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Relaying is a protocol extension for cellular wireless computer networks; in order to utilize radio resources more efficiently, several hops are allowed within one cell. This paper investigates the principle potential of relaying by casting transmission scheduling as a mathematical optimization problem, namely, a linear program. We analyze the throughput gains showing that, irrespective of the concrete scheduling algorithm, performance gains of up to 30\% on average for concrete example networks are achievable.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The perturbation and ADAE index of a degenerated hyperbolic system modelling a heat exchanger
(2007)
The heat exchanger in a heat pump can be modelled by the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.
The periodic QR algorithm is a strongly backward stable method for computing the
eigenvalues of products of matrices, or equivalently for computing the eigenvalues of
block cyclic matrices. The main purpose of this paper is to show that this algorithm
is numerically equivalent to the standard QR algorithm. It will be demonstrated
how this connection may be used to develop a better understanding of the periodic
QR algorithm.
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable
first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later.
We introduce the novel concept of an Online Target Date Assignment Problem (OnlineTDAP) as a general framework for online problems with this nature. Requests for the OnlineTDAP become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.g., an appointment with a customer for a
washing machine repair). The cost at a target date is given by the downstream cost, the optimal cost of processing all requests
at that date w.r.t. some fixed downstream offline optimization problem (e.g., the cost of an optimal dispatch for service
technicians). We provide general competitive algorithms for the OnlineTDAP independently of the particular downstream problem,
when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the particular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
We consider simple models of financial markets with less and better
informed investors described by a smaller and a larger filtration on a
general stochastic basis that describes the market dynamics, including
continuous and jump components. We study the relation between different forms of non existance of arbitrage and the characteristics of the stochastic basis under the different filtrations. This is achieved through the analysis of the properties of the numéraire portfolio. Furthermore, we focus on the problem of calculating the additional logarithmic utility of the better informed investor in terms of the Shannon antropy of is additional information. The information drift, i.e. the drift to eliminate in order to preserved the martingale property in the larger filtration terms out to be the crucial quantity needed to tackle these problems. We show that the expected
ed logarithmic utility increment due to better information equals its Shannon
entropy also in case of a pure jump basis with jumps that are quadratically
hedgeable, and so extend a similar result known for bases consisting of
continuous semimartingales. An example illustrates that the equality may
not persist if both continuous and jump components are present in the
underlying.
In the planning process of railway companies, we propose to integrate important
decisions of network planning, line planning, and vehicle scheduling into the task of periodic
timetabling. From such an integration, we expect to achieve an additional potential for
optimization.
Models for periodic timetabling are commonly based on the Periodic Event Scheduling
Problem (PESP). We show that, for our purpose of this integration, the PESP has to be extended
by only two features, namely a linear objective function and a symmetry requirement.
These extensions of the PESP do not really impose new types of constraints, because practitioners
have already required them even when only planning timetables autonomously without
interaction with other planning steps.
The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces
(2012)
The paper continues the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr\"odinger operators and their
exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order $\vartheta$ for $\vartheta<3/4$. The bound $3/4$ is best possible and can neither be reached nor surpassed. Such results are important for the study
of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.
The minimization of an L^{\infty}-functional subject to an elliptic PDE and state constraints
(2008)
We study the optimal control of a maximum-norm objective functional subjected to an elliptic-type PDE and pointwise state constraints. The problem is transformed into a problem where the non-differentiable L^{\infty}-norm in the functional will be replaced by a scalar variable and additional state constraints. This problem is solved by barrier methods. We will show the existence and convergence of the central path for a class of barrier functions. Numerical experiments complete the presentation.
Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We
identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of SCIP and CPLEX. We make use of the complemented mixed integer rounding framework (cMIR) but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of MIPs coming from network design problems
by a factor of two on average.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
This paper introduces the line connectivity
problem, a generalization of the Steiner tree problem and a
special case of the line planning problem. We study its complexity and
give an IP formulation in terms of an exponential number of
constraints associated with "line cut constraints". These inequalities
can be separated in polynomial time. We also generalize the Steiner
partition inequalities.
We derive a full asymptotic expansion for call option prices and a third order approximation for implied volatility in the large-time, large log-moneyness regime for a general exponential Levy model, by extending the saddlepoint argument used in Forde,Jacquier & Mijatovic for the Heston model. As for the Heston model, there are two special log-moneyness values where the call option asymptotics are qualitatively different, and we use an Edgeworth expansion to deal with these cases. We also characterise the behaviour of the implied volatility skew at large-maturities; in particular we show that the derivative of the dimensionless implied variance with respect to log-moneyness exists and is less than or equal to 4 in the large-maturity limit, which is consistent with the bound on the right and left-side derivative given in Rogers&Tehranchi.
We investigate properties of a certain countably infinite graph called the
infinite locally random graph, written R_N. The graph R_N arises in the study
of models for massive, self-organizing networks like the web-graph. We
characterize the isomorphism type of R_N as the limit of a random process, and
via a domination elimination ordering. We prove that R_N satisfies vertex
deletion properties generalizing inexhaustibility. As is the case for the
infinite random graph R, R_N has a universal automorphism group and
endomorphism monoid. Unlike R, R_N isometrically embeds all finite graphs.
The hypergraph assignment problem (HAP) is the generalization of assignments
from directed graphs to directed hypergraphs. It serves, in particular,
as a universal tool to model several train composition rules in vehicle rotation
planning for long distance passenger railways. We prove that even for problems
with a small hyperarc size and hypergraphs with a special partitioned structure
the HAP is NP-hard and APX-hard. Further, we present an extended integer
linear programming formulation which implies, e. g., all clique inequalities.
In this paper we consider the first exit problem of an overdamped
Lévy driven particle in a confining potential. We survey results
obtained in recent years from our work on the Kramers' times for
dynamical systems of this type with Lévy perturbations containing
heavy, and exponentially light jumps, and compare them to the well
known case of dynamical systems with Gaussian perturbations. It
turns out that exits induced by Lévy processes with jumps are
always essentially faster than Gaussian exits.
The goal of this paper is to show how the heat treatment of steel can be modelled in terms of a mathematical optimal
control problem. The approach is applied to laser surface hardening and the cooling of a steel slab including
mechanical effects. Finally, it is shown how the results can be utilized in industrial practice by a coupling with
machine-based control.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a
spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs--Thomson law in a weak generalized BV-formulation.
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems
for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation.
A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for the classification for the computational complexity of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism.
We prove that this result remains valid for countably categorical structures, and prove that every countably categorical structure has a core, which is unique up to isomorphism, and which is again countably categorical. We thus reduced the classification for the complexity of constraint satisfaction problems with countably categorical templates to the classifiaction for constraint satisfaction problems where the templates are countably categorical cores. We also show that a core of a countably categorical structure Gamma is model complete, and therefore universal-existential axiomatizable. If Gamma contains all primitive positive definable relations, then the core of Gamma admits quantifier elimination. We discuss consequences for constraint satisfaction with countably categorical templates.
The paper addresses primal interior point method for state constrained PDE optimal
control problems. By a Lavrentiev regularization, the state constraint is transformed to a mixed
control-state constraint with bounded Lagrange multiplier. Existence and convergence of the central
path are established, and linear convergence of a short-step pathfollowing method is shown. The
behaviour of the regularizations are demonstrated by numerical examples.
In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution.
Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.
Single-hop WDM networks with a central Passive Star Coupler (PSC), as well as single-hop networks with
a central Arrayed-Waveguide Grating (AWG) and a single transceiver at each node, have been extensively
studied as solutions for the quickly increasing amounts of unicast and multicast traffic in the metropolitan
area. The main bottlenecks of these networks are the lack of spatial wavelength reuse in the studied PSC
based networks and the single transceiver in the studied AWG based metro WDM networks. In this paper
we develop and evaluate the FT EE ???? FREE AWG network, which is based on a central AWG and has arrays
of fixed-tuned transmitters and receivers at each node. Transceiver arrays are a mature technology, making
the proposed network practical. In addition, the transmitter arrays allow for high speed signaling over the
AWG while the receiver arrays relieve the receiver bottleneck arising from multicasting in conjunction with
spatial wavelength reuse on the AWG. Our results from probabilistic analysis and simulation indicate that
the FTEE ???? FREE AWG network gives particularly good throughput-delay performance for a mix of unicast
and multicast traffic.
Single–hop WDM networks with a central Passive
Star Coupler (PSC), as well as single–hop networks with a central
Arrayed–Waveguide Grating (AWG) and a single transceiver at
each node, have been extensively studied as solutions for the
quickly increasing amounts of unicast and multicast traffic in
the metropolitan area. The main bottlenecks of these networks
are the lack of spatial wavelength reuse in the studied PSC based
networks and the single transceiver in the studied AWG based
metro WDM networks. In this paper we develop and evaluate
the FT? ? FR? AWG network, which is based on a central
AWG and has arrays of fixed–tuned transmitters and receivers
at each node. Transceiver arrays are a mature technology, making
the proposed network practical. In addition, the transmitter
arrays allow for high speed signaling over the AWG while
the receiver arrays relieve the receiver bottleneck arising from
multicasting in conjunction with spatial wavelength reuse on the
AWG. Our results from probabilistic analysis and simulation
indicate that the FT? ? FR? AWG network gives particularly
good throughput–delay performance for multicast traffic with
small multicast group sizes or localized destination nodes, as
well as for a mix of unicast and multicast traffic.
Starting from the logical description of gene regulatory networks developed by R.~Thomas, we introduce an enhanced modelling approach
based on timed automata. We obtain a refined qualitative description of the dynamical behaviour by exploiting not only information on ratios of kinetic parameters related to synthesis and decay, but also constraints on the time delays associated with the operations of the system. We develop a formal framework for handling such temporal constraints using timed automata, discuss the relationship with the original Thomas formalism, and demonstrate the potential of our approach by analysing an illustrative gene regulatory network of bacteriophage~$\lambda$.
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a~priori $L^1$ term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the $L^1$ right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.
Periodic timetabling for railway networks is usually modeled by the Periodic Event Scheduling
Problem (PESP). This model permits to express many requirements that practitioners impose
on periodic railway timetables. We discuss a requirement practitioners are asking for, but which,
so far, has not been the topic of mathematical studies: the concept of symmetry.
Several motivations why symmetric timetables might seem promising will be given. However,
we provide examples showing that symmetry leads to suboptimality.
To integrate symmetry into the graph model of the PESP, there are many obstacles to overcome.
Nevertheless, adding symmetry requirements to mixed-integer programming formulations
explicitly, enables MIP solvers such as CPLEX
to terminate earlier with good solutions.
We consider the design of a logical network topology, together with node hardware, link capacities, and a survivable routing of demands. In addition to all single node failures, the routing must also survive multiple logical link failures caused by single failures in the underlying physical network. Furthermore, the number of logical links supported by a physical link is bounded. We propose an integer linear programming model for this design problem, together with a branch-and-cut based solution approach combined with column generation. The model and algorithm are tested on three real-world test instances, and preliminary results are given.
Supporting Global Numerical Optimization of Rational Functions by Generic Symbolic Convexity Tests
(2010)
Convexity is an important property in nonlinear optimization since it allows to apply efficient local methods for finding global solutions. We propose to apply symbolic methods to prove or disprove convexity of rational functions over a polyhedral domain. Our algorithms reduce convexity questions to real quantifier elimination problems. Our methods are implemented and publicly available in the open source computer algebra system REDUCE. Our long term goal is to integrate REDUCE as a ``workhorse'' for symbolic computations into a numerical solver.
A thorough convergence analysis of the Control Reduced Interior Point
Method in function space is performed. This recently proposed method is a
primal interior point pathfollowing scheme with the special feature, that the
control variable is eliminated from the optimality system. Apart from global
linear convergence we show, that this method converges locally almost quadratically,
if the optimal solution satisfies a certain non-degeneracy condition. In
numerical experiments we observe, that a prototype implementation of our
method behaves as predicted by our theoretical results.
An optimal control problem for a 2-d elliptic equation is investigated with pointwise control constraints.
This paper is concerned with discretization of the control by piecewise constant functions. The state and
the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous
optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these
approximations have convergence order h2.
Second-order sufficient optimality conditions are established for the optimal control
of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In
contrast to former publications on this subject, the cone of critical directions is the smallest possible in the
sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory
is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic
boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one,
respectively.
In this paper we study the shape and growth of structured pseudospectra
for small matrix perturbations of the form $A \leadsto
A_\Delta=A+B\Delta C$, $\Delta \in \DD$, $\|\Delta\|\leq \delta$.
It is shown that the properly scaled pseudospectra components converge
to non-trivial limit sets as $\delta$ tends to 0.
We discuss the relationship of these limit sets with $\mu$-values and
structured eigenvalue condition numbers for multiple eigenvalues.
Let $\lambda$ be a nonderogatory eigenvalue of $A \in \C^{n \times
n}$. The sensitivity of $\lambda$ with respect to matrix
perturbations
$A \leadsto A+\Delta,\Delta \in \DD$, is measured by the structured
condition number $\kappa_\DD(A,\lambda)$. Here $\DD$ denotes the set
of admissible perturbations. However, if $\DD$ is not a vector space
over $\C$ then $\kappa_\DD(A,\lambda)$ provides only incomplete
information about the mobility of $\lambda$ under small
perturbations from $\DD$. The full
information is then given by a certain set $K_\DD(x,y)\subset \C$
which depends on $\DD$ and
a pair of normalized right and left eigenvectors $x,y$. In this paper
we study the sets $K_\DD(x,y)$ and obtain methods for computing
them.
In particular we show that $K_\DD(x,y)$ is an ellipse in some
important cases.
Structured eigenvalue conditioning and backward error of a class of polynomial eigenvalue problems
(2007)
Characterisations of simple eigenvalues of complex matrix polynomials with *-even/odd and *-palindromic/antipalindromic structures that have the same normwise condition number with respect to structure preserving and arbitrary perturbations are obtained. Here * denotes either the transpose T or the conjugate tranpose *. In the process we obtain formulae for the normwise structured condition number of simple eigenvalues of T-palindromic/antipalindromic and *-even/odd polynomials. Moreover, conditions under which the normwise structured backward error of approximate eigenvalues of such polynomials is equal to the unstructured error are also derived. These lead to complete characterisations of approximate eigenvalues that have the same structured and unstructured backward errors for the *-even/odd and T-even/odd polynomials.
This paper investigates the effect of structure-preserving perturbations on the eigenvalues
of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to
structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product.
Bounds and computable expressions for structured eigenvalue condition numbers are derived for
these classes of matrices, which include complex symmetric, pseudo symmetric, persymmetric, skewsymmetric,
Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under mild
assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers
are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of
incorporating structure varies greatly with the structure. We identify Lie algebras for which structure
does not affect the eigenvalue condition number.
Structured eigenvalue backward errors of matrix pencils and polynomials with palindromic structures
(2014)
We derive formulas for the backward error of an approximate eigenvalue of a *-palindromic
matrix polynomial with respect to *-palindromic perturbations. Such formulas are also obtained
for complex T-palindromic pencils and quadratic
polynomials. When the T-palindromic polynomial is real, then we derive the backward error
of a real number considered as an approximate eigenvalue of the matrix polynomial with
respect to real T-palindromic perturbations.
In all cases the corresponding minimal structure preserving perturbations are obtained as well.
The results are illustrated by numerical experiments. These show that there is
significant difference between the backward errors with respect to structure
preserving and arbitrary perturbations in many cases.
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.