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While seemingly straightforward in principle, the reliable estimation of rate constants is seldom easy in practice. Numerous issues, such as the complication of poor reaction coordinates, cause obvious approaches to yield unreliable estimates. When a reliable order parameter is available, the reactive flux theory of Chandler allows the rate constant to be extracted from the plateau region of an appropriate reactive flux correlation function. However, when applied to real data from single- molecule experiments or molecular dynamics simulations, the reactive flux correlation function requires the numerical differentiation of a noisy empirical correlation function, which can result in an unacceptably poor estimate of the rate and pathological dependence on the sampling interval. We present a modified version of this theory which does not require numerical derivatives, allowing rate constants to be robustly estimated from the time-correlation function directly. We illustrate the approach using single-molecule passive force spectroscopy measurements of an RNA hairpin.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
In this review article we discuss different techniques to solve numerically the
time-dependent Schrödinger equation on unbounded domains.
We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works.
We conclude with several numerical examples from
different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.
A Lavrentiev type regularization technique for
solving elliptic boundary control problems with pointwise state
constraints is considered. The main concept behind this
regularization is to look for controls in the range of the adjoint
control-to-state mapping. After investigating the analysis of the
method, a semismooth Newton method based on the optimality
conditions is presented. The theoretical results are confirmed by
numerical tests. Moreover, they are validated by comparing the
regularization technique with standard numerical codes based on the
discretize-then-optimize concept.
This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.
This paper focuses on
rate-independent damage in elastic bodies. Since the driving energy is nonconvex,
solutions may have jumps as a function of time, and in this situation it is known that the classical concept
of energetic solutions for rate-independent systems
may fail to accurately describe the
behavior of the system at jumps.
Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling
and approximate the model by its viscous regularization.
In fact, the analysis of the latter PDE system presents
remarkable difficulties, due to its highly nonlinear character.
We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with
careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable.
Hence, for the viscous problem we conclude the existence of weak solutions satisfying a
suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis.
The latter leads to the notion of (weak) parameterized
solution to our rate-independent system,
which encompasses the influence of viscosity in the description of the jump regime.
Quasi-Newton methods based on least change secant updating
formulas that solve linear equations $Ax=b$ in $n=\dim(x)=\dim(b)$ steps
can be expected to solve corresponding smooth nonlinear
systems $n$-step quadratically, i.e. with an $r$-order
of $\rho = 2^{1/n} = 1 + 1/n +O(1/n^2)$. The best rate one can
possibly expect on general problems is given by the positive root
$\rho_n$ of $\rho^n(\rho -1)=1$, for which
$\rho_n-1 = \ln(n)/n + O(1/n^2)$. To show that this upper bound is
actually achieved one usually has to impose a priori some kind of
linear independence condition on the sequence of steps taken by the
quasi-Newton iteration in question. Without any such assumptions we
establish in this paper the convergence order $\rho_n$ for the
two-sided rank one formula proposed by Schlenkrich et al in \cite{SGW06}.
It requires the evaluation of adjoint vectors, is invariant with respect
to linear transformations on the variable domain and combines the
properties of bounded deterioration and heredity.
We study a stationary Schrödinger-Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.
In this paper we develop a QR-like algorithm for the palindromic eigenvalue problem $Ax=\lambda A^\adj x$.
We will discuss the two cases that $A^\adj$ denotes the transpose or the conjugate transpose of $A\in\C^{n,n}$.
It is shown that this so-called palindromic QR iteration is equivalent to applying the standard QR algorithm to $A^{-\adj}A$.
Also the concepts of deflation, shifting, and exploiting the invariance of a Hessenberg-type form are adapted.
Moreover, we analyze the problem of reducing a general square matrix to the mentioned Hessenberg-type form
and establish analogies to the Hamiltonian eigenvalue problem.
Finally, we present concrete Hessenberg-type reduction algorithms for special cases.
A posteriori error estimators for non-symmetric eigenvalue model problems are discussed in [Heuveline and Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems, 2001] in the context of the dual-weighted residual method (DWR). This paper directly analyses the variational formulation rather than the non-linear ansatz of Becker and Rannacher for some convection-diffusion model problem and presents error estimators for the eigenvalue error based on averaging techniques. In the case of linear P1 finite elements and globally constant coefficients, the error estimates of the residual and averaging error estimators are refined. Moreover, several postprocessing techniques attached to the DWR paradigm plus two new dual-weighted error estimators are compared in numerical experiments. The first new estimator utilises an auxiliary Raviart-Thomas mixed finite element method and the second exploits an averaging technique in combination with ideas of DWR.
This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as well as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method.
We study a frictionless dynamical contact problem between both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries.
Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.
Large-area organic light-emitting diodes are thin-film multilayer devices that show pronounced self-heating and brightness inhomogeneities at high currents. As these high currents are typical for lighting applications, a deeper understanding of the mechanisms causing these inhomogeneities is necessary. We discuss the modeling of the interplay between current flow, self-heating, and heat transfer in such devices using a system of partial differential equations of thermistor type, that is capable of explaining the development of luminance inhomogeneities. The system is based on the heat equation for the temperature coupled to a p(x)-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p(x)-Laplacian allows to take into account non-Ohmic electrical behavior of the different organic layers. Moreover, we present analytical results on the existence, boundedness, and regularity of solutions to the system. A numerical scheme based on the finite-volume method allows for efficient simulations of device structures.
The line planning problem is one of the fundamental problems in strategic
planning of public and rail transport. It consists in finding lines
and corresponding frequencies in a transport network such that a given
travel demand can be satisfied. There are (at least) two objectives. The
transport company wishes to minimize operating costs, the passengers
want to minimize travel times. We propose a new multi-commodity
ow model for line planning. Its main features, in comparison to existing
models, are that the passenger paths can be freely routed and
that the lines are generated dynamically. We discuss properties of this
model and investigate its complexity. Results with data for the city of
Potsdam, Germany, are reported.
In this paper we study a certain cardinality constrained packing integer program which is motivated by the problem of dimensioning a cut in a two-layer network. We prove NP-hardness and consider the facial structure of the corresponding polytope. We provide a complete description for the smallest nontrivial case and develop two general classes of facet-defining inequalities. This approach extends the
notion of the well known cutset inequalities to two network layers.
UMTS radio network evaluation and design are currently important issues for telecommunication operators.
We present a novel view on network evaluation. The recent dimension reduction approach is
generalized to an analytical approximation of the network's general performance based on average traffic.
The pivot is an average coupling matrix that captures the essential coverage and cell coupling properties of
the radio network. Based on this new evaluation method, we present new optimization methods, namely
a new optimization model based on designing the generalized average coupling matrix and an efficient
1-opt local search. We give preliminary computational results that show the potential of our methods on
realistic data.
This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in [2] and describes how to circumvent it. This completes the proof of Corollary 2.4 in [2] and hence of the main result in [2], which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity.
Recently, Khuller, Moss and Naor presented a greedy algorithm for the budgeted maximum coverage problem. In this note, we observe that this algorithm also approximates a special case of set-union knapsack problem within a constant factor. In the special case, an element is a member of less than a constant number of subsets. This guarantee naturally extends to densest k-subgraph problem on graphs of bounded degree.
We extend earlier work on scenario reduction by relying directly on Fortet-Mourier metrics instead of using upper bounds given in terms of mass transportation problems. The importance of Fortet-Mourier metrics for quantitative stability of two-stage models is reviewed and some numerical results are also provided.
We review some known results for POD model reduction applied to ODEs. Then, these
results are generalized to several types of DAEs. We provide algorithms for the
model reduction and error bounds for the reduced order models. Some limits of
the approach are pointed out and alternative methods for reduced order subspace approximation
are presented. The POD approach is tested and evaluated for a medium sized DAE example
from multibody dynamics.
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-calles Wentzell boundary condition. First, we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable Lojasiewicz-Simon-type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms $f$, $g$ are real analytic. Moreover, we provide an estimate for the convergence rate.
A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity
(2012)
In this paper, we prove the existence
and global boundedness from above for a solution to an
integrodifferential model for nonisothermal multi-phase
transitions under nonhomogeneous third type boundary conditions.
The system couples a quasilinear internal energy balance
ruling the evolution of the absolute temperature with a vectorial
integro-differential inclusion governing the vectorial
phase-parameter dynamics. The specific heat and the heat
conductivity k are allowed to depend both on the order parameter
$\chi$ and on the absolute temperature $\teta$ of the system, and
the convex component of the free energy may or may not be
singular. Uniqueness and continuous data dependence are
also proved under additional assumptions.
We prove the existence, uniqueness, thermodynamic consistency,
global boundedness from both above and below, and continuous data
dependence for a strong solution to an
integrodifferential model for nonisothermal phase transitions
under nonhomogeneous mixed boundary conditions.
The specific heat is allowed to depend on the order parameter,
and the convex component of the free energy may or may not
be singular.
For a nice Markov process such as Brownian motion on a bounded domain, we introduce a non-linear potential operator defined in terms of running
suprema, and we prove a non-linear Riesz representation of a given function as
the sum of a harmonic function and a non-linear potential. The proof involves
a family of optimal stopping problems in analogy to the general construction
of Bank and El Karoui [3], but here the analysis is carried out in terms of
probabilistic potential theory.
The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster near the zero eigenvalues. For
the solution of the eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which uses a hybrid method that combines the Jacobi-Davidson, the shift-invert residual Arnoldi and nonlinear Arnoldi methods to compute the clustered eigenvalues. Numerical results illustrate that the method is robust even for the case of computing many eigenvalues in very large problems.
In this paper we propose a new finite element realization of the Perfectly Matched
Layer method (PML-method). Our approach allows to deal with a wide class of
polygonal domains and with certain types of inhomogeneous exterior domains.
Among the covered inhomogeneities are open waveguide structures playing an essential
role in integrated optics. We give a detailed insight into implementation
aspects. Numerical examples show exponential convergence behavior to the exact
solution with the thickness of the PML sponge layer.
Modelling incompressible ideal fluids as a finite collection of
vortex filaments is important in physics (super-fluidity, models for the
onset of turbulence) as well as for numerical algorithms used in computer
graphics for the real time simulation of smoke. Here we introduce
a time-discrete evolution equation for arbitrary closed polygons in 3-
space that is a discretisation of the localised induction approximation of
filament motion. This discretisation shares with its continuum limit the
property that it is a completely integrable system. We apply this polygon
evolution to a significant improvement of the numerical algorithms
used in Computer Graphics.
A generalization of the method of Chu, Liu and Mehrmann
for the computation of the Hamiltonian real Schur form is presented.
The new method avoids some of the difficulties that may arise when
a Hamiltonian matrix has tightly clustered groups of eigenvalues.
A detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method.
O&D revenue management (RM)
– either leg-based or PNR-based – has become
a standard in the airline industry. In this paper,
we present a new approach to O&D RM which
does not make any assumptions on demand distributions
or on the correlations of the booking
process. Protection levels are determined for all
origin destination itineraries, fare classes, points
of sale and data collection points (DCPs). This
approach to the seat inventory problem is modelled
as a multistage stochastic program, where
its stages correspond to the DCPs of the booking
horizon. The stochastic passenger demand
process is approximated by a scenario tree generated
from historical data by a recursive scenario
reduction procedure. The stochastic program
represents a specially structured large scale
LP that may be solved by standard LP software
(e.g. CPLEX). Preliminary numerical experience
is reported.
Flux coupling analysis is a method to identify blocked and coupled reactions
in a metabolic network at steady state. We present a new approach to
flux coupling analysis, which uses a minimum set of generators of the steady state
flux cone. Our method does not require to reconfigure the network by splitting
reversible reactions into forward and backward reactions.
By distinguishing different types of reactions (irreversible, pseudo-irreversible,
fully reversible), we show that reaction coupling relationships can only hold
between certain reaction types. Based on this mathematical analysis, we propose
a new algorithm for flux coupling analysis.
We propose a new approach to competitive analysis by introducing the novel concept of online approximation schemes. Such scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our method can also be applied to many other problems and yields a completely new and interesting view on online algorithms.
Dominance constraints are logical descriptions of trees. Efficient algorithms for the subclass of normal dominance constraints were recently proposed. We present a new and simpler graph algorithm solving these constraints more efficiently, in quadratic time per solved form. It also applies to weakly normal dominance constraints as needed for an application to computational linguistics. Subquadratic running time can be achieved employing decremental graph biconnectivity algorithms.
A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on an adapted mesh which consists of triangles and parametric quadrilaterals in 2D. The number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the distance of the neighboring inclusions. While being related to network methods, the approach can tackle more general settings. We present an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. Moreover, it is demonstrated that the assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.
We consider the linear-quadratic optimal control problem for a controlled differential
algebraic equation (DAE). Under minimal assumptions on the DAE concerning index and regularity
it will be possible to prove that the sufficient optimality condition given in papers of G. Kurina
and R. März is also a necessary condition. This condition includes the solution of an appropriate
boundary value problem and, in the special case of an explicit ordinary differential equation, the
condition is equal to the well known necessary and sufficient condition of the classical linear-quadratic
optimal control problem.
The purpose of the paper is to apply monotone multigrid methods
to static and dynamic biomechanical contact problems.
In space, a finite element method involving a mortar
discretization of the contact conditions is used.
In time, a new contact--stabilized Newmark scheme is presented.
Numerical experiments for a two body Hertzian contact problem
and a biomechanical knee problem are reported.
In this paper we discuss the numerical
solution of projected generalized Lyapunov
equations using the matrix sign function
method. Such equations arise in stability
analysis and control problems for
descriptor systems including model reduction
based on balanced truncation. It is known
that the matrix sign function method applied
to a matrix pencil $\lambda E-A$ converges
if and only if $\lambda E-A$ is of index at
most two. The convergence is quadratic if
$E$ is nonsingular, and it is linear,
otherwise. We will propose a modification
of the matrix sign function method that
converges quadratically for pencils of
arbitrary index. Numerical examples will be
presented to demonstrate the properties of
the modified method.
We describe a prototype web service for model reduction of very large-scale linear systems, with
dimension in the order of millions of states, that includes a user-friendly interface designed so that the computation
can be easily performed via the HTTP protocol. Access via a web browser isolates the user of the service from the
complexities of installing and using the parallel model reduction codes and the maintenance of the hardware. In case
the routines are found to be appropriate for the problem the user needs to solve, the library can be then downloaded
and installed on the user’s own computing resources.
This paper illustrates the major issues of the access procedure by means of graphical examples, and describes
the structure and implementation of the remote model reduction service. The service is offered in a cluster of Linux
machines.
We present a thermodynamically consistent model to describe the austenite-ferrite phase transition in steel. We consider the influence of the mechanical displacement field due to eigenstrains caused by volumetric expansions. The model equations are derived in a systematical framework. They are based on the conservation laws for mass and momentum and the second law of thermodynamics. By means of numerical computations for a simplified interface-controlled model, we examine the influence of the mechanical contributions to the transformation kinetics and the equilibrium states.
We propose a model for phase transformations that are driven by changes in the temperature. We consider the temperature as a prescribed prescribed quantity like an applied load. The model is based on the energetic formulation for rate-independent systems and thus allows for finite-strain elasticity. Time-dependent Dirichlet boundary conditions can be treated by decomposing the deformation as a composition of a given deformation satisfying the time-dependent boundary conditions and a part coinciding with the identity on the Dirichlet boundary.
In this paper, a model for (joint) dynamic chance constraints is proposed and applied to an optimization problem in water reservoir management. The model relies on discretization of the decision variables but keeps the probability distribution
continuous. Our approach relies on calculating probabilities of rectangles which is particularly useful in the presence of independent random variables but works for a moderate number of stages equally well in case of correlated variables. Numerical results are provided for two and three stages.
The nonlinear Schrödinger equation based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function such that the resonances are naturally accounted for. This approach allows us to derive a simple non-envelope model for short pulses propagating in one spatial dimension. This model is further investigated numerically and analytically.
A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints
(2013)
We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.
Markov State Models (MSMs) are widely used to represent molecular
conformational changes as jump-like transitions between subsets of the conformational
state space. However, the simulation of peptide folding in explicit water is
usually said to be unsuitable for the MSM framework. In this article, we summarize
the theoretical background of MSMs and indicate that explicit water simulations do
not contradict these principles. The algorithmic framework of a meshless conformational
space discretization is applied to an explicit water system and the sampling
results are compared to a long-term molecular dynamics trajectory. The meshless
discretization approach is based on spectral clustering of stochastic matrices (MSMs)
and allows for a parallelization of MD simulations. In our example of Trialanine we
were able to compute the same distribution of a long term simulation in less computing
time.
Automatic, or algorithmic, differentiation addresses the need for the accurate
and efficient calculation of derivative values in scientific computing. To this
end procedural programs for the evaluation of problem-specific functions are
transformed into programs that also compute the required derivative values
at the same numerical arguments in floating point arithmetic. Disregarding
many important implementation issues, we examine in this article complexity
bounds and other more mathematical aspects of the program transformation
task sketched above.
This study presents a differential equation model for the feedback mechanisms between
Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing
Hormone (LH), development of follicles and corpus luteum, and the production of estradiol
(E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual
cycle. In contrast to other models, this model does not involve delay differential equations
and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster
simulation times and efficient parameter identification. These steps were essential to tackle
the task of developing a mathematical model for the administration of GnRH analogues. The
focus of this paper is on model development for GnRH receptor binding and the integration of
a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH
antagonist Cetrorelix into the menstrual cycle model. The final mathematical model describes
the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women.
Moreover, it correctly predicts the changes in the cycle following single and multiple dose
administration of Nafarelin or Cetrorelix at different stages in the cycle.
A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to
render the steel austenitic, followed by quenching to form
a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled
with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions.
We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.
Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be
solved in polynomial time. The weighted matching problem is to find a matching in an
edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm + n2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3 . This improves the previously best performance ratio of 1/2. Our algorithm is not only of theoretical interest but because it is easy to implement and the constants involved are quite small it is also useful in practice.
We study the effect of investor inertia on stock price fluctuations with a market microstructure
model comprising many small investors who are inactive most of the time.
It turns out that semi-Markov processes are tailor made for modelling inert investors.
With a suitable scaling, we show that when the price is driven by the market imbalance,
the log price process is approximated by a process with long range dependence
and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently,
investor inertia may lead to arbitrage opportunities for sophisticated market
participants. The mathematical contributions are a functional central limit theorem for
stationary semi-Markov processes, and approximation results for stochastic integrals
of continuous semimartingales with respect to fractional Brownian motion.
Constraint-based analysis of metabolic networks has become a widely used approach in computational systems biology.
In the simplest form, a metabolic network is represented by a stoichiometric matrix and thermodynamic information on the irreversibility of certain reactions.
Then one studies the set of all steady-state flux vectors satisfying these stoichiometric and thermodynamic constraints.
We introduce a new lattice-theoretic framework for the computational analysis of metabolic networks, which focuses on the support of the flux vectors,
i.e., we consider only the qualitative information whether or not a certain reaction is active, but not its specific flux rate.
Our lattice-theoretic view includes classical metabolic pathway analysis as a special case, but turns out to be much more flexible and general,
with a wide range of possible applications.
We show how important concepts from metabolic pathway analysis, such as blocked reactions, flux coupling, or elementary modes, can be generalized to arbitrary lattice-based models. We develop corresponding general algorithms and present a number of computational results.
We consider the problem of designing a network that employs a non-bifurcated shortest path routing
protocol. The network's nodes and the set of potential links are given together with a set of forecasted endto-
end traffc demands. All relevant hardware components installable at links or nodes are considered. The
goal is to simultaneously choose the network's topology, to decide which hardware components to install
on which links and nodes, and to find appropriate routing weights such that the overall network cost is
minimized.
In this paper, we present a mathematical optimization model for this problem and an algorithmic solution
approach based on a Lagrangian relaxation. Computational results achieved with this approach for several
real-world network planning problems are reported.
Stewart's recently introduced Krylov-Schur algorithm
is a modification of the implicitly restarted Arnoldi algorithm which
employs reordered Schur decompositions to perform restarts and de-
ations in a numerically reliable manner. This paper describes a variant
of the Krylov-Schur algorithm suitable for addressing eigenvalue
problems associated with products of large and sparse matrices. It
performs restarts and de
ations via reordered periodic Schur decompositions
and, by taking the product structure into account, it is
capable to achieve qualitatively better approximations to the eigenvalues
of small magnitude.
Stewart's recently introduced Krylov-Schur algorithm
is a modification of the implicitly restarted Arnoldi algorithm which
employs reordered Schur decompositions to perform restarts and de-
ations in a numerically reliable manner. This paper describes a variant
of the Krylov-Schur algorithm suitable for addressing eigenvalue
problems associated with products of large and sparse matrices. It
performs restarts and de
ations via reordered periodic Schur decompositions
and, by taking the product structure into account, it is
capable to achieve qualitatively better approximations to the eigenvalues
of small magnitude.
An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the "regularity" of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.
Given a directed graph D = (V;A), we consider its cycle space CD, i.e. the vector
subspace of Q|A| spanned by the incidence vectors of the oriented cycles of D. An
oriented cycle of D is just any cycle of the underlying undirected graph of D along
with an orientation; its incidence vector is 0 on the arcs not included, while, for the
included arcs, it is +1 on the arcs oriented according to the orientation and -1 on
the arcs going backward. Assume a nonnegative weight wa ? R+ is associated to
each arc a of D. We can extend the weighting w to subsets F of A and to families F
of such subsets by dening w(F) := ?f?F w(f) and w(F) := ?F?F w(F). Given
the pair (D;w), we are interested in computing a minimum weight basis of CD.
This problem is strongly related to the classical problem of computing a minimum
cycle basis of an undirected graph. In 1987, Horton developed the first polynomial
time algorithm for computing a minimum cycle basis of an undirected graph. As for
directed graphs, the first algorithm for computing a minimum directed cycle basis
is due to Kavitha and Mehlhorn. Its asymptotic complexity is ~O (m4n).
In this paper, we show how the original approach of Horton can be actually pursued
also in the context of directed graphs, while retaining its simplicity. This both
allows for a practical ~O(m4n) adaptation of Horton's original algorithm requiring
only minor modifications in the actual code and for a more involved ~O(mw+1n)
solution. At the end, we discuss the applicability of this approach to more specialized
classes of directed cycle bases, namely, integral cycle bases and generalized
fundamental cycle bases.
We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems
with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces.
The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.
We consider a particle constrained to a submanifold ? of the
configuration space Rm. Using that the notion of holonomic constraints coincides
with integrability of the corresponding vector field, we show how this property
naturally determines local coordinates on ?. We give a rigorous justification for
the calculation of the mean force along a constrained coordinate, and we provide
a concise geometrical interpretation of the different contributions to the mean
force in terms of the unconstrained vector field and extrinsic curvature properties
of ? in Rm. Our approach gives rise to a Hybrid Monte-Carlo based algorithm
that can be used to compute the mean force acting on selected coordinates in the
context of thermodynamic free energy statistics.
A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework
(2007)
We give a mathematically rigorous definition of a grid for algorithms solving
partial differential equations. Unlike previous approaches, our grids have a
hierarchical structure. This makes them suitable for geometric multigrid
algorithms and hierarchical local grid refinement. The description is also
general enough to include geometrically nonconforming grids. The definitions
in this article serve as the basis for an implementation of an abstract grid
interface as C++ classes in the DUNE.
In a companion paper [Matheon-Preprint 403] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this
definition we derive an efficient interface specification as a set of C++ classes.
This interface separates the applications from the grid data structures.
Thus, user implementations become independent of the underlying grid
implementation. Modern C++ template techniques are used to provide an
interface implementation without big performance losses.
The implementation is realized as part of the
software environment DUNE.
Numerical tests demonstrate the flexibility and the efficiency of our approach.
We propose a generalization of the Structured Doubling Algorithm (SDA) to compute invariant subspaces
of structured matrix pencils
that arise in the context of solving linear quadratic optimal control problems.
The new algorithm is
designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because
the stable and unstable invariant subspaces are not well separated (due to eigenvalues near or on the imaginary
axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence
of the method and compare the new method with the classical SDA algorithm as well as some recent structured QR-methods.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
We introduce a forward scheme to simulate backward SDEs. Compared
to existing schemes, we avoid high order nestings of conditional
expectations backwards in time. In this way the error, when
approximating the conditional expectation, in dependence of the
time partition is significantly reduced. Besides this generic
result, we present an implementable algorithm and provide an error
analysis for it. Finally, we demonstrate the strength of the new
algorithm by solving some financial problems numerically.
We present a new solver for large-scale two-body contact problems in nonlinear elasticity. It is based on an SQP-trust-region approach.
This guarantees global convergence to a first-order critical point of
the energy functional. The linearized contact conditions are
discretized using mortar elements. A
special basis transformation known from linear contact problems
allows to use a monotone multigrid solver for the inner quadratic programs.
They can thus be solved with multigrid complexity. Our algorithm
does not contain any regularization or penalization parameters,
and can be used for all hyperelastic material models.
A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static
elasto-plasticity with kinematic or isotropic hardening is considered. The associated path is induced
by a combined Moreau-Yosida / Tichonov regularization of the dual problem.
The sequence of solutions to the regularized problems is shown to converge strongly to the optimal
displacement-stress-strain triple of the original elasto-plastic contact problem in the space-continuous setting.
This property relies on the density of the intersection of certain convex sets which is shown as well.
It is also argued that the mappings associated with the resulting problems are Newton- or slantly differentiable.
Consequently, each regularized subsystem can be solved mesh-independently at a local
superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and
a numerical validation of the theoretical results is given.
This paper concerns the shortest path problem for a network in which arc costs can vary with
time, each arc has a transit time, parking with a corresponding
time-varying cost is allowed at the nodes, and time is modeled as a continuum. The resulting problem is called the {\em continuous-time dynamic shortest path problem}, which is well studied in the literature. However, the problem appears as a subproblem when one wishes to test, via an algorithm for dynamic shortest paths, the presence of negative cycles in the residual network in order to develop continuous-time analogues of several well-known optimality conditions for continuous-time dynamic network flow problems. But, in general, the residual network contains arcs with negative transit times and hence the results in the literature are useless for these purposes since all results are based on the assumption of positive transit times.
In this paper, we relax this condition to allow negative transit times. We study a corresponding linear program in space of measures and prove the existence of an optimal extreme point solution. Moreover, we define a dual problem and establish a strong duality result that shows under certain assumptions that the value of the linear program equals the value of the dual problem and both values are attained. We also present counterexamples to show that strong duality only holds under these assumptions.
A duality based semismooth Newton framework for solving variational inequalities of the second kind
(2010)
In an appropriate function space setting, semismooth Newton methods are proposed
for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the
second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which
is regularized if necessary. In the latter case, consistency of the regularization with respect to the
original problem is studied. The application of the general framework to specific model problems
including Bingham flows, simplified friction, or total variation regularization in mathematical imaging
is described in detail. Finally, numerical experiments are presented in order to verify the theoretical
results.
We prove an optimal regularity result in the two dimensional theory of soft ferromagnetic films. The associate Euler-Lagrange equation
is given by a microlocally degenerate variational inequality involving
fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a H\"older estimate for the uniquely determined gradient projection of the magnetization field.
In incompressible flows with vanishing
normal velocities at the boundary, irrotational forces in the momentum
equations should be balanced
completely by the pressure gradient.
Unfortunately, nearly all available discretization methods for incompressible flows violate this property.
The origin of the problem is that discrete velocity approximations
of incompressible flows are usually not
divergence-free. Hence, the use of divergence-free velocity reconstructions is
proposed wherever an $L^2$ scalar product appears in the discrete
variational formulation.
The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids.
It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element
increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic and holomorphic functions, discrete mean curvature, and discrete minimal surfaces.
We discuss the solution of linear second order differential-algebraic equations with variable coefficients.
Since index reduction and order reduction for higher order higher index differential-algebraic systems do not commute, appropriate index reduction methods for higher order DAEs are required.
We present an index reduction method based on derivative arrays that allows to determine
an equivalent second order system of lower index in a numerical computable way.
For such an equivalent second order system
an appropriate order reduction method allows to formulate a suitable first order DAE system of low index
that has the same solution components as the original second order system.
We consider the problem of satisfying the maximum number of constraints
of an instance of the Periodic Event Scheduling Problem (PESP). This is
a key issue in periodic railway timetable construction, and has many other applications,
e.g. for traffic light scheduling.
We generalize two (in-) approximability results, which are known for MAXIMUM-
K-COLORABLE-SUBGRAPH. Moreover, we present a deterministic combinatorial
polynomial time algorithm. Its output violates only very few constraints
for five real-world instances.
A coupling of discrete and continuous optimization to solve kinodynamic motion planning problems
(2014)
This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Cea. A theorem is formulated
which allows a rigorous proof of the shape differentiability without the usage of material derivative;
the domain expression is automatically obtained and the boundary expression is easy to derive.
Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission
problem. Using a Gagliardo penalization the existence of optimal shapes is established.
A mathematical model for instationary magnetization
processes is considered, where the underlying spatial domain
includes electrically conducting and nonconducting regions. The
model accounts for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil. By a theorem of Showalter on degenerate parabolic
equations, theorems on existence, uniqueness, and regularity of the
solution to the associated Maxwell integrodifferential system are
proved.
As an alternative to Newton's method an approach based on singularity
theory can be exploited for computing a simple eigenvalue and corresponding
eigenvectors of a nonnormal matrix $A$ in a stable way, see
[Schwetlick/L\"osche, 2000].
In the paper it is shown by constructing a counterexample with an singular
singularity system, that a straightforward extension of this technique
to the computation of invariant subspaces of dimension $p>1$
will not work, in general. The finding of this counterexample required a
detailed study of the linear block singularity operator.
A primal interior point method for control constrained optimal control problems
with PDE constraints is considered. Pointwise elimination of the control
leads to a homotopy in the remaining state and dual variables, which is addressed
by a short step pathfollowing method. The algorithm is applied to the
continuous, infinite dimensional problem, where discretization is performed
only in the innermost loop when solving linear equations. The a priori elimination
of the least regular control permits to obtain the required accuracy with
comparatively coarse meshes. Convergence of the method and discretization
errors are studied, and the method is illustrated at two numerical examples.
We derive a continuous time approximation of the evolutionary market selection model of Blume &
Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous
time approximation equals the solution of an integral equation
in a random environment. For constant asset returns, the integral equation reduces to an autonomous
ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to
Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.
This paper proposes a new mathematical model for the open pit mine planning problem,
based on continuous functional analysis. The traditional models for this problem have been
constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial
and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach
which allows for a refined imposition of slope constraints associated with geotechnical stability.
The model introduced here is posed in a suitable functional space, essentially the
real-valued functions that are Lipschitz continuous on a given two dimensional bounded region.
We derive existence results and investigate some qualitative properties of the solutions
The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors,
as the diffusion coefficient in the equation governing
the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove
a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.
A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control
(2008)
This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.
We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLIB we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.
We consider real or complex palindromic pencils, i.e., pencils of the form $A-\lambda A^\star$,
where $A^{\star}$ denotes either the transpose $A^T$ or the conjugate transpose $A^*$.
Structured canonical forms for these pencils are derived that reveal complete spectral information.
Moreover, this canonical form is used to provide necessary and sufficient conditions for the existence of palindromic factorizations $B=A^{-\star}A$ of a given square matrix $B$.
In particular, we answer the questions when symplectic matrices allow palindromic factorization and when a matrix having a palindromic factorization is similar to a symplectic one.
Using a standard first-order optimality condition for nonsmooth optimization problems, a general framework for a descent method is developed. This setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived. Global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments.
We solve Skorokhod's embedding problem for Brownian mostion with linear drift $(W_t + \kappa t)_{t\ge 0}$ by means of techniques of stochastic control theory. The search for a stopping time $T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ processing the first moment is based on solutions of backward stochastic differential equations of quadratic type. Theis new approach generalizes an approach by Bass [BAS] of the classical version of Skorokhod's embedding problem using martingale representation techniques.
We solve Skorokhod's embedding problem for Brownian motion with linear drift $(W_t+ \kappa t)_{t\geq 0}$ by means of techniques of stochastic control theory. The search for a stopping time
$T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ possessing the first
moment is based on solutions of backward stochastic differential equations of quadratic
type. This new approach generalizes an approach by Bass [Bas] of the classical version of
Skorokhod's embedding problem using martingale representation techniques.
A Branch-Cut-And-Price Algorithm for a Combined Buy-at-Bulk Network Design Facility Location Problem
(2014)
In the combined buy-at-bulk network design facility location
problem we are given an undirected network with a set of potential facilities and a set of clients with demands. In addition, we are also provided a set of cable types with each cable type having a capacity and cost per unit length. The cost to capacity ratio decreases from large to small
cables following economies of scale. A network planner is expected to determine a set of facilities to open and install cables along the edges of the network in order to route the demands of the clients to some open facility. The capacities of the installed cable must be able to support the demands of the clients routed along that edge.The objective is to minimize the cost that is paid for opening facilities and installing cables. We model the problem as an integer program and propose a branch-cut-and-price algorithm for solving it. We study the effect of two family of valid inequalities that naturally emerge from the model. We present the results of our implementation that were tested on a set of large real world instances.
We develop a behavioural approach to linear, time-varying, differential algebraic systems.
The analysis is \almost everywhere" in the sense that the statements hold on R T, where
T is a discrete set. Controllability, observability and autonomy is introduced and related to
the behaviour of the system. Classical results on the behaviour of time-invariant systems are
studied in the context of time-varying systems.
Diffusion processes are relevant for a variety of phenomena in the natural sciences, including
diffusion of cells or biomolecules within cells, diffusion of molecules on a membrane or surface,
diffusion of a molecular conformation within a complex energy landscape. Many experimental
tools exist now to track such diffusive motions in single cells or molecules, including high-resolution
light microscopy, optical tweezers, fluorescence quenching, and Förster resonance energy transfer
(FRET). Experimental observations are most often indirect and incomplete: (1) They do not
directly reveal the potential or diffusion constants that govern the diffusion process, (2) they have
limited time and space resolution, and (3) the highest-resolution experiments do not track the
motion directly but rather probe it stochastically by recording single events, such as photons,
whose properties depend on the state of the system under investigation.
Here, we propose a general Bayesian framework to model diffusion processes with nonlinear
drift based on incomplete observations as generated by various types of experiments. A maximum
penalized likelihood estimator is given as well as a Gibbs sampling method that allows to estimate
the trajectories that have caused the measurement, the nonlinear drift or potential function and
the noise or diffusion matrices, as well as uncertainty estimates of these properties. The approach
is illustrated on numerical simulations of FRET experiments where it is shown that trajectories,
potentials and diffusion constants can be efficiently and reliably estimated even in cases with little
statistics or non-equilibrium measurement conditions.
We consider a one–dimensional coupled stationary Schrödinger drift–diffusion model for quantum
semiconductor device simulations. The device domain is decomposed into a part with large quantum
effects (quantum zone) and a part where quantum effects are negligible (classical zone). We give
boundary conditions at the classic–quantum interface which are current preserving. Collisions within
the quantum zone are introduced via a Pauli master equation. To illustrate the validity we apply the
model to three resonant tunneling diodes
We present a globally convergent method for the solution of frictionless large deformation contact problems involving hyperelastic materials.
For the discretisation we apply the dual mortar method which is known to be more stable than node-to-segment approaches. The resulting non-convex constrained minimisation problems are solved using a filter–trust-region scheme.
This method combines several techniques from non-linear optimisation to achieve global convergence towards first-order optimal points. To speed up the method inexact linearisations of the non-penetration constraint are used
whenever the current iterate is far away from a critical point. A monotone multigrid method is applied for the fast solution of the constrained Newton
problems.
We introduce a behavioural approach to linear, time-varying, differential algebraic
(descriptor) systems. The analysis is \almost global" in the sense that the analysis is
not restricted to an interval I R but is allowed for the \time axis" RnT, where T is
a discrete set of critical points, at which the solution may exhibit a finite escape time.
Controllable, observable, autonomous, and adjoint behaviour for linear time-varying
descriptor systems is introduced and characterized.