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We derive a formula for the backward error of a complex number $\lambda$ when considered as an approximate eigenvalue
of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate
eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, $*$-even and $*$-odd.
Numerical experiments suggest that in many cases there is a significant difference between the backward
errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.
Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical
systems. One of the major advantages of the DH formulation is that the system encodes system
properties in an algebraic way in the system. Making use of the structure,
it is easy to see that DH systems are stable. In this paper
the question is discussed when a linear constant coefficient DH system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well-understood. In this paper,
explicit formulas for this distance under structure preserving perturbations are determined.
It is also shown (via numerical examples) that under structured perturbations the asymptotical
stability of a DH system is much more robust than for unstructured perturbations, since the
distance can be much larger.
Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.
The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations. We derive such a deflation procedure and apply this new technique to symmetric, Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils. We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples.
The paper provides a structural analysis of the feasible set defined by linear probabilistic constraints. Emphasis is laid on single (individual) probabilistic constraints. A classical convexity result by Van de Panne/Popp and Kataoka is extended to a broader class of distributions and to more general functions of the decision vector. The range of probability levels for which convexity can be expected is exactly identified. Apart from convexity, also nontriviality and compactness of the
feasible set are precisely characterized at the same time. The relation between feasible sets with negative and with nonnegative right-hand side is revealed. Finally, an existence result is formulated for the more difficult case of joint probabilistic constraints.
Actin is a major structural protein of the eukaryotic cytoskeleton and enables cell motility.
Here, we present a model of the actin filament (F-actin) that incorporates the global structure
of the recently published model by Oda et al. but also conserves internal stereochemistry. A
comparison is made using molecular dynamics simulation of the model with other recent F-
actin models. A number of structural determents such as the protomer propeller angle, the
number of hydrogen bonds and the structural variation among the protomers are analyzed.
The MD comparison is found to reflect the evolution in quality of actin models over the last
six years. In addition, simulations of the model are carried out in states with both ADP or
ATP bound and local hydrogen-bonding differences characterized. The results point to the
significance of a direct interaction of Gln137 with ATP for activation of ATPase activity after
the G-to-F-actin transition.
Diffusion Weighted Imaging has become and will certainly continue to be an important tool in medical research and diagnostics. Data obtained with Diffusion Weighted Imaging are characterized by a high noise level. Thus, estimation of quantities like anisotropy indices or the main diffusion direction may be significantly compromised by noise in clinical or neuroscience applications. Here, we present a new package dti for R, which provides functions for the analysis of diffusion weighted data within the diffusion tensor model. This includes smoothing by a recently proposed structural adaptive smoothing procedure based on the Propagation-Separation approach in the context of the widely used Diffusion Tensor Model. We extend the procedure and show, how a correction for Rician bias can be incorporated. We use a heteroscedastic nonlinear regression model to estimate the diffusion tensor. The smoothing procedure naturally adapts to different structures of different size and thus avoids oversmoothing edges and fine structures. We illustrate the usage and capabilities of the package through some examples.
Functional Magnetic Resonance Imaging inherently involves noisy measurements and a severe multiple test
problem. Smoothing is usually used to reduce the effective number of multiple
comparisons and to locally integrate the signal and hence increase the
signal-to-noise ratio. Here, we provide a new structural adaptive segmentation
algorithm (AS)
that naturally combines the signal detection with noise reduction in one procedure.
Moreover, the new method
is closely related to a recently proposed structural adaptive smoothing
algorithm and preserves shape and spatial extent of activation areas without
blurring the borders.
In this paper, we consider the characterization of strong stationary solutions to
equilibrium problems with equilibrium constraints (EPECs). Assuming that the underlying
generalized equation satisfies strong regularity in the sense of Robinson, an explicit
multiplier-based stationarity condition can be derived. This is applied then
to an equilibrium model arising from ISO-regulated electricity spot markets.
Research on flows over time has been conducted mainly in two separate and mainly independent approaches, namely \emph{discrete} and \emph{continuous} models, depending on whether a discrete or continuous representation of time is used. Recently, Borel flows have been introduced to build a bridge between these two models.
In this paper, we consider the maximum Borel flow problem formulated in a network where capacities on arcs are given as Borel measures and storage might be allowed at the nodes of the network. This problem is formulated as a linear program in a space of measures. We define a dual problem and prove a strong duality result. We show that strong duality is closely related to a MaxFlow-MinCut Theorem.
We present a novel algorithm for automatic parameterization of tube-like surfaces of arbitrary genus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.
In the first part of this article, we have shown how time-dependent optimal control for partial
differential equations can be realized in a modern high-level modeling and simulation package. In this second part we extend our approach to (state) constrained problems. "Pure" state constraints in a function space
setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel
measures. This will be overcome by different regularization techniques.
To implement inequality constraints, active set methods and interior point methods (or barrier methods) are widely in use. We show how these techniques can be realized in the modeling and simulation package Comsol
Multiphysics.
In contrast to the first part, only the one-shot-approach based on space-time elements is considered. We implemented a projection method based on active sets as well as a barrier method and compare these methods
by a specialized PDE optimization program, and a program that optimizes the discrete version of the given problem.
We show how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. We summarize the general formulation for distributed and boundary control for initial-boundary value problems for parabolic PDEs and derive the optimality system including the adjoint equation. The main difficulty therein is that the latter has to be integrated backwards in time. This implies that complicated implementation effort is necessary to couple state and adjoint equations to compute an optimal solution. Furthermore a large amount of computational effort or storage is required to provide the needed information (i.e the trajectories) of the state and adjoint variables. We show how this can be realized in the modeling and simulation package COMSOL MULTIPHYSICS, taking advantage of built-in discretization, solver and post-processing technologies and thus minimizing the implementation effort. We present two strategies: The treatment of the coupled optimality system in the space-time cylinder, and the iterative approach by sequentially solving state and adjoint system and updating the controls. Numerical examples show the elegance of the implementation and the efficiency of the two strategies.
Boolean modeling frameworks have long since proved their worth for capturing and analyzing essential characteristics of complex systems.
Hybrid approaches aim at exploiting the advantages of Boolean formalisms while refining expressiveness. In this paper, we present a formalism that augments Boolean models with stochastic aspects. More specifically, biological reactions effecting a system in a given state are associated
with probabilities, resulting in dynamical behavior represented as a Markov chain. Using this approach, we model and analyze the cytokinin
response network of Arabidopsis thaliana with a focus on clarifying the character of an important feedback mechanism.
We present a mixed-integer multistage stochastic programming model for the short term unit commitment of a hydro-thermal power system under uncertainty in load, inflow to reservoirs, and prices for fuel and delivery contracts. The model is implemented for uncertain load and tested on realistic data from a German power utility. Load scenario trees are generated by a procedure consisting of two steps: (i) Simulation of load scenarios using an explicit respresentation of the load distribution and (ii) construction of a tree out of these scenarios. The dimension of the corresponding mixed-integer programs ranges up to 200,000 binary and 350,000 continuous variables. The model is solved by a Lagrangian-based decomposition strategy exploiting the loose coupling structure. Solving the Lagrangian dual by a proximal bundle method leads to a successive decomposition into single unit subproblems, which are solved by specific algorithms. Finally, Lagrangian heuristics are used to construct nearly optimal first stage decisions.
We consider the preemptive and non-preemptive problems of scheduling jobs with precedence constraints on parallel machines with the
objective to minimize the sum of~(weighted) completion times. We investigate an online model in which the scheduler learns about a
job when all its predecessors have completed. For scheduling on a single machine, we show matching lower and upper bounds of~$\Theta(n)$ and~$\Theta(\sqrt{n})$ for jobs with general and equal weights, respectively. We also derive corresponding results on parallel machines.
Our result for arbitrary job weights holds even in the more general stochastic online scheduling model where, in addition to the limited information about the job set, processing times are uncertain. For a
large class of processing time distributions, we derive also an improved performance guarantee if weights are equal.
We consider a non-preemptive, stochastic parallel machine
scheduling model with the goal to minimize the weighted completion
times of jobs. In contrast to the classical stochastic model where jobs
with their processing time distributions are known beforehand, we assume
that jobs appear one by one, and every job must be assigned
to a machine online. We propose a simple online scheduling policy for
that model, and prove a performance guarantee that matches the currently
best known performance guarantee for stochastic parallel machine
scheduling. For the more general model with job release dates we derive
an analogous result, and for NBUE distributed processing times we
even improve upon the previously best known performance guarantee for
stochastic parallel machine scheduling. Moreover, we derive some lower
bounds on approximation.
We consider empirical approximations of two-stage stochastic mixed-integer linear programs and derive central
limit theorems for the objectives and optimal values. The limit theorems are based on empirical process theory
and the functional delta method. We also show how these limit theorems can be used to derive confidence intervals
for optimal values via a certain modification of the bootstrapping method.
We analyze an interactive model of credit ratings where external shocks, initially
affecting only a small number of firms, spread by a contagious chain reaction to the
entire economy. Counterparty relationships along with discrete adjustments of credit
ratings generate a transition mechanism that allows the financial distress of one firm
to spill over to its business partners. Such a contagious infectious of financial distress
constitutes a source of intrinsic risk for large portfolios of credit sensitive securities that
cannot be “diversified away.” We provide a complete characterization of the fluctuations
of credit ratings in large economies when adjustments follow a threshold rule. We also
analyze the effects of downgrading cascades on aggregate losses of credit portfolios. We
show that the loss distribution has a power-law tail if the interaction between different
companies is strong enough.
Stochastic Optimization of Electricity Portfolios: Scenario Tree Modeling and Risk Management
(2008)
We present recent developments in the field of stochastic programming with regard to application in power management. In particular we discuss issues of scenario tree modeling, i.e., appropriate discrete approximations of the underlying stochastic parameters. Moreover, we suggest risk avoidance strategies via the incorporation of
so-called polyhedral risk functionals into stochastic programs. This approach, motivated through tractability of the resulting problems, is a constructive framework providing particular flexibility with respect to the dynamic aspects of risk.
A strategy for controlling the stepsize in the numerical integration of stochastic
differential equations (SDEs) is presented. It is based on estimating the p-th mean of
local errors. The strategy leads to deterministic stepsize sequences that are identical
for all paths. For the family of Euler schemes for SDEs with small noise we derive
computable estimates for the dominating term of the p-th mean of local errors
and show that the strategy becomes efficient for reasonable stepsizes. Numerical
experience is reported for test examples including scalar SDEs and a stochastic
circuit model.
The package fmri is provided for analysis of single run functional
Magnetic Resonance Imaging data. It implements structural adaptive smoothing
methods with signal detection for adaptive noise reduction which avoids blurring
of edges of activation areas. fmri provides fmri analysis from time series
modeling to signal detection and publication-ready images.
New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the
monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function.
Using phase space analysis for the corresponding fifth-order dynamical
system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input.
Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.
We give sufficient conditions for a non-zero sum discounted stochastic game with
compact and convex action spaces and with norm-continuous transition probabilities,
but with possibly unbounded state space, to have a Nash equilibrium in homogeneous
Markov strategies that depends in a Lipschitz continuous manner on the current state. If
the underlying state space is compact this yields the existence of a stationary equilibrium.
Stochastic games with weakly interacting players provide a probabilistic framework within
which to study strategic behavior in models of non-market interactions.
We discuss a parallel library of efficient algorithms for model reduction of largescale
systems with state-space dimension up to O(104). We survey the numerical
algorithms underlying the implementation of the chosen model reduction methods.
The approach considered here is based on state-space truncation of the system
matrices and includes absolute and relative error methods for both stable and unstable
systems. In contrast to serial implementations of these methods, we employ
Newton-type iterative algorithms for the solution of the major computational tasks.
Experimental results report the numerical accuracy and the parallel performance of
our approach on a cluster of Intel Pentium II processors.
A state-constrained optimal control problem arising in the context of sublimation crystal growth is considered. The presence of pointwise state-constraints and nonlocal radiation interface conditions
constitutes the major issue of this problem. A regularity result of the state is presented that allows to
derive the optimality condition.
We consider a control- and state-constrained optimal control problem
governed by a semilinear
elliptic equation with nonlocal interface conditions.
These conditions occur during the
modeling of diffuse-gray conductive-radiative heat transfer.
The nonlocal radiation interface condition and the pointwise state-constraints
represent the particular features of this problem. To deal with the
state-constraints, continuity of the state is shown which allows to
derive first-order necessary conditions. Afterwards, we establish second-order
sufficient conditions that account for strongly active sets and
ensure local optimality in an $L^2$-neighborhood.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.
We discuss the eigenvalue problem for
general and structured matrix polynomials which may
be singular and may have eigenvalues at infinity.
We derive staircase
condensed forms that allow deflation of the infinite eigenvalue and
singular structure of the matrix polynomial.
The remaining reduced order staircase form leads to
new types of linearizations which determine the finite eigenvalues and
and corresponding eigenvectors. The new linearizations
also simplify the construction of structure preserving linearizations.
Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest to apply an aggregation/disaggregation method which only addresses wellconditioned sub-problems and thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples.
The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.
This work studies the stability and the stochastic properties of neural activity evoked by external
stimulation. The underlying model describes the spatiotemporal dynamics of neural populations
involving both synaptic delay and axonal transmission delay. We show, that the linear model
recasts to a set of affne delay differential equations in spatial Fourier space. Besides a stability
study for general kernels and general external stimulation, the power spectrum of evoked activity
is derived analytically in case of external Gaussian noise. Further applications to specific kernels
reveal critical
uctuations at Hopf- and Turing bifurcations and allow the numerical detection of
1/f fluctuations near the stability threshold.
We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical
systems. In this paper we analyze when such a system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the \emph{real distance to instability (real stability radius)} has been a challenging problem. In this paper, we analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical
stability of a DH system is much more robust than for unstructured perturbations.
Classical stability properties of solutions
that are well-known for ordinary differential
equations (ODEs) are generalized to differential-algebraic equations (DAEs).
A new test equation is derived for the analysis of numerical methods applied
to DAEs with respect to the stability of the numerical approximations.
Morevover, a stabilization technique is developed to improve the stability of classical DAE integration methods. The stability regions for these stabilized discretization methods are determined and it is shown that they much better reproduce the stability properties known for the ODE case
than in the unstabilized form.
Movies that depict the stability regions for several methods are included for interactive use.
We analyse stability aspects of linear multistage stochastic programs with polyhedral risk measures in the objective. In particular, we consider sensitivity of the optimal value with respect perturbations of the underlying stochastic input process. An existing stability result for multistage stochastic programs with expectation objective is carried forward to the case of polyhedral risk-averse objectives. Beside Lr-distances these results also involve filtration distances of the perturbations of the stochastic process. We discuss additional requirements for the
polyhedral risk measures such that the problem dependent filtration distances can be bounded by problem independent ones. Stability and such bounds are the basis for scenario tree approximation techniques used in practical problem solving.
Quantitative stability of linear multistage stochastic programs is studied. It
is shown that the infima of such programs behave (locally) Lipschitz continuous
with respect to the sum of an Lr-distance and of a distance measure for the filtrations
of the original and approximate stochastic (input) processes. Various issues
of the result are discussed and an illustrative example is given. Consequences for
the reduction of scenario trees are also discussed.
Stability of Linear Stochastic Difference Equations in Strategically Controlled Random Environments
(2004)
We consider the stochastic sequence fYtgt2N defined recursively by the linear relation
Yt+1 = AtYt+Bt in a random environment. The environment is described by the stochastic
process f(At;Bt)gt2N and is under the simultaneous control of several agents playing a
discounted stochastic game. We formulate sufficient conditions on the game which ensure
the existence of Nash equilibrium in Markov strategies which has the additional property
that, in equilibrium, the process fYtgt2N converges in distribution to a stationary regime.
The Lang-Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues grows to infinity. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.
An analysis of convex stochastic programs is provided if the underlying probability distribution is subjected to (small) perturbations. It is shown, in particular, that epsilon-approximate solution sets of convex stochastic programs behave Lipschitz continuous with respect to certain distances of probability distributions that are generated by the relevant integrands. It is shown that these results apply to linear two-stage stochastic programs with random recourse. Consequences are discussed on associating Fortet-Mourier metrics to two-stage models and on the asymptotic behavior of empirical estimates of such models, respectively.
We consider convex optimization problems with $k$th order stochastic dominance constraints for $k\ge 2$. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets in terms of a suitably selected probability metrics.Moreover, we provide conditions ensuring that the optimal value function is Hadamard directionally differentiable. Finally, we discuss some implications of the results for empirical (Monte Carlo,
sample average) approximations of dominance constrained optimization models.
Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints
(2007)
We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type.
Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional
derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the
dominance constraints.
By extending the stability analysis of [17] for multistage stochastic programs we show that their solution sets behave stable with respect to the sum of an Lr-distance and a filtration distance. Based on such stability results we suggest a scenario tree generation method for the (multivariate) stochastic input process. It starts with a fan of individual scenarios and consists of a recursive deletion and branching procedure which is controlled by bounding the approximation error. Some numerical experience for generating scenario trees in electricity portfolio management is reported.
A stability analysis is presented for neural field equations in the presence
of axonal delays and for a general class of connectivity kernels and synap-
tic properties. Sufficient conditions are given for the stability of equilibrium
solutions. It is shown that the delays play a crucial role in non-stationary
bifurcations of equilibria, whereas the stationary bifurcations depend only on
the kernel. Bounds are determined for the frequencies of bifurcating periodic
solutions. A perturbative scheme is used to calculate the types of bifurca-
tions leading to spatial patterns, oscillatory solutions, and traveling waves.
For high transmission speeds a simple method is derived that allows the de-
termination of the bifurcation type by visual inspection of the Fourier trans-
forms of the connectivity kernel and its first moment. Results are numerically
illustrated on a class of neurologically plausible second order systems with
combinations of Gaussian excitatory and inhibitory connections.
In this paper, we discuss stability properties of positive descriptor systems in the continuous-time as well as in the discrete-time case. We present different characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems. We show that if the spectral projector onto the right finite deflating subspace of the matrix pair $(E,A)$ is non-negative, then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case. Furthermore, we provide sufficient conditions that guarantee entry-wise non-negativity along with positive semi-definiteness of solutions of generalised projected Lyapunov equations. As an application of the framework established throughout this paper, we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case.
In this paper we discuss the stability and model order reduction of coupled linear
time-invariant systems. Sufficient conditions for a closed-loop system to be asymptotically stable are
given. We present a model reduction approach for coupled systems based on reducing the order of the
subsystems and coupling the reduced-order subsystems by the same interconnection matrices as for
the original model. Such an approach allows to obtain error bounds for the reduced-order closed-loop
system in terms of the errors in the reduced-order subsystems. Model reduction of coupled systems
with unstable subsystems is also considered. Numerical examples are given.
The paper introduces an identification problem arising in modern regional hyperthermia, a cancer
therapy aiming at heating the tumor by microwave radiation. The task is to identify the highly
individual perfusion, which affects the resulting temperature distribution, from MR measurements.
The identification problem is formulated as an optimization problem. Existence of a solution and
optimality conditions are analyzed. Different regularizations and problem variants are considered. For
the numerical solution, a standard SQP method is used. Sufficient conditions for the convergence of
the method are derived. Finally, numerical examples on artificial as well as clinical data are presented.
This work studies dynamical properties of spatially extended neu-
ronal ensembles. We first derive an evolution equation from tem-
poral properties and statistical distributions of synapses and somata.
The obtained integro-differential equation considers both synaptic and
axonal propagation delay, while spatial synaptic connectivities ex-
hibit gamma-distributed distributions. This familiy of connectivity
kernels also covers the cases of divergent, finite, and negligible self-
connections. The work derives conditions for both stationary and
nonstationary instabilities for gamma-distributed kernels.It turns out
that the stability conditions can be formulated in terms of the mean spatial interaction ranges and the mean spatial interaction times. In
addition, a numerical study examines the evoked spatiotemporal re-
sponse activity caused by short local stimuli and reveals maximum
response activity after the mean interaction time at a distance from
stimulus offset location equal to the mean interaction range. These
findings propose new insights to neuronal mechanisms of experimen-
tally observed evoked brain activity.
Chimera states are particular trajectories
in systems of phase oscillators with non-local coupling
that display a spatio-temporal pattern of coherent and incoherent motion.
We present here a detailed analysis
of the spectral properties for such trajectories.
First, we study numerically their Lyapunov spectrum
and its behavior for an increasing number of oscillators.
The spectra demonstrate the hyperchaotic nature of the chimera states
and show a correspondence of the Lyapunov dimension
with the number of incoherent oscillators.
Then, we pass to the thermodynamic limit equation
and present an analytic approach
to the spectrum of a corresponding linearized evolution operator.
We show that in this setting, the chimera state is neutrally stable
and that the continuous spectrum coincides with the limit
of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying
graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
Polzehl and Spokoiny (2000) introduced the adaptive weights smoothing
(AWS) procedure 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 assumption
can be rather restrictive for a number of potential applications. Here we
present a new method based on the ideas of propagation and separation
which extends the AWS procedure to the case of an arbitrary local linear
parametric structure. We also establish some important results about
properties of the new ‘propagation-separation’ procedure including rate
optimality in the pointwise and global sense. The performance of the
procedure is illustrated by examples for local polynomial regression and
by applications to artificial and real images.
Three properties of matrices: the spark, the mutual incoherence and the restricted isometry property have recently been introduced in the context of compressed sensing. We study these properties for matrices that are Kronecker products and show how these properties relate to those of the factors. For the mutual incoherence we also
discuss results for sums of Kronecker products.
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely
representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to nonzero expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.
A state-constrained optimal boundary control problem governed by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater assumption has to be made that restricts the theory to the two-dimensional case. This difficulty is overcome by a source representation of the control and combined with a Lavrentiev type regularization. Optimality conditions for the regularized problem are derived, where the corresponding Lagrange multipliers have $L^2$-regularity. By the spectral theorem for compact and normal operators, the convergence result is shown. Moreover, the convergence for vanishing regularization parameter of the adjoint state associated with the regularized problem is shown. Finally, the uniform boundedness of the regularized Lagrange multipliers in $L^1(\O)$ is verified by a maximum principle argument.
The purpose of this paper is the analysis of dynamic iteration methods for
the numerical integration of coupled systems of ODEs and DAEs.
We will investigate convergence of these methods and put special emphasis
on the {\sc Jacobi}- and {\sc Gauss-Seidel} methods. Furthermore, the
fundamental difference in the convergence behaviour of coupled ODEs and DAEs
is pointed out. This difference is used to explain why certain relaxation methods
for coupled DAEs may fail. Finally, a remedy to this undesirable
effect is proposed that makes use of a so-called {\em preconditioned dynamic
iteration} strategy. This regularization also allows significant reduction of
dynamic iteration steps.
Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
(2012)
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.
Some aspects of reachability for parabolic boundary control problems with control constraints
(2009)
A class of one-dimensional parabolic optimal boundary control problems
is considered. The discussion includes Neumann, Robin, and Dirichlet
boundary conditions. The reachability of a given target state in final
time is discussed under box constraints on the control. As a mathematical
tool, related exponential moment problems are investigated. Moreover,
based on a detailed study of the adjoint state, a technique is presented
to find the location and the number of the switching points of optimal
bang-bang controls. Numerical examples illustrate this procedure.
Solving Time-Dependent Optimal Control Problems in Comsol Multiphyiscs ba Space-Time Discretizations
(2009)
We use COMSOL Multiphysics to solve time-dependent optimal control problems for par-
tial differential equations whose optimality conditions can be formulated as a PDE. For a
class of linear-quadratic model problems we summarize known analytic results on existence
of solutions and first order optimality conditions that exhibit the typical feature of time-dependent control problems, namely the fact that a part of the optimality system has to be
integrated backward in time. We present a strategy that is based on the treatment of the
coupled optimality system in the space-time cylinder. A brief motivation of this approach is
given by showing that the optimality system is elliptic in some sence. Numerical examples
show advantages and limits of the usage of COMSOL Multiphysics and of our approach.
The stochastic dynamics of a well-stirred mixture of molecular species
interacting through different biochemical reactions can be
accurately modelled by the chemical master equation (CME). Research in
the biology and scientific computing community has
concentrated mostly on the development of numerical techniques to
approximate the solution of the CME via many realizations of the associated
Markov jump process. The domain of exact and/or efficient methods for
directly solving the CME is still widely open, which is due to its
large dimension that grows exponentially with the number of molecular
species involved. In this article, we present an exact solution
formula of the CME for arbitrary initial conditions in the case where
the underlying system is governed by monomolecular reactions. The
solution can be expressed in terms of the convolution of multinomial
and product Poisson distributions with time-dependent parameters
evolving according to the traditional reaction-rate equations. This
very structured representation allows to deduce any property of the
solution. The model class includes many interesting examples and may
also be used as the starting point for the design of new numerical
methods for the CME of more complex reaction systems.
Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field.
In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.
This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.
We discuss solvers for Sylvester, Lyapunov, and Stein equations
that are available in the SLICOT Library (Subroutine
Library In COntrol Theory). These solvers offer improved
efficiency, reliability, and functionality compared to corresponding
solvers in other computer-aided control system design
packages. The performance of the SLICOT solvers is
compared with the corresponding MATLAB solvers.
In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6,7,8].
In the case of the equidistant discretization of the Airy differential equation (\discrete
Airy equation") the exact solution can be found explicitly. This fact is used
to derive a discrete transparent boundary condition (TBC) for a Schroedinger
equation with linear varying potential, which can be used in \parabolic equation"
simulations in (underwater) acoustics and for radar propagation in the troposphere.
We propose different strategies for the discrete TBC and show an efficient implementation.
Finally a stability proof for the resulting scheme is given. A numerical
example in the application to underwater acoustics shows the superiority of the new
discrete TBC.
Propagation of short optical pulses in a nonlinear dispersive medium is considered without the use of slow envelope and
unidirectional propagation approximations. The existence of uniformly moving solitary solutions is predicted in the anomalous
dispersion domain. A four-parametric family of such solutions is found that contains the classical envelope soliton in the limit of
large pulse durations. In the opposite limit we get another family member, which in contrast to the envelope soliton strongly depends on nonlinearity model and represents the shortest and the most intense pulse which can propagate in a stationary manner.
Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.
Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
(2013)
It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers
(parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
In this paper we introduce the notion of smoothed competitive analysis of online
algorithms. Smoothed analysis has been proposed by Spielman and Teng [22] to explain
the behaviour of algorithms that work well in practice while performing very poorly
from a worst case analysis point of view. We apply this notion to analyze the Multi-
Level Feedback (MLF) algorithm to minimize the total flow time on a sequence of
jobs released over time when the processing time of a job is only known at time of
completion.
The initial processing times are integers in the range [1, 2K ]. We use a partial bit
randomization model, where the initial processing times are smoothened by changing
the k least significant bits under a quite general class of probability distributions. We
show that MLF admits a smoothed competitive ratio of O(max((2k /σ)3 , (2k /σ)2 2K−k )),
where σ denotes the standard deviation of the distribution. In particular, we obtain a
competitive ratio of O(2K−k ) if σ = Θ(2k ). We also prove an Ω(2K−k ) lower bound for
any deterministic algorithm that is run on processing times smoothened according to
the partial bit randomization model. For various other smoothening models, including
the additive symmetric smoothening model used by Spielman and Teng [22], we give a
higher lower bound of Ω(2K ).
A direct consequence of our result is also the first average case analysis of MLF. We
show a constant expected ratio of the total flow time of MLF to the optimum under
several distributions including the uniform distribution.
Many applications give rise to matrix polynomials whose coefficients have
a kind of reversal symmetry, a structure we call palindromic.
Several properties of scalar palindromic polynomials are derived,
and together with properties of compound matrices, used to
establish the Smith form of regular and singular T-palindromic matrix polynomials,
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials are characterized,
and necessary conditions for the
existence of structured linearizations established.
In the odd degree case, a constructive procedure for building
palindromic linearizations shows that the necessary conditions are sufficient as well.
The Smith form for *-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of
characteristic two are presented.
Using Freidlin-Wentzell sample path large deviations theory, we characterise the small-time behaviour of probabilities of a process following an uncorrelated local-stochastic volatility model.
As a corollary, we determine the small-maturity behaviour of the implied volatility under this class of processes.
We characterize the Smith form of skew-symmetric matrix polynomials
over an arbitrary field $\F$,
showing that all elementary divisors occur with even multiplicity.
Restricting the class of equivalence transformations to unimodular congruences,
a Smith-like skew-symmetric canonical form
for skew-symmetric matrix polynomials is also obtained.
These results are used to analyze the eigenvalue and elementary divisor structure
of matrices expressible as products of two skew-symmetric matrices,
as well as the existence of structured linearizations
for skew-symmetric matrix polynomials.
By contrast with other classes of structured matrix polynomials
(e.g., alternating or palindromic polynomials),
every regular skew-symmetric matrix polynomial
is shown to have a structured strong linearization.
While there are singular skew-symmetric polynomials of even degree
for which a structured linearization is impossible,
for each odd degree we develop a skew-symmetric companion form
that uniformly provides a structured linearization
for every regular and singular skew-symmetric polynomial
of that degree.
Finally, the results are applied to the construction of minimal
symmetric factorizations of skew-symmetric rational matrices.
The classical singular value decomposition for a matrix $A\in\Cmn$ is a
canonical form for $A$ that also displays the eigenvalues
of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop
a corresponding decomposition for $A$ that provides the Jordan canonical forms
for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider
the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$
are invertible and either complex symmetric and complex skew-symmetric, and we
provide a canonical form under transformations of the form
$(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.
Mehta, Roughgarden, and Sundararajan recently introduced a new class of cost sharing mechanisms called acyclic mechanisms. These mechanisms achieve a slightly weaker notion of truthfulness than the well-known Moulin mechanisms, but provide additional freedom to improve budget balance and social cost approximation guarantees. In this paper, we investigate the potential of acyclic mechanisms for combinatorial optimization problems. In particular, we study a subclass of acyclic mechanisms which we term singleton acyclic mechanisms. We show that every rho-approximate algorithm that is partially increasing can be turned into a singleton acyclic mechanism that is weakly group-strategyproof and rho-budget balanced. Based on this result, we develop singleton acyclic mechanisms for parallel machine scheduling problems with completion time objectives, which perform extremely well both with respect to budget balance and social cost.
We study a planning problem arising in SDH/WDM multi-layer telecommunication network design. The goal is to find a minimum cost
installation of link and node hardware of both network layers such that traffic demands can be realized via grooming and a survivable routing. We present a mixed-integer programming formulation that takes many practical side constraints into account, including node hardware, several bitrates, and survivability against single physical node or link failures. This model is solved using a branch-and-cut approach with problem-specific preprocessing and cutting planes based on either of the two layers. On several realistic two-layer planning scenarios, we show that these cutting planes are still useful in the multi-layer context,
helping to increase the dual bound and to reduce the optimality gaps.
Bei der Produktions- und Handelsplanung treffen Energieversorgungsunternehmen eine Reihe von Entscheidungen unter unsicheren Randbedingungen. Ein Optimierungsmodell für einen mittelfristigen Planungshorizont muss diese Unsicherheiten berücksichtigen, etwa durch Einbeziehung von statistischen Modellen für die zufallsbehafteten Eingangsdaten. Dadurch ist es prinzipiell möglich, Risikobetrachtungen direkt in die Optimierung zu integrieren. Wir demonstrieren in dieser Arbeit die Möglichkeit, spezielle dynamische Risikomaße, so genannte polyedrische Risikomaße, in die Zielfunktion der Optimierung mit aufzunehmen. Im Gegensatz zu vielen anderen Ansätzen wird dadurch die Komplexität des Problems nicht wesentlich erhöht. Das vorgestellte Modell stellt ein Werkzeug zur Entscheidungsunterstützung für kleinere Marktteilnehmer hinsichtlich der Beschaffungsplanung dar. Dabei werden insbesondere konkrete mittelfristig bindende Bezugsverträge mit der Möglichkeit verglichen, die Versorgung in erster Linie auf der Basis von Spot- und Futuremarkt zu planen.
This paper demonstrates simulation tools for edge-emitting multi quantum well (MQW) lasers.
Properties of the strained MQW active region are simulated by eight-band kp calculations. Then, a 2D
simulation along the transverse cross section of the device is performed based on a drift-diffusion model,
which is self-consistently coupled to heat transport and equations for the optical field. Furthermore, a
method is described, which allows for an efficient quasi 3D simulation of dynamic properties of multisection
edge-emitting lasers.
Im Zentrum der Arbeiten soll die Chaos- und Kohärenzkontrolle von Halbleiterlasern mit gegenseitiger optischer Kopplung stehen. Diese Fragestellung ist von erheblicher praktischer Relevanz, da Rauschen und chaotisches Verhalten generelle Probleme in der optischen Hochgeschwindigkeitskommunikation sind. Die Kontrolle von optischen Systemen mit komplexer Selbstorganisation stellt aber auch aus grundsätzlicher Sicht Neuland dar. Hierfür geeignete Konzepte sind bisher weder überzeugend theoretisch beschrieben noch experimentell umgesetzt.
We study the exact recovery of signals from quantized frame coefficients. Here, the basis of the quantization
is hard thresholding, and we present a simple algorithm for the recovery of reconstructable signals. The set of
non-reconstructable signals is shown to be star-shaped and symmetric with respect to the origin. Moreover, we
provide a criterion on the frame for the boundedness of this set. In this case, we also give a priori bounds.
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
We consider scheduling on a single machine with one non-availability period to minimize the weighted sum of completion times. We provide a preemptive algorithm with an approximation ratio arbitrarily close to the Golden Ratio,~$(1+\sqrt{5})/2+\eps$, which improves on a previously best known~$2$-approximation. The non-preemptive version of the same algorithm yields a~$(2+\eps)$-approximation.
We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and
Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the
remaining faces. Under natural smoothness assumptions on the data we obtain
$u\in L^\infty((0,T);H^{3/2-\delta}(\Omega))$ for the displacements and
$z\in L^\infty((0,T);H^{1/2-\delta}(\Omega))$ for the internal variables.
The proof is based on a difference quotient technique and a reflection argument.
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece
changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.
The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and de- velopment of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more com- plicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.
A class of optimal control problems for a semilinear parabolic partial differential equation
with control and mixed control-state constraints is considered.
For this problem, a projection formula is derived
that is equivalent to the necessary optimality
conditions. As main result, the superlinear convergence of a semi-smooth Newton method is shown.
Moreover we show the numerical treatment and several numerical experiments.
Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ. 6, 69–83, 2004] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A 460, 169–198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm [Appl. Math. Fin. 15, 107–121, 2008], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.
The LSW model with encounters has been suggested by Lifshitz and
Slyozov as a regularization of their classical mean-field model for
domain coarsening to obtain universal self-similar long-time
behavior. We rigorously establish that an exponentially decaying
self-similar solution to this model exist, and show that this
solutions is isolated in a certain function space. Our proof relies
on setting up a suitable fixed-point problem in an appropriate
function space and careful asymptotic estimates of the solution to a
corresponding homogeneous problem.
We consider a control constrained optimal control problem governed by a semilinear
elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of
diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, secondorder
sufficient conditions are derived that account for strongly active sets. These conditions ensure
local optimality in a Ls-neighborhood whereby the underlying analysis allows to use weaker norms
than L?.
We derive formulae for the second-order subdifferential of polyhedral norms. These formulae are fully explicit in terms of initial data. In a first step we rely on the explicit formula for the coderivative of normal cone mapping to polyhedra. Though being explicit, this formula is quite involved and difficult to apply. Therefore, we derive simple formulae for the 1-norm and - making use of a recently obtained formula for the second-order subdifferential of the maximum function - for the maximum norm.
This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.
We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming
(SDDP) algorithm can be used to obtain approximations of
the corresponding risk-averse recourse functions. This allows us to define a risk-averse nonanticipative feasible policy for thestochastic linear program. Formulas for the cuts that approximate the recourse functions are given.
この論文ではソフトウェア・パッケージSCIP Optimization Suite を紹介し,その3つの構成要素:モデリン グ言語Zimpl, 線形計画(LP: linear programming) ソルバSoPlex, そして,制約整数計画(CIP: constraint integer programming) に対するソフトウェア・フレームワークSCIP, について述べる.本論文では,この3つの 構成要素を利用して,どのようにして挑戦的な混合整数線形計画問題(MIP: mixed integer linear optimization problems) や混合整数非線形計画問題(MINLP: mixed integer nonlinear optimization problems) をモデル化 し解くのかを説明する.SCIP は,現在,最も高速なMIP,MINLP ソルバの1つである.いくつかの例により, Zimpl, SCIP, SoPlex の利用方法を示すとともに,利用可能なインタフェースの概要を示す.最後に,将来の開 発計画の概要について述べる.
We study two related problems in non-preemptive scheduling and packing of malleable tasks with precedence constraints to minimize the makespan. We distinguish the scheduling variant, in which we allow the free choice of processors, and the packing variant, in which a task must be assigned to a contiguous subset of processors.
For precedence constraints of bounded width, we completely resolve the complexity status for any particular problem setting concerning width bound and number of processors, and give polynomial-time algorithms with best possible performance. For both, scheduling and packing malleable tasks, we present an FPTAS for the NP-hard problem variants and exact algorithms for all remaining special cases. To obtain the positive results, we do not require the common monotonous penalty assumption on processing times, whereas our hardness results hold even when assuming this restriction.
With the close relation between contiguous scheduling and strip packing, our FPTAS
is the first (and best possible) constant factor approximation for (malleable) strip packing under special precedence constraints.
A framework for the reduction of scenario trees as inputs of (linear) multistage stochastic programs is provided such that optimal values and approximate solution sets remain close to each other. The argument is based on upper bounds of the Lr-distance and the filtration distance, and on quantitative stability results for multistage stochastic programs. The important difference from scenario reduction in two-stage models consists in incorporating the filtration distance. An algorithm is presented for selecting and removing nodes of a scenario tree such that a prescribed error tolerance is met. Some numerical experience is reported.
An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, forward and backward approaches are developed for generating scenario trees out of an initial fan of individual scenarios. Both approaches are motivated by the recent stability result in [15] for optimal values of multistage stochastic programs. They are based on upper bounds for the two relevant ingredients of the stability estimate, namely, the probabilistic and the filtration distance, respectively. These bounds allow to control the process of recursive scenario reduction [13] and branching. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.