators with capacity one in a transport environment with ` floors, an idealized
version of a pallet elevator system in a large distribution center of the Herlitz PBS AG
in Falkensee. Each elevator in the idealized system has an individual waiting queue of
infinite capacity. On each floor, requests arrive over time in global waiting queues of
infinite capacity. The goal is to find a policy that, without any knowledge about future
requests, assigns an elevator to each request and a schedule to each elevator so that certain
expected cost functions (e.g., the average or the maximal flow times) are minimized. We
show that a reoptimization policy for minimizing average squared waiting times can be
implemented to run in real-time (1 s) using dynamic column generation. Moreover, in
discrete event simulations with Poisson input it outperforms other commonly used policies
like multi-server variants of greedy and nearest neighbor.
We propose an online model for general demand cost sharing games and identify critical properties for group-strategyproofness and weak group-strategyproofness of cost sharing mechanisms for these games. We define incremental online cost sharing mechanisms which can be derived from competitive algorithms.
Based on our general results, we develop online cost sharing mechanisms for several binary demand and general demand cost sharing games derived from network design and scheduling problems. Our results complement the work on incremental mechanisms by Moulin.
We consider the scheduling problem of minimizing the average-weighted completion time on identical parallel machines when jobs are arriving over time. For both the preemptive and the nonpreemptive setting, we show that straightforward extensions of Smith's ratio rule yield smaller competitive ratios than the previously best-known deterministic on-line algorithms.
Two optimal control problems for instationary magnetization
processes are considered in 3D spatial domains that
include electrically conducting and nonconducting regions. The magnetic
fields are generated by induction coils. In the first model, the induction coil
is considered as part of the conducting region and the electrical current is taken
as control. In the second, the coil is viewed as part of the nonconducting region and the
electrical voltage is the control. Here, an integro-differential equation accounts
for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil.
We derive first-order necessary
optimality condition for the optimal controls of both problems. Based on them,
numerical methods of gradient type are applied. Moreover, we report on the application
of model reduction by POD that lead to tremendous savings. Numerical tests are
presented for academic 3D geometries but also for a real-world application.
A linear-quadratic elliptic control problem with pointwise box constraints on the
state is considered. The state-constraints are treated by a Lavrentiev type regularization. It is
shown that the Lagrange multiplier associated with the regularized state-constraints are functions
in L2. Moreover, the convergence of the regularized controls is proven for regularization parameter
tending to zero. To solve the problem numerically, an interior point method and a primal-dual active
set strategy are implemented and treated in function space.
We apply network flow techniques to find good exit selections for evacuees in an emergency evacuation. More precisely, we present two algorithms for computing exit distributions using both classical flows and flows over time which are well known from combinatorial optimization. The performance of these new proposals is compared to a simple shortest path approach and to a best response dynamics approach by using a
cellular automaton model.
The role of larger bulges in the QR algorithm is controversial. Large bulges are infamous
for having a strong, negative influence on the convergence of the implicit shifted QR algorithm.
This paper provides a new explanation of this shift blurring effect by connecting the computation of
the first column of the shift polynomial to the notoriously ill-conditioned pole placement problem.
To avoid shift blurring, modern variants of the QR algorithm employ chains of tightly coupled tiny
bulges instead of one large bulge. It turns out that larger bulges still play a positive role in these
variants; a slight increase of the bulge sizes often results in considerable performance improvements.
We prove a necessary and sufficient criterion for the exponential
stability of periodic solutions of delay differential equations with
large delay. We show that for sufficiently large delay the Floquet
spectrum near criticality is characterized by a set of curves, which we
call asymptotic continuous spectrum, that is independent on the
delay.
This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.
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.
This paper discusses some relationships between ILU factorization techniques and factored sparse approximate inverse techniques. While ILU factorizations compute approximate LU factors of the coefficient matrix A, approximate inverse techniques aim at building triangular matrices Z and W such that $W^\top AZ$ is approximately diagonal. The paper shows that certain forms of approximate inverse techniques amount to approximately inverting the triangular factors obtained from some variants of ILU factorization of the original matrix. A few useful applications of these relationships will be discussed.
Grad-div stabilization has been proved to be a very useful tool in discretizations
of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of
finite element spaces predicts that the stabilization parameter should be optimally chosen
to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis
of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure.
It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle
than known so far in the literature. It depends on the used norm,
the solution, the family of finite
element spaces, and the type of mesh. Depending on the situation, the
optimal
stabilization parameter might range from being very small to very large.
The analytic results
are supported by numerical examples.
In this Note we consider a Lipschitz backward stochastic
differential equation (BSDE) driven by a continuous martingale $M$. We
prove (in Theorem \ref{theorem:main}) that if $M$ is a strong Markov
process and if the BSDE has regular data then the unique solution
$(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, \textit{i.e.} the
orthogonal martingale $N$ is equal to zero, showing that in a Markovian
setting the "usual" solution $(Y,Z)$ (of a BSDE with regular data) has
not to be completed by a strongly orthogonal component even if $M$ does
not enjoy the martingale representation property.
Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility functionals
and convex risk measures. Explicit solutions are found for quantile-based coherent
risk measures and related utility functionals. Typically, these solutions exhibit a critical
phenomenon: If the capital constraint is below some critical value, then the solution will
coincide with a classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For general risk measures
and utility functionals, it is shown that there exists a solution that can be written
as a deterministic increasing function of the price density.
We revisit the problem of the linear response of a constrained mechanical system. In doing so we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville’s theorem and the Green-Kubo relations naturally follow.
We introduce an optimization model for the line planning problem in a public transportation system that aims at minimizing operational costs while ensuring a given level of quality of service in terms of available transport capacity. We discuss the computational complexity of the model for tree network topologies and line structures that arise in a real-world application at the Trolebus Integrated System in Quito. Computational results for this system are reported.
We derive a-priori estimates on the length of the primal-dual path that results from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the problem. Comparison with numerical results indicates that these bounds are sharp and are attained for the case of a single active point.
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 present a new incomplete LU decomposition which is based on
an existing sparse direct solver. In contrast to many incomplete LU decompositions
this ILU incorporates information about the inverse factors L-1 and U-1 which
have direct in
uence on the dropping strategy. We demonstrate in several large scale
examples that this implementation constructs a robust preconditioner.
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium.
We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines mixes
the method of matched asymptotic expansions and the method of periodic surface homogenization.
Abstract. Let Xd,n be an n-element subset of {0, 1}d chosen uniformly
at random, and denote by Pd,n := conv Xd,n its convex hull. Let ∆d,n
be the density of the graph of Pd,n (i.e., the number of one-dimensional
faces of Pd,n divided by n ). Our main result is that, for any function 2
n(d), the expected value of ∆d,n(d) converges (with d → ∞) to one if, √
for some arbitrary ε < 0, n(d) ≤ ( 2 − ε)d holds for all large d, while it √
converges to zero if n(d) ≥ ( 2 + ε)d holds for all large d.
This paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.
The goal of this paper is a mathematical investigation of dilatometer experiments. These are used to detect the kinetics
of solid-solid phase transitions in steel upon cooling from the high temperature phase. Usually, the data are only used for measuring
the start and end temperature of the phase transition. In the case of several coexisting product phases, expensive microscopic
investigations have to be performed to obtain the resulting fractions of the different phases. In contrast, it is shown in this paper that in the case of at most two product phases the complete phase transition kinetics including the final phase fractions are uniquely determined by the dilatometer data. Numerical results confirm the theoretical result.
In this paper we introduce a new method for the computation of KKT matrices that arise from solving constrained, nonlinear optimization problems. This method requires updating of null-space factorizations after a low rank modification. The update procedure has the advantage that it is significantly cheaper than a re-factorization of the system at each new iterate. This paper focuses on the cheap update of a rectangular LU decomposition after a rank-1 modification.
Two different procedures for updating the LU factorization are presented in detail and compared regarding their costs of computation and their stability. Moreover we will introduce an extension of these algorithms which further improves the computation time. This turns out to be an excellent alternative to algorithms based on orthogonal transformations.
On the Efficient Generation of Taylor Expansions for DAE Solutions by Automatic Differentiation
(2005)
Under certain conditions the signature method suggested by
Pantiledes and Pryce facilitates the local expansion of DAE solutions
by Taylor polynomials of arbitrary order. The successive calculation of
Taylor coefficients involves the solution of nonlinear algebraic equations
by some variant of the Gauss-Newton method. Hence, one needs to evaluate
certain Jacobians and several right hand sides. Without advocating
a particular solver we discuss how this information can be efficiently
obtained using ADOL-C or similar automatic differentiation packages.3
For regular matrix pencils the distance in norm to the nearest singular pencil
under low rank perturbation is studied. Characterizations of this distance are derived via the Weyl function of the perturbation. Special attention is paid to the Hermitian pencil case.
Estimates for the distance of a given pencil to the set of singular pencils are obtained.
We provide conditions for convergence of polyhedral surfaces and their
discrete geometric properties to smooth surfaces embedded in R^3. The
notion of totally normal convergence is shown to be equivalent to the convergence
of either one of the following: surface area, intrinsic metric, and
Laplace-Beltrami operators. We further show that totally normal convergence
implies convergence results for shortest geodesics, mean curvature,
and solutions to the Dirichlet problem. This work provides the justifi-
cation for a discrete theory of differential geometric operators defined on
polyhedral surfaces based on a variational formulation.
We investigate the condition number for a complex eigenvalue of a real matrix under
real perturbations. Based on an explicit formula, it is shown that this number is never
smaller than 1/
p
2 times the corresponding condition number with respect to complex
perturbations. This result can be generalized to the condition number of an arbitrary
complex-valued function under real perturbations. This extends to related condition
numbers.
We provide a computational study of the performance of a state-of-the-art solver for nonconvex mixed-integer quadratically constrained programs (MIQCPs). Since successful general-purpose solvers for large problem classes necessarily comprise a variety of algorithmic techniques, we focus especially on the impact of the individual solver components. The solver SCIP used for the experiments implements a branch-and-cut algorithm based on linear outer approximation to solve MIQCPs to global optimality. Our analysis is based on a set of 86 publicly available test instances.
Invariant measures of dynamical systems generated e. g. by difference equations
can be computed by discretizing the originally continuum state space, and
replacing the action of the generator by the transition mechanism of a Markov
chain. In fact they are approximated by stationary vectors of these Markov
chains. Here we extend this well known approximation result and the underlying
algorithm to the setting of random dynamical systems, i.e. dynamical systems
on the skew product of a probability space carrying the underlying stationary
stochasticity and the state space, a particular non-autonomous framework. The
systems are generated by difference equations driven by stationary random processes
modelled on a metric dynamical system. The approximation algorithm
involves spatial discretizations and the definition of appropriate random Markov
chains with stationary vectors converging to the random invariant measure of
the system.
This survey concerns optimization problems arising in the design of survivable communication networks. It turns out that such problems can be modeled in a natural way as non-compact linear programming formulations based on multicommodity flow network models. These non-compact formulations involve an exponential number of path flow variables, and therefore require column generation to be solved to optimality. We consider several path-based survivability mechanisms and present results, both known and new, on the complexity of the corresponding column
generation problems (called the pricing problems). We discuss results for the case of the single link (or node) failures
scenarios, and extend the considerations to multiple link failures. Further, we classify the design problems corresponding to different survivability mechanisms according to the structure of their pricing problem. Finally, we show that almost all encountered pricing problems are hard to solve for scenarios admitting multiple failures.
The paper deals with co-derivative formulae for normal cone mappings to smooth
inequality systems. Both the regular (Linear Independence Constraint Qualification satisfied)
and nonregular (Mangasarian-Fromovitz Constraint Qualification satisfied) cases are considered.
A major part of the results relies on general transformation formulae previously obtained by
Mordukhovich and Outrata. This allows one to derive exact formulae for general smooth, regular and polyhedral, possibly
nonregular systems. In the nonregular, nonpolyhedral case a generalized transformation formula by
Mordukhovich and Outrata applies, however, a major difficulty consists in
checking a calmness condition of a certain multivalued mapping. The paper provides a translation
of this condition in terms of much easier to verify constraint qualifications. The final section is
devoted to the situation where the calmness condition is violated. A series of examples
illustrates the use and comparison of the presented formulae.
The Bottleneck Shortest Path Problem is a basic problem
in network optimization. The goal is to determine the limiting capacity of any path between two specified vertices of the network. This is
equivalent to determining the unsplittable maximum flow between the
two vertices. In this note we analyze the complexity of the problem, its
relation to the Shortest Path Problem, and the impact of the underlying
machine/computation model.
We show that it is not possible to approximate the minimum Steiner tree problem within 1+1/162 unless RP=NP. The currently best known lower bound is 1+ 1/400. The reduction is from Hastad’s nonapproximability result for maximum satisfiability of linear equation modulo 2. The improvement on the nonapproximability ratio is mainly based on the fact that our reduction does not use variable gadgets. This idea was introduced by Papadimitriou and Vempala.
Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between
different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable
voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality conditions are derived by
Pontryagin's principle and a Lagrange technique. In the case of a linear system, the principal structure of time-optimal controls is
discussed. The associated optimality systems are solved by a one-shot strategy
using a multigrid software package. Various numerical examples are discussed.
In this paper we present a strategy to solve parabolic optimal control problems
using available specialized elliptic PDE solvers. We aim at an indirect solution
approach, i.e. developing optimality conditions in function spaces that are then
discretized and solved. Classes of problems where optimality conditions can be derived
as coupled systems of parabolic partial differential equations are considered.
We consider a simultaneous space-time discretization. We verify that for our model
problems the parabolic forward-backward system of PDEs can equivalently be expressed
by a single elliptic boundary value problem in the space-time domain. This
fact has been used as a motivation for space-time-multigrid solution approaches,
which may also be an option in our context.
The theoretical base developed for the example problems then allows to apply
specialized elliptic PDE solvers to the optimality system without much implementational
effort. Numerical experiments for some example problems are conducted and
underline the applicability of this approach.
This paper deals with the computation of regular coderivatives of
solution maps associated with a frequently arising class of generalized equations.
The constraint sets are given by (not necessarily convex) inequalities,
and we do not assume linear independence of gradients to active constraints.
The achieved results enable us to state several versions of sharp necessary optimality
conditions in optimization problems with equilibria governed by such
generalized equations. The advantages are illustrated by means of examples.
On probabilistic constraints induced by rectangular sets and multivariate normal distributions
(2009)
In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided
inequalities for the underlying normally distributed random vector. As a main step
for an algorithmic solution of such problems, we derive a derivative formula for (normal) probabilities
of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus
of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a
similar well-known statement for multivariate normal distribution functions. As an application, we consider
a problem from water reservoir management. One of the outcomes of the problem solution is that the
(still frequently encountered) use of simple individual probabilistic can completely fail. In contrast, the
(more difficult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned
before yields very reasonable and robust solutions over the whole time horizon considered.
We study in detail Schroedinger-type operators on a bounded interval of the real axis with dissipative boundary conditions. The characteristic function of such operators is computed, its minimal self-adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.
We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition ist bounded and the generator fulfills the usual continuity and boundedness conditions, we show the measure solutions with equivalent measures just reinterpret classical ones. In case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in cas of non-uniqueness classical solutions that fail to be measure solutions can coexists with different measure solution.
Recently, the format of TT tensors
\cite{hackbuschHT,osele1,tyrtosele2,tyrtosele3} has turned out to be
a promising new format for the approximation of solutions of high
dimensional problems. In this paper, we prove some new results for
the TT representation of a tensor $U \in \R^{n_1\times \ldots\times
n_d}$ and for the manifold of tensors of TT-rank $\underline{r}$.\As a first result, we prove that the TT (or compression) ranks $r_i$
of a tensor $U$ are unique and equal to the respective separation
ranks of $U$ if the components of the TT decomposition are required to
fulfil a certain maximal rank condition. We then show that the set
$\mathcal{T}$ of TT tensors of fixed rank $\underline{r}$ forms an embedded
manifold in $\R^{n^d}$, therefore preserving the essential theoretical
properties of the Tucker format, but often showing an improved scaling
behaviour. Extending a similar approach for matrices \cite{conte_lub},
we introduce certain gauge conditions to obtain a unique
representation of the tangent space $\cT_U\mathcal{T}$ of $\mathcal{T}$
and deduce a
local parametrization of the TT manifold. The parametrisation of
$\cT_{U}\mathcal{T}$ is often crucial for an algorithmic treatment of
high-dimensional time-dependent PDEs and minimisation problems
\cite{lubuch_blau}. We conclude with remarks on those applications and
present some numerical examples.
We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures that constitute the two components of a Lévy process. In this new type of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDE, we prove existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the variational or Malliavin sense and derive equations that are satisfied by the Malliavin gradient processes. On the chosen stochastic basis this addresses smoothness both with respect to the continuous part of our Lévy process in terms of the classical Malliavin derivative for Hilbert space valued random variables, as well as with respect to the pure jump component for which it takes the form of an increment quotient operator related to the Picard difference operator.
Modeling several competitive leaders and followers acting in an electricity market
leads to coupled systems of mathematical programs with equilibrium constraints,
called equilibrium problems with equilibrium constraints (EPECs). We consider
a simplified model for competition in electricity markets under uncertainty of demand
in an electricity network
as a (stochastic) multi-leader-follower game. First order necessary conditions are
developed for the corresponding stochastic EPEC based on a result of Outrata.
For applying the general result an explicit representation of the co-derivative of
the normal cone mapping to a polyhedron is derived. Later the
co-derivative formula is used for verifying constraint qualifications and for identifying
$M$-stationary solutions of the stochastic EPEC if the demand is represented by a
finite number of scenarios.
Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity.
We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for
simplicial balls and spheres. The main corollaries from this study are: (1.) Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.)
(2.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.)
(3.) All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.)
(4.) Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)
Recent research has shown that
in some practically relevant situations like multi-physics flows[11]
divergence-free mixed finite elements may have a significantly
smaller discretization error than standard non-divergence-free
mixed finite elements. In order to judge the overall performance of
divergence-free mixed finite elements, we
investigate linear solvers for the saddle point linear systems arising in $((P_k)^d,P_{k-1}^{disc})$ Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods.
Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like
Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H-LU preconditioners with GMRES, on both full and statically condensed systems.
Several numerical experiments are provided that show these classes of solvers are well suited for use with Scott-Vogelius elements, and could deliver an interesting overall performance in several applications.
In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nano-antennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate the near-field from an impinging plane wave with
second order absorbing boundary conditions. The magnetic field amplitude on the surface of one object is compared with a computation based on a multiple multipole expansion. Moreover, the convergence behavior of p-FEM with absorbing boundary conditions motivate an adaptive strategy of polynomial degree enhancement and enlargement of the domain.
{We suggest hierarchical a posteriori error estimators for time-discretized
Allen-Cahn and Cahn-Hilliard equations with logarithmic potential and investigate
their robustness numerically.
We observe that the associated effectivity ratios seem to saturate for decreasing mesh size
and are almost independent of the temperature.
We study the approximation of Wasserstein gradient structures
by their finite-dimensional analog. We show that simple
finite-volume discretizations of the linear Fokker-Planck
equation exhibit the recently established entropic gradient-flow
structure for reversible Markov chains. Then we reprove the
convergence of the discrete scheme in the limit of vanishing
mesh size using only the involved gradient-flow structures.
In particular, we make no use of the linearity of the equations
nor of the fact that the Fokker-Planck equation is of second order.
The paper proposes goal-oriented error estimation and mesh refinement
for optimal control problems with elliptic PDE constraints using the value
of the reduced cost functional as quantity of interest. Error representation,
hierarchical error estimators, and greedy-style error indicators are derived and
compared to their counterparts when using the all-at-once cost functional as
quantity of interest. Finally, the efficiency of the error estimator and generated
meshes are demonstrated on numerical examples.
Genome-scale metabolic networks are useful tools for achieving a system-level understanding of metabolism. However, due to their large size, analysis of such networks may be difficult and algorithms can be very slow. Therefore, some authors have suggested to analyze subsystems instead of the original genome-scale models.
Flux coupling analysis (FCA) is a well-known method for detecting functionally related reactions in metabolic networks.
In this paper, we study how flux coupling relations may change if we analyze a subsystem instead of the original network. We show mathematically that a pair of fully, partially or directionally coupled reactions may be detected as uncoupled in certain subsystems. Interestingly, this behavior is the opposite of the flux coupling changes that may occur due to missing reactions, or equivalently, deletion of reactions.
Computational experiments suggest that the analysis of plastid (but not mitochondrial) subsystems may significantly influence the results of FCA. Therefore, the results of FCA for subsystems, especially plastid subsystems, should be interpreted with care.
We introduce a new technique for solving several sequencing problems. We consider Gilmore and Gomory's variant of the Traveling Salesman Problem and two variants of no-wait flowshop scheduling, the classical makespan minimization problem and a new problem arising in the multistage production process in steel manufacturing.
Our technique is based on an intuitive interpretation of sequencing problems as Eulerian Extension Problems. This view reveals new structural insights and leads to elegant and simple algorithms and proofs for this ancient type of problems. As a major effect, we compute not only a single solution; instead, we represent the entire space of optimal solutions. For the new flowshop scheduling problem we give a full complexity classification for any machine configuration.
In this article we study capacitated network design problems. We unify and extend polyhedral results for directed, bidirected and undirected link capacity models. Based on valid inequalities for a network cut we show that regardless of the link capacity model, facets of the polyhedra associated with such a cut translate to facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Our investigation of the facial structure of the cutset polyhedra allows to complement existing polyhedral results for the three variants by presenting facet-defining flow-cutset inequalities in a unifying way. In addition, we present a new class of facet-defining inequalities, showing as well that flow-cutset inequalities alone do not suffice to give a complete description for single-commodity, single-module cutset polyhedra in the bidirected and undirected case – in contrast to a known result for the directed case. The practical importance of the theoretical investigations is highlighted in an extensive computational study on 27 instances from the Survivable Network Design Library (SNDlib).
Canonical forms are developed for several sets of complex matrices that are normal
with respect to an indefinite inner product induced by a nonsingular Hermitian,
symmetric, or skew-symmetric matrix. The most general result covers the case of
polynomially normal matrices, i.e., matrices whose adjoint with respect to the indefinite
inner product is a polynomial of the original matrix. From this result, canonical
forms for matrices that are selfadjoint, skewadjoint, or unitary with respect to the
given indefinite inner product are derived.
In this article we compare two different calmness conditions which are
widely used in the literature on bilevel programming and on mathematical
programs with equilibrium constraints. In order to do so, we consider convex
bilevel programming as a kind of intersection between both research areas.
The so-called partial calmness concept is based on the function value
approach for describing the lower level solution set. Alternatively,
calmness in the sense of multifunctions may be considered for perturbations
of the generalized equation representing the same lower level solution set.
Both concepts allow to derive first order necessary optimality conditions
via tools of generalized differentiation introduced by Mordukhovich. They
are very different, however, concerning their range of applicability and the
form of optimality conditions obtained. The results of this paper seem to
suggest that partial calmness is considerably more restrictive than calmness
of the perturbed generalized equation. This fact is also illustrated by
means of a dicretized obstacle control problem.
The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the
computation of the Schur form of a general complex matrix is investigated.
Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a
sequence of rotations that annihilate a pivot element in the strict lower triangular
part of the matrix until convergence to the Schur form of the matrix is
achieved.
In this paper, it is shown that the cyclic nonsymmetric Jacobi method converges
locally and asymptotically quadratically under mild hypotheses if special ordering
schemes are chosen, namely ordering schemes that lead to so-called northeast directed sweeps.
The theory is illustrated by the help of numerical experiments. In particular, it is shown that
there are ordering schemes that lead to asymptotic quadratic convergence for the cyclic symmetric
Jacobi method, but only to asymptotic linear convergence for the cyclic nonsymmetric
Jacobi method. Finally, a generalization of the nonsymmetric Jacobi method
to the computation of the Hamiltonian Schur form for Hamiltonian matrices is introduced
and investigated.
Discrete Laplace--Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace--Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace--Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace--Beltrami operators.
The terminal Steiner tree problem is a special version of
the Steiner tree problem, where a Steiner minimum tree has to be found
in which all terminals are leaves. We prove that no polynomial time approximation
algorithm for the terminal Steiner tree problem can achieve
an approximation ratio less than (1 - o(1)) ln n unless NP has slightly superpolynomial
time algorithms. Moreover, we present a polynomial time
approximation algorithm for the metric version of this problem with a performance
ratio of 2 , where denotes the best known approximation ratio
for the Steiner tree problem. This improves the previously best known
approximation ratio for the metric terminal Steiner tree problem of +2.
Here we analyze algebraic multilevel methods applied to non-symmetric M-matrices.
We consider two types of multilevel approximate block factorizations.
The first one is related to the AMLI method.
The second method
is the multiplicative counterpart of the AMLI approach which we call
multiplicative algebraic multilevel method, the MAMLI method. The MAMLI method is closely related to certain geometric
and algebraic multigrid methods like the AMGr method. Although these
multilevel methods work very well
in practice for many problems, there is not that much known about
theoretical convergence
properties for non-symmetric problems.
Here, we establish
convergence results and comparison results between AMLI and MAMLI
multilevel methods
applied to non-symmetric M-matrices.
We establish theoretical comparison results for algebraic multi-level
methods applied
to nonsingular non-symmetric M-matrices.
We consider two types of multi-level approximate block factorizations or AMG methods, the
AMLI and the MAMLI method.
We compare
the spectral radii
of the iteration matrices of these methods. This comparison
shows, that the spectral radius of the MAMLI method is less than or equal to
the spectral radius of the AMLI method.
Moreover, we establish how the quality of the approximations
in the block factorization effects the spectral radii of the
iteration
matrices. We prove comparisons results
for different approximation of the fine grid block as well as for the
used Schur
complement. We also establish a theoretical comparison between the
AMG methods and the classical block Jacobi and block Gauss-Seidel methods.
We investigate a thermomechanical model of phase transitions in steel. The strain is assumed to be additively decomposed into an
elastic and a thermal part as well as a contribution from transformation induced plasticity. The resulting model can be viewed
as an extension of quasistatic linear thermoelasticity. We prove existence of a unique solution and conclude with some numerical simulations.
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.
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.
We study a mathematical model for laser-induced thermotherapy,
a minimally invasive cancer treatment. The model consists of a
diffusion approximation of the radiation transport equation
coupled to a bio-heat equation and a model to describe the
evolution of the coagulated zone. Special emphasis is laid on
a refined model of the applicator device, accounting for the
effect of coolant flow inside. Comparisons between experiment
and simulations show that the model is able to predict the
experimentally achieved temperatures reasonably well.
In this paper, we investigate the decay rate of stabilization of the solution of the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a viscous stress. The corresponding free energy is assumed in Landau-Ginzburg form and nonconvex as a function of the order parametr. We prove that for appropriate constants, which appear in the above-mentioned model, we can decide upon the exponencial decrease of the solution to its attractor for time tending to infinity.
Deformable surface models are often represented as triangular meshes in image segmentation applications. For a fast and easily regularized deformation onto the target object boundary, the vertices of the mesh are commonly moved along line segments (typically surface normals). However, in case of high mesh curvature, these lines may intersect with the target boundary at “non-corresponding” positions, or may not intersect at all. Consequently, certain deformations cannot be achieved. We propose omnidirectional displacements for deformable surfaces (ODDS) to overcome this limitation. ODDS allow each vertex to move not only along a line segment but within a surrounding sphere, and achieve globally optimal deformations subject to local regularization con-
straints. However, allowing a ball-shaped instead of a linear range of motion per vertex significantly increases runtime and memory. To alleviate this drawback, we propose a hybrid approach, fastODDS, with improved runtime and reduced memory requirements. Furthermore, fastODDS can also cope with simultaneous segmentation of multiple objects. We show the theoretical benefits of ODDS with experiments on synthetic data, and evaluate ODDS and fastODDS quantitatively on clinical image data of the mandible and the hip bones. There, we assess both the global segmentation accuracy as well as local accuracy in high curvature regions, such as the tip-shaped mandibular coronoid processes and the ridge-shaped acetabular rims of
the hip bones.
Motivated by an obstacle problem for a membrane
subject to cohesion forces, constrained minimization problems involving
a non-convex and non-differentiable objective functional
representing the total potential energy are considered. The associated
first order optimality system leads to a hemi-variational inequality,
which can also be interpreted as a special complementarity
problem in function space. Besides an analytical investigation
of first-order optimality, a primal-dual active set solver is introduced.
It is associated to a limit case of a semi-smooth Newton
method for a regularized version of the underlying problem class.
For the numerical algorithms studied in this paper, global as well as
local convergence properties are derived and verified numerically.
The CreditRisk model launched by CSFB in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit
risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk, equivalent to an algorithm recently proposed by
Giese, based on well-known expansions of the logarithm and the exponential
of a power series. We show that it is advantageous to the Panjer recursion
advocated in the original CreditRisk
document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the
computational complexity of the resulting algorithm is stated.
A class of optimal control problem for a semilinear elliptic partial differential equation
with control constraints is considered. It is well known that
sufficient second-order conditions ensure the stability of optimal solutions, the convergence of
numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically).
We will propose a new approach: Starting with a numerical solution for a fixed mesh we will
show the existence of a local minimizer of the continuous problem. Moreover, we will prove that
this minimizer satisfies the sufficient second-order conditions.
We present the mathematical theory of general over- and underdetermined
hybrid (switched) systems of differential-algebraic equations
(HDAEs). We give a systematic formulation of HDAEs and discuss existence
and uniqueness of solutions, the numerical computation of the switch
points and how to perform consistent initialization at switch points. We
show how numerical solution methods for DAEs can be adapted for HDAEs
and present a comparison of these methods for the real world example of
simulating an automatic gearbox.
A finite volume scheme suitable for nonlinear heat transfer in materials with anisotropic thermal conductivity is formulated, focussing on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. The discretization is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, it is assumed that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. Numerical simulations verify the accuracy of the method in two test cases where a closed-form solution is available. Further results demonstrate the effectiveness of the method in computing the heat transfer in a complex growth apparatus used in crystal growth.
Coarsening in solder alloys is a widely accepted indicator for
possible failure of joints in electronic devices.
Based on the well-established Cahn--Larch\'e model
with logarithmic chemical energy density (see, e.g., Dreyer and Mueller 2001),
we present a numerical environment for the efficient
and reliable simulation of coarsening in binary alloys.
Main features are adaptive mesh refinement based on hierarchical error estimates,
fast and reliable algebraic solution by multigrid and Schur--Newton multigrid methods,
and the quantification of the coarsening speed by the temporal growth of mean phase radii.
We provide a detailed description and a numerical assessment of the algorithm and its different components,
together with a practical application to a eutectic AgCu brazing alloy.
The computation of guided modes in photonic crystal wave-guides is a key issue in the process of designing devices in photonic communications. Existing methods, such as the super-cell method, provide an efficient computation of well-confined modes. However, if the modes are not well-confined, the modelling error of the super-cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed. In this work we demonstrate the numerical realization of a recently proposed Dirichlet-to-Neumann approach and compare the results with those of the super-cell method. For the resulting non-linear eigenvalue problem we propose an iterative solution based on Newton's method and a direct solution using Chebyshev interpolation of the non-linear operator. Based on the Dirichlet-to-Neumann approach, we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave-guides.
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 survey of methods from numerical linear algebra for linear constant coefficient differential-algebraic equations (DAEs) and descriptor control systems is presented. We discuss numerical methods to check the solvability properties of DAEs as well as index reduction and regularization techniques. For descriptor systems we discuss controllability and observability properties and how these can be checked numerically. These methods are based on staircase forms and derivative arrays, transformed with real orthogonal transformations that are discussed in detail. Then we use the reformulated problems in several control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We discuss these applications and give a systematic overview over the theory and the numerical solution methods. In particular, we show that all these applications can be treated with a common approach that is based on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows to generalize and improve several techniques that are currently in use in systems and control.
We present a unified computational framework for matching 3d geometric objects (points, lines, surfaces, volumes) of highly varying shape. Our approach is based on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) method acting on $m$-currents. After stating an optimization algorithm in the function space of admissible morph generating velocity fields, two innovative aspects in this framework are presented: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Secondly, we directly compute the temporal evolution of discrete $m$-current attributes. Several numerical experiments demonstrate the effectiveness of this approach.
Traveling wave equations are used to model the dynamics of multisection semiconductor lasers. To perform a bifurcation analysis of this system of 1-D partial differential equations its low dimensional approximations are constructed and considered. Along this paper this analysis is used for the extensive study of the pulsations in a three section distributed feedback laser. Namely, stability of pulsations, different bifurcation scenaria, tunability of the pulsation frequency and its locking by the frequency of electrical modulation are considered. All these pulsation qualities are highly important when applying lasers in optical communication systems.
We consider a non-isothermal multi-phase field model.
We subsequently discretize implicitly in time and with
linear finite elements. The arising algebraic problem is
formulated in two variables where one is the multi-phase
field, and the other contains the inverse temperature field.
We solve this saddle point problem numerically by a
non-smooth Schur-Newton approach using truncated
non-smooth Newton multigrid methods. An application in
grain growth as occurring in liquid phase crystallization
of silicon is considered.
A capillary surface in a negative gravitational field describes the shape of the surface of a hanging drop in a capillary tube with wetting material on the bottom. Mathematical modeling leads to the volume- and obstacle-constrained minimization of a nonconvex nonlinear energy functional of mean curvature type which is unbounded from below. In 1984 Huisken proved the existence and regularity of local minimizers of this energy under the condition on gravitation being sufficiently weak. We prove convergence of a first order finite element approximation of these minimizers. Numerical results demonstrating the theoretic convergence order are given.
A general abstract approximation scheme for rate-independent processes in the
energetic formulation is proposed and its convergence is proved under various
rather mild data qualifications. The abstract theory is illustrated on several
examples: plasticity with isotropic hardening, damage, debonding,
magnetostriction, and two models of martensitic transformation in
shape-memory alloys.
In this work we are interested in the numerical solution of a coupled model of
differential algebraic equations (DAEs) and partial differential equations (PDEs).
The DAEs describe the behavior of an electrical circuit that contains semiconductor
devices and the partial differential equations constitute drift-diffusion equations
modelling the semiconductor devices in the circuit.
After space discretization using a finite element method, the coupled system results
in a differential-algebraic system with a properly stated leading term. We
investigate the structure and the properties of this DAE system. In particular, we
develop structural criteria for the DAE index. This is of basic interest since DAE
properties like stability, existence and uniqueness of solutions depend strongly on
its index.
We consider linear differential-algebraic m-input m-output systems with positive
strict relative degree or proper inverse transfer function; in the single-input single-output case these
two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties
- such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero
dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the
system classes and secondly, to solve the output regulation problem by funnel control. The funnel
controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in
particular, the transient behaviour of the output error can be specified and the funnel controller does
neither incorporate any internal model for the reference signals nor any identification mechanism, it
is simple in its design. The results are illuminated by position and velocity control of a mechanical
system encompassing springs, masses, and dampers.
We introduce nonsmooth Schur-Newton methods for the solution of the nonlinear discrete saddle-point problems arising from discretized vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The discrete problems are obtained by semi-implicit discretization in time and a first order finite element discretization in space. We incorporate the linear constraints that enforce solutions to stay on the Gibbs simplex using Lagrangian multipliers and prove existence of these multipliers under the assumption of a non-trivial initial condition for the order parameters.
We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
We propose a model for non-isothermal phase
transitions with non-conserved order parameter driven by
a spatially nonlocal free energy with respect to both the
temperature and the order parameter. The resulting system of equations
is shown to be thermodynamically consistent and to admit a strong
solution.
In this paper a nonlocal phase-field model for non-isothermal phase transitions
with a non-conserved order parameter is studied. The paper complements
recent investigations by S. Zheng and the second author and treats
the case when the part of the free energy density forcing the order parameter
to attain values within the physically meaningful range [0; 1] is not given
by a logarithmic expression but by the indicator function of [0; 1] . The resulting
field equations form a system of integro-partial differential inclusions
that are highly nonlinearly coupled. For this system, results concerning global
existence, uniqueness and large-time asymptotic behaviour are derived. The
main results are proved by first transforming the system of inclusions into an
equivalent system of equations in which hysteresis operators occur, and then
employing techniques similar to those recently developed by the authors for
phase-field systems involving hysteresis operators.
This work deals with the efficient numerical solution of
the two-dimensional one-way Helmholtz equation
posed on an unbounded domain.
In this case one has to introduce
artificial boundary conditions to confine the computational domain.
Here we construct with the Z-transformation
so-called discrete transparent
boundary conditions
for higher-order parabolic equations schemes.
These methods are Pade ``Parabolic'' approximations of
the one-way Helmholtz equation
and frequently used in integrated optics and (underwater) acoustics.
Pseudo-Boolean problems lie on the border between satisfiability problems, constraint programming, and integer programming. In particular, nonlinear constraints in pseudo-Boolean optimization can be handled by methods arising in these different fields: One can either linearize them and work on a linear programming relaxation or one can treat them directly by propagation. In this paper, we investigate the individual strengths of these approaches and compare their computational performance. Furthermore, we integrate these techniques into a branch-and-cut-and-propagate framework, resulting in an efficient nonlinear pseudo-Boolean solver.
We discuss the state of the art in numerical solution methods for large scale polynomial or
rational eigenvalue problems. We present the currently available solution methods such as
the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties.
We briefly introduce a new linearization technique and demonstrate how it can be used to
improve structure preservation and with this the accuracy and efficiency of linearization based
methods. We present several recent applications where structured and unstructured nonlinear
eigenvalue problems arise and some numerical results.
We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency
response problems that arise in the analysis, simulation and optimization of acoustic fields.
We report about the cooperation with the company SFE in Berlin. We present the challenges
in the current industrial problems and the state-of-the-art of current methods. The difficulties
that arise with current off-the-shelf methods are discussed and several industrial examples are presented. It is documented that industrial cooperation is by no means a one-way street
of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.