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Stochastic optimization techniques are highly relevant for applications in electricity production and trading since, in particular after the deregulations of many electricity markets, there is a high number of uncertainty factors (e.g., demand, spot prices) to be considered that can be described reasonably by statistical models. Here, we want to highlight two aspects of this approach: scenario tree approximation and risk aversion. The former is a procedure to replace a general statistical model (probability distribution), which makes the optimization problem intractable, suitably by a finite discrete distribution (scenarios). This is typically an indispensable first step towards a solution of a stochastic optimization model. On the other hand, this is a highly sensitive concern, in particular if dynamic decision structures are involved (multistage stochastic programming). Then, the approximate distribution must exhibit tree structure. Moreover, it is of interest to get by with a moderate number of scenarios to have the resulting problem tractable. In any case, it has to be relied on suitably stability results to ensure that the obtained results are indeed related to the original (infinite dimensional) problem. These stability results involve probability distances and, for the multistage case, a filtration distance that evaluates the information increase over time. We present respective approximation schemes relying on Monte Carlo sampling and scenario reduction and combining techniques. The second topic of this talk is risk aversion. Namely, we present the approach of polyhedral risk measures which are given as (the optimal values of) certain simple stochastic programs. Well-known risk measures such as CVaR and expected polyhedral utility belong to this class and, moreover, multiperiod risk measures for multistage stochastic programs are suggested. For stochastic programs incorporating polyhedral risk measures it has been shown that numerical tractability as well as stability results known for classical (non-risk-averse) stochastic programs remain valid. In particular, the same scenario approximation methods can be used. Finally, we present illustrative numerical results from an electricity portfolio optimization model for a municipal power utility.
Stochastic programming problems appear as mathematical models for optimization problems under stochastic uncertainty. Most computational approaches for solving such models are based on approximating the underlying probability distribution by a probability measure with finite support. Since the computational complexity for solving stochastic programs gets worse when increasing the number of atoms (or scenarios), it is sometimes necessary to reduce their number. Techniques for scenario reduction often require fast heuristics for solving combinatorial subproblems. Available techniques are reviewed and open problems are discussed.
Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario
sets. The relevant distances of (multivariate) probability
distributions for deriving quantitative stability results for such stochastic programs are $\mathcal{B}$-discrepancies, where the class $\mathcal{B}$ of Borel sets depends on their structural properties.
Hence, the optimal scenario reduction problem for such models is stated with respect to $\mathcal{B}$-discrepancies. In this paper,
upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop
heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or
Kolmogorov metric) in some detail and provide some numerical experience.
Portfolio and risk management problems of power
utilities may be modeled by multistage stochastic programs. These
models use a set of scenarios and corresponding probabilities
to model the multivariate random data process (electrical load,
stream flows to hydro units, and fuel and electricity prices). For
most practical problems the optimization problem that contains
all possible scenarios is too large. Due to computational complexity
and to time limitations this program is often approximated by
a model involving a (much) smaller number of scenarios. The proposed
reduction algorithms determine a subset of the initial scenario
set and assign new probabilities to the preserved scenarios.
The scenario tree construction algorithms successively reduce the
number of nodes of a fan of individual scenarios by modifying the
tree structure and by bundling similar scenarios. Numerical experience
is reported for constructing scenario trees for the load
and spot market prices entering a stochastic portfolio management
model of a German utility
Stability-based methods for scenario generation in stochastic programming are reviewed. In particular, we briefly discuss Monte Carlo sampling, Quasi-Monte Carlo methods, quadrature rules based on sparse grids and optimal quantization. In addition, we provide some convergence results based on recent developments in multivariate integration. The method of optimal scenario reduction and techniques for scenario trees generation are also reviewed.
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.
Scalable Frames
(2012)
Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.
We estimate potential energy savings in IP-over-WDM networks achieved by switching off router line cards in low-demand hours. We compare three approaches to react on dynamics in the IP traffic over time, FUFL,
DUFL and DUDL. They provide different levels of freedom in adjusting the routing of lightpaths in the WDM layer and the routing of demands in the IP layer. Using MILP models based on realistic network topologies and node architectures as well as realistic demands, power, and cost values, we show that already a simple monitoring of the lightpath utilization in order to deactivate empty line cards (FUFL) brings substantial
benefits. The most significant savings, however, are achieved by rerouting traffic in the IP layer (DUFL), which allows emptying and deactivating lightpaths together with the corresponding line cards. A
sophisticated reoptimization of the virtual topologies and the routing in the optical domain for every demand scenario (DUDL) yields nearly no additional profits in the considered networks.
We define a risk averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk averse formulation of the problem.
This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures.
To implement the proposed policy, the approximation of the risk averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk averse problem.
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 network such that a giv en demand can be satisfied. There are two objectives. passengers want to minimize travel times, the transport company wishes to minimize operating costs. We investigate three variants of a multi-commo dity flow model for line planning that differ with respect to passenger routings. The first model allows arbitrary routings, the second only unsplittable routings, and the third only shortest path rou tings with respect to the network. We compare these models theoretically and computationally on data for the city of Potsdam.
We study the dynamics of a ring of unidirectionally coupled autonomous
Duffing oscillators. Starting from a situation where the individual
oscillator without coupling has only trivial equilibrium dynamics,
the coupling induces complicated transitions to periodic, quasiperiodic,
chaotic, and hyperchaotic behavior. We study these transitions in
detail for small and large numbers of oscillators. Particular attention
is paid to the role of unstable periodic solutions for the appearance
of chaotic rotating waves, spatiotemporal structures and the Eckhaus
effect for a large number of oscillators. Our analytical and numerical
results are confirmed by a simple experiment based on the electronic
implementation of coupled Duffing oscillators.
Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB~2010 benchmark instances.
This paper presents some weighted H2-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., highly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of thermal conductivity of the material as the volume fraction of the particles is close to the jammed state. We proof that the characteristic percolation behavior of the material is well captured by standard conforming finite element models.
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
We investigate the problem of maximizing the robust utility functional inf QEQ EQu(X).
We give the dual characterization for its solution for both a complete and an incomplete
market model. To this end, we introduce the new notion of reverse f-projections and
use techniques developed for f-divergences. This is a suitable tool to reduce the robust
problem to the classical problem of utility maximization under a certain measure: the
reverse f-projection. Furthermore, we give the dual characterization for a closely related
problem, the minimization of expenditures given a minimum level of expected utility in
a robust setting and for an incomplete market.
Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite element method (FEM) can be avoided by using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses $d$, where the accuracy is best possible independent of the conductivity or the frequency being small or large -- this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs which are accurate up to $O(d^2)$.
We consider scheduling to minimize the weighted sum of completion
times on a single machine that may experience unexpected changes in
processing speed or even full breakdowns. We design a polynomial
time deterministic algorithm that finds a robust prefixed scheduling
sequence with a solution value within~$4$ times the value
an optimal clairvoyant algorithm can achieve, knowing the
disruptions in advance and even being allowed to interrupt jobs at
any moment. A randomized version of this algorithm attains in
expectation a ratio of~$e$ w.r.t. a clairvoyant optimum.
We show that such a ratio can never be achieved by any deterministic
algorithm by proving that the price of robustness of any such
algorithm is at least~$1+\sqrt{3} \approx 2.73205>e$.
As a direct consequence of our results, the question whether a
constant approximation algorithm exists for the problem with given
machine unavailability periods is answered affirmatively. We
complement this result by an FPTAS for the preemptive and non-preemptive special case with a single
non-available period.
The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.
Traffic in communication networks fluctuates heavily over time. Thus, to avoid capacity bottlenecks,
operators highly overestimate the traffic volume during network planning. In this paper we consider
telecommunication network design under traffic uncertainty, adapting the robust optimization approach
of Bertsimas and Sim [21]. We present three different mathematical formulations for this problem, provide valid inequalities,
study the computational implications, and evaluate the realized robustness.
To enhance the performance of the mixed-integer programming solver we derive robust cutset inequalities generalizing their deterministic counterparts. Instead of a single cutset inequality for every
network cut, we derive multiple valid inequalities by exploiting the extra variables available in the robust formulations. We show that these inequalities define facets under certain conditions and that they
completely describe a projection of the robust cutset polyhedron if the cutset consists of a single edge.
For realistic networks and live traffic measurements we compare the formulations and report on the
speed up by the valid inequalities. We study the “price of robustness” and evaluate the approach by
analyzing the real network load. The results show that the robust optimization approach has the potential
to support network planners better than present methods.
Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
We present new residual estimates based on Kato's square root theorem for spectral approximations of diagonalizable non-self-adjoint differential operators of convection-diffusion-reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations, where it is shown that the
resulting a posteriori error estimates are reliable. Provided experiments demonstrate the efficiency and reliability of our approach.
We consider a sorting problem from railway optimization
called train classification: incoming trains are split up into their single
cars and reassembled to form new outgoing trains. Trains are subject
to delay, which may turn a prepared sorting schedule infeasible for the
disturbed situation. The classification methods applied today deal with
this issue by completely disregarding the input order of cars, which provides
robustness against any amount of disturbance but also wastes the
potential contained in the a priori knowledge about the input.
We introduce a new method that provides a feasible sorting schedule for
the expected input and allows to
flexibly insert additional sorting steps
if the schedule has become infeasible after revealing the disturbed input.
By excluding disruptions that almost never occur from our consideration,
we obtain a classification process that is quicker than the current railway
practice but still provides robustness against realistic delays. In fact, our
algorithm allows
flexibly trading off fast classification against high degrees
of robustness depending on the respective need. We further explore
this
flexibility in experiments on real-world traffic data, underlining our
algorithm improves on the methods currently applied in practice.
The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
To address the plurality of interpretations of the subjective notion of risk, we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity. Our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets. This representation covers most instruments related to risk and allow for a differentiated interpretation depending on the underlying context which is illustrated in different settings: For random variables, risk perception can be interpreted as model risk, and we compute among others the robust representation of the economic index of riskiness. For lotteries, risk perception can be viewed as distributional risk and we study the "Value at Risk". For consumption patterns, which excerpt an intertemporality dimension in risk perception, we provide an interpretation in terms of discounting risk and discuss some examples.
We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional $\sigma$-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular we show how ``bubbles'' may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.
Revlex-Initial 0/1-Polytopes
(2005)
Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.
We consider the problem of numerical approximation for forward-backward stochastic
differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance
of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial
derivative using correlated assets. For the convergence of numerical approximation schemes for
such systems of stochastic equations, path regularity of the solution processes is instrumental.
We present a method based on the truncation of the driver, and explicitly exhibit error estimates
as functions of the truncation height. We discuss a reduction method to FBSDE with globally
Lipschitz continuous drivers, by using the Cole-Hopf exponential transformation. We finally
illustrate our numerical approximation methods by giving simulations for prices and optimal
hedges of simple insurance derivatives.
Resolving the apparent gap in complexity between
simulated and measured kinetics of biomolecules
(2012)
Molecular simulations of biomolecules often reveal a complex picture of the their kinetics,
whereas kinetic experiments typically seem to indicate considerably simpler two- or three-state
kinetics. Markov state models (MSM) provide a tool to link between simulation and experi-
ment, and to resolve this apparent contradiction.
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
Under high load, the automated dispatching of service vehicles for
the German Automobile Association (ADAC) must reoptimize a dispatch for
100{150 vehicles and 400 requests in about ten seconds to near optimality. In
the presence of service contractors, this can be achieved by the column generation
algorithm ZIBDIP. In metropolitan areas, however, service contractors
cannot be dispatched automatically because they may decline. The problem:
a model without contractors yields larger optimality gaps within ten seconds.
One way out are simplified reoptimization models. These compute a shortterm
dispatch containing only some of the requests: unknown future requests
will in
uence future service anyway. The simpler the models the better the
gaps, but also the larger the model error. What is more significant: reoptimization
gap or reoptimization model error? We answer this question in
simulations on real-world ADAC data: only the new models ShadowPrice and
ZIBDIPdummy can keep up with ZIBDIP.
RENS – the optimal rounding
(2012)
This article introduces RENS, the relaxation enforced neighborhood search, a large neighborhood search algorithm for mixed integer nonlinear programming (MINLP) that uses a sub-MINLP to explore the set of feasible roundings of an optimal solution x' of a linear or nonlinear relaxation. The sub-MINLP is constructed by fixing integer variables x_j with x'_j in Z and bounding the remaining integer variables to x_j in {floor(x'_j), ceil(x'_j)}. We describe two different applications of RENS: as a standalone algorithm to compute an optimal rounding of the given starting solution and as a primal heuristic inside a complete MINLP solver.
We use the former to compare different kinds of relaxations and the impact of cutting planes on the roundability of the corresponding optimal solutions. We further utilize RENS to analyze the performance of three rounding heuristics implemented in the branch-cut-and-price framework SCIP. Finally, we study the impact of RENS when it is applied as a primal heuristic inside SCIP.
All experiments were performed on three publically available test sets of mixed integer linear programs (MIPs), mixed integer quadratically constrained programs (MIQCPs), and MINLPs, using solely software which is available in source code.
It turns out that for these problem classes 60% to 70% of the instances have roundable relaxation optima and that the success rate of RENS does not depend on the percentage of fractional variables. Last but not least, RENS applied as primal heuristic complements nicely with existing root node heuristics in SCIP and improves the overall performance.
In the recent years, a couple of quite successful large neighborhood search heuristics for mixed integer programs has been published.
Up to our knowledge, all of them are improvement heuristics.
We present a new start heuristic for general MIPs working in the spirit of large neighborhood search.
It constructs a sub-MIP which represents the space of all feasible roundings of some fractional point - normally the optimum of the LP-relaxation of the original MIP.
Thereby, one is able to determine whether a point can be rounded to a feasible solution and which is the best possible rounding.
Furthermore, a slightly modified version of RENS proves to be a well-performing heuristic inside the branch-cut-and-price-framework SCIP.
Relating Attractors and Singular Steady States in the Logical Analysis of Bioregulatory Networks
(2007)
In 1973 R. Thomas introduced a logical approach to modeling and analysis of
bioregulatory networks. Given a set of Boolean functions describing the
regulatory interactions, a state transition graph is constructed that captures
the dynamics of the system. In the late eighties, Snoussi and Thomas extended
the original framework by including singular values corresponding to
interaction thresholds. They showed that these are needed for a refined
understanding of the network dynamics.
In this paper, we study systematically singular steady states, which are
characteristic of feedback circuits in the interaction graph, and relate them
to the type, number and cardinality of attractors in the state transition
graph. In particular, we derive sufficient conditions for regulatory networks
to exhibit multistationarity or oscillatory behavior, thus giving a partial
converse to the well-known Thomas conjectures.
The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of
pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida
regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions
towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization
strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is
carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each
regularized problem a semi-smooth Newton method is applied.
A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.
Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames
(2014)
Regularization techniques for the numerical solution of nonlinear inverse scattering
problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions.
Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Tikhonov method.
We examine both directly the nonlinear problem and a linearized problem obtained by
the Born approximation technique. As problem classes we study the acoustic inverse
scattering problem and the electromagnetic inverse scattering problem. We show that
this approach introduces a sparse regularization for the nonlinear setting and we present
a result describing the behavior of the local regularity of a scatterer under linearization,
which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by numerical examples for the acoustic inverse scattering problem that highlight the effectiveness of this approach.
The chemical master equation is a fundamental equation in chemical kinetics. It is an adequate substitute for the classical reaction-rate equations whenever stochastic effects become relevant. In the present paper we give a simple argument showing that the solutions of a large class of chemical master equations, including all those in which elementary reactions between two and more molecules do not generate a larger number of molecules than existed before,
are bounded in weighted $\ell_1$-spaces. As an illustration for the implications of this kind of regularity we analyze the effect of truncating the state space. This leads to an error analysis of the finite state projection of the chemical master equation, an approximation that underlies many numerical methods.
The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.
Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints
(2004)
A class of quadratic optimization problems in Hilbert spaces is considered, where
pointwise box constraints and constraints of bottleneck type are given. The main focus is to prove the
existence of regular Lagrange multipliers in L2-spaces. This question is solved by investigating the
solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control
problems for elliptic and parabolic partial differential equations with mixed pointwise control-state
constraints.
In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.
We consider a time-dependent optimal control problem, where the state
evolution is described by an ODE. There is a variety of methods for the treatment
of such problems. We prefer to view them as boundary value problems and apply to
them the Riccati approach for non-linear BVPs with separated boundary conditions.
There are many relationships between multiple shooting techniques, the Riccati
approach and the Pantoja method, which describes a computationally efficient
stage-wise construction of the Newton direction for the discrete-time optimal control
problem.
We present an efficient implementation of this approach. Furthermore, the wellknown
checkpointing approach is extended to a `nested checkpointing` for multiple
transversals. Some heuristics are introduced for an efficient construction of nested
reversal schedules. We discuss their benefits and compare their results to the optimal
schedules computed by exhaustive search techniques.
In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.
In this paper we investigate two different recoverable robust models to deal with cost uncertainties in a shortest path problem. Recoverable robustness extends the classical concept of robustness to deal with uncertainties by incorporating limited recovery actions after the
full data are revealed. Our first model focuses on the case where the recovery actions are quite restricted: after a simple path is fixed in the first stage, in the second stage, after all data are revealed, any path containing at most k new arcs may be chosen.
Thus, the parameter k can be interpreted as a mediator between
robust optimization - no changes allowed - and optimization
on the fly - an arbitrary solution can be chosen. Considering three
classical scenario sets, which model uncertainties in the cost function,
we show that this new problem is strongly NP-hard in all
these cases and is not approximable, unless P=NP.
This is in contrast to the robust shortest path problem, where, for
example, an optimal solution can be computed efficiently for interval
and Gamma-scenarios. For series-parallel graphs and interval scenarios,
we present a polynomial time algorithm for this recoverable robust
setting.
In our second model the recovery set, i.e., the set of paths selectable
in the second stage is not limited, but deviating from the previous
choice comes at extra cost. Thus, a path chosen in the first stage
produces renting costs modeled as an alpha-fraction of the scenario
cost. For an arc taken in the second stage the remaining cost needs
to be paid in addition to some extra inflation cost modeled by a beta-fraction
of the scenario cost, if the arc was not reserved beforehand. The
complexity status of this problem is similar to the robust case. Yet,
for Gamma-scenarios the problem is again strongly NP-hard,
but can be approximated.
The knapsack problem is one of the basic problems in combinatorial optimization. In real-world applications it is often part of a more complex problem. Examples are machine capacities in production planning or bandwidth restrictions in telecommunication network design. Due to unpredictable future settings or erroneous data, parameters of such a subproblem are subject to uncertainties.
In high risk situations a robust approach should be chosen to deal with these uncertainties.
Unfortunately, classical robust optimization outputs solutions with little profit by prohibiting any adaption of the solution when the actual realization of the uncertain parameters is known.
This ignores the fact that in most settings minor changes to a previously determined solution are possible. To overcome these drawbacks we allow a limited recovery of a previously fixed item set as soon as the data are known by deleting at most k items and adding up to l new items.
We consider the complexity status of this recoverable robust knapsack problem and extend the classical concept of cover inequalities to obtain stronger polyhedral descriptions. Finally, we present two extensive computational studies to investigate the influence of parameters k and l to the objective and evaluate the effectiveness of our new class of valid inequalities.
In this paper, we investigate the recoverable robust knapsack problem,
where the uncertainty of the item weights follows the approach of Bertsimas and
Sim. In contrast to the robust approach, a limited recovery action is allowed,
i.e., up to k items may be removed when the actual weights are known. This problem
is motivated by the assignment of traffic nodes to antennas in wireless network
planning. Starting from an exponential min-max optimization model, we derive an
integer linear programming formulation of quadratic size. In a preliminary computational
study, we evaluate the gain of recovery using realistic planning data.
We present recent developments in two-stage mixed-integer stochastic
programming with regard to application in power production planning.
In particular, we review structural properties, stability issues, scenario
reduction and decomposition algorithms for two-stage models. Furthermore,
we describe an application to stochastic thermal unit commitment.
Given a set of service requests (events), a set of guided servers (units),
and a set of unguided service contractors (conts), the vehicle dispatching problem
VDP is the task to find an assignment of events to units and conts as well as tours
for all units starting at their current positions and ending at their home positions
(dispatch) such that the total cost of the dispatch is minimized.
The cost of a dispatch is the sum of unit costs, cont costs, and event costs. Unit
costs consist of driving costs, service costs and overtime costs; cont costs consist of
a fixed cost per service; event costs consist of late costs linear in the late time, which
occur whenever the service of the event starts later than its deadline.
The program ZIBDIP based on dynamic column generation and set partitioning
yields solutions on heavy-load real-world instances (215 events, 95 units) in less
than a minute that are no worse than 1% from optimum on state-of-the-art personal
computers.
We introduce new elevator group control algorithms that can be implemented to be real-time compliant on embedded microcontrollers. The algorithms operate a group of elevators in a destination call system, i.e. passengers specify the destination floor instead of the travel direction only. The aim is to achieve small waiting and travel times for the passengers. We provide evidence, using simulation, that the algorithms offer good performance. One of our algorithms has been implemented by our industry partner and is used in real-world systems.
A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. In the present note we provide new analytic insights into the asymptotic behavior of local volatility in the wings. We present a general approximation formula and specialize it to the Heston model, showing that local variance is linear in the wings. This further justifies the choice of certain local volatility parametrizations.
In this paper we consider the rational interpolation problem consisting in finding a rational matrix-valued function that
interpolates a given set of parameters. We briefly describe two different numerical methods for solving this problem. These are
the vector fitting and the frequency domain subspace identification method. Several numerical examples are given that compare
the properties of these methods. Furthermore, we discuss the computation of a (minimal) state space realization of a rational
function. Model order reduction methods such as modal approximation and balanced truncation are also presented. These
methods can be used to compute a reduced-order approximation of the realized dynamical system.
We consider general economies in which rational agents interact locally. The local aspect
of the interactions is designed to represent in a simple abstract way social interactions, that
is, socioeconomic environments in which markets do not mediate all of agents' choices, and
each agent's choice might be in part determined, for instance, by family, peer group, or ethnic
group effects. We study static as well as dynamic infinite horizon economies; we allow for
economies with incomplete information, and we consider jointly global and local interactions,
to integrate e.g., global externalities and markets with peer and group effects. We provide
conditions under which such economies have rational expectations equilibria.
We illustrate the effects of local interactions when agents are rational by studying in detail
the equilibrium properties of a simple economy with quadratic preferences which captures, in
turn, local preferences for conformity, habit persistence, and preferences for status or adherence
to aggregate norms of behavior.
Learning during search allows solvers for discrete optimization problems to remember parts of the search that they have already performed and avoid revisiting redundant parts. Learning approaches pioneered by the SAT and CP communities have been successfully incorporated into the SCIP constraint integer programming platform. In this paper we show that performing a heuristic constraint programming search during root node processing of a binary program can rapidly learn useful nogoods, bound changes, primal solutions, and branching statistics that improve the remaining IP search.
Rapid Branching
(2012)
We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.
The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.
Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data.
We consider quickest flows within a new model that is based on transportation applications. In contrast to other models, it is forbidden to store flow in nodes and to cross nodes with more than one flow unit simultaneously. We work on undirected graphs. Grid graphs are of special interest because they typically arise in practice. Our model allows to close edges temporarily by time windows, and considers waiting on edges.
We solve several quickest s,t–flow problems without time windows polynomially. We prove that time windows make these problems NP–hard and even not approximable. In a multicommodity environment, all quickest flow variants are shown to be NP–hard even in grid graphs with uniform edge transit times. An alternative proof shows NP-hardness already for a small number of commodities in the case that waiting is not allowed and transit times are edge–specific. Finally, we propose two approximation algorithms in the case that time windows do not occur.
Quasistatic small-strain plasticity in the limit of small hardening and its numerical approximation
(2011)
The quasistatic rate-independent evolution of the Prager-Ziegler-type model of linearized plasticity with hardening is shown to converge to the rate-independent evolution of the Prandtl-Reuss elastic/perfectly plastic model. Based on the concept of energetic solutions we study the convergence of the solutions in the limit for hardening coefficients converging to 0 by using the abstract method of Gamma-convergence for rate-independent systems. An unconditionally convergent numerical scheme is devised and 2D and 3D numerical experiments are presented. A two-sided energy inequality is a posteriori verified to document experimental convergence rates.
Mixed-integer two-stage stochastic programs with fixed recourse matrix, random recourse costs, technology matrix, and right-hand sides are considered. Quantitative continuity properties of its optimal value and solution set are derived when the underlying probability distribution is perturbed with respect to an appropriate probability metric.
We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying
probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin's property of the solution set mapping with respect to the change of probability measure. The established results are
applied to stability analysis of stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints.
Flux coupling analysis (FCA) has become a useful tool for aiding metabolic reconstructions and guiding genetic manipulations. Originally, it was introduced for constraint-based models of metabolic networks that are based on the steady-state assumption. Recently, we have shown that the steady-state assumption can be replaced by a much weaker lattice-theoretic property related to the supports of metabolic fluxes. In this paper, we further extend our approach and delevelop an efficient algorithm for general qualitative flux coupling analysis (QFCA). We illustrate our method by thermodynamic flux coupling analysis (tFCA), which allows studying steady-state metabolic models with loop-law thermodynamic constraints. These models do not satisfy the lattice-theoretic properties required in our previous work. For a selection of genome-scale metabolic network reconstructions, we discuss both theoretically and practically, how thermodynamic constraints strengthen the coupling results that can be obtained with classical FCA.
The extent to which catastrophic weather events occur strongly depends on global climate conditions such as average sea surface temperatures (SST) or sea level pressures. Some of the factors can be predicted up to a year in advance, and should therefore be taken into account in any reasonable management of weather related risk. In this paper we first set up a risk model that integrates climate factors. The we show how variance minimizing hedging strategies explicitly depend on the factors' prediction. Our analysis is based on a detailed study of the predictable representation property on the combined Poisson and Wiener spaces. Using tools of the stochastic calculus of variations we derive a representation formula of the Clark-Ocone type. Finally, we exemplify the theory developed in a case study of US hurricane risk. We derive hedging strategies taking into account that US hurricane activity strongly depends on the SST of the Pacific Ocean.
In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for
linear differential-algebraic equations.
Furthermore, a perturbation and error analysis for these methods is presented. We
investigate how errors in the data and in the numerical integration affect the
accuracy of the approximate spectral intervals. Although we need to integrate
numerically some differential-algebraic systems on usually very long
time-intervals, under certain assumptions, it is shown that the error of the
computed spectral intervals can be controlled by the local error of numerical
integration and the error in solving the algebraic constraint.
Some numerical examples are presented to illustrate the theoretical results.
Can OR methods help the public transport industry to break even?
How would you build a public transport system? For example, have a look at
Berlin. The BVG, Berlin's public transport company, maintains a network
of 2,423 km, operates 197 lines with 3,286 stops, using 1,554 busses, 1,391
subway cars, and 599 trams from 12 depots, and has 13,409 employees [7].
The BVG currently transports about 800 million passengers per year and
covers about 40% of the total non-pedestrian traffic volume of the city [18].
Does Berlin have a "reasonable" public transportation network? Does
the BVG run a "good" transportation system? Is it "efficient"?
These are difficult questions. In fact, politicians, transportation managers,
customers, taxpayers, etc. frequently employ judgments such as "good"
and "efficient", but nobody can give a defiition what this exactly means.
Since almost every public transportation system in the world is in the red,
the cheapest system is no public transportation at all. On the other hand,
the most convenient system for the passenger - a stop in front of every house
with direct connections to everywhere - is much too expensive. What is the
right compromise? Operations Research has no good answer either - so far.
But OR can improve aspects of public transportation significantly, as we
want to demonstrate in the following.
One-shot optimization aims at attaining feasibility and optimality simultane-
ously, especially on problems where even the linearized constraint equations
cannot be resolved economically. Here we consider a scenario where forming
and factoring the active Jacobian is out of the question, as is for example the
case when the constraints represent some discretization of the Navier Stokes
equation. Assuming that the 'user' provides us with a linearly converging solver
that gradually restores feasibility after each change in the design variables, we
derive a corresponding adjoint iteration and attach an optimization (sub)step.
The key question addressed is how the approximate reduced gradient generated
by the adjoint iteration should be preconditioned in order to achieve overall
convergence at a reasonable speed. An eigenvalue analysis yields necessary
conditions on the preconditioning matrix, which are typically not satised by
the familiar reduced Hessian. Some other projection of the Lagrangian Hessian
appears more promising and is found to work very satisfactorily on a nonlinear
test problem.
The analyzed approach is one-step in that the normal, dual and design variables
are always updated simultaneously on the basis of one function evaluation and
its adjoint. Multi-step variants are promising but remain to be investigated.
Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe
long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great
amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into
microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data
to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current
perspective of Markov models and explain in short their theoretical basis and assumptions. We
describe transition path theory which allows the entire ensemble of protein folding pathways to be
investigated and that combines naturally with Markov models. Experimental observations can be
naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally
observable timescales can be equipped with an understanding of the structural rearrangement
processes that take place at these timescales. The concepts of this paper are illustrated by a
simple kinetic model of protein folding.
Discrete-state Markov (or master equation) models provide a useful simplified representation for
characterizing the long-time statistical evolution of biomolecules in a manner that allows direct
comparison with experiments as well as the elucidation of mechanistic pathways for an inherently
stochastic process. A vital part of meaningful comparison with experiment is the characterization of
the statistical uncertainty in the predicted experimental measurement, which may take the form of
an equilibrium measurement of some spectroscopic signal, the time-evolution of this signal following
a perturbation, or the observation of some statistic (such as the correlation function) of the equilib-
rium dynamics of a single molecule. Without meaningful error bars (which arise due to the finite
quantity of data used to construct the model), there is no way to determine whether the deviations
between model and experiment are statistically meaningful. Previous work has demonstrated that
a Bayesian method that enforces microscopic reversibility can be used to characterize the correlated
uncertainties in state-to-state transition probabilities (and functions thereof) for a model inferred from
molecular simulation data. Here, we extend this approach to include the uncertainty in observables
that are functions of molecular conformation (such as surrogate spectroscopic signals) characteriz-
ing each state, permitting the full statistical uncertainty in computed spectroscopic experiments to be
assessed. We test the approach in a simple model system to demonstrate that the computed uncer-
tainties provide a useful indictor of statistical variation, and then apply it to the computation of the
fluorescence autocorrelation function measured for a dye-labeled peptide previously studied by both
experiment and simulation.
Large-scale stochastic models are relevant in many different fields such as com- putational biology, finance, social sciences, communication and traffic networks. In order to both efficiently simulate and analyze such models and to understand the essential properties of the sys- tem, it is desirable to have model reduction techniques that much reduce the dimensionality of the model while at the same time preserving the system’s essential dynamical properties. In this paper, a general model reduction technique for the class of discrete space and time Hidden Markov Models is presented, thereby also including the more special class discrete Markov Chains. The method is illustrated on some model applications.
Primal-dual linear Monte Carlo algorithm for multiple stopping - An application to flexible caps
(2012)
In this paper we consider the valuation of Bermudan callable derivatives with
multiple exercise rights. We present in this context a new primal-dual linear
Monte Carlo algorithm that allows for ecient simulation of lower and upper price
bounds without using nested simulations (hence the terminology). The algorithm
is essentially an extension of a primal{dual Monte Carlo algorithm for standard
Bermudan options proposed in Schoenmakers et al. (2011), to the case of multiple
exercise rights. In particular, the algorithm constructs upwardly a system of dual
martingales to be plugged into the dual representation of Schoenmakers (2010).
At each level the respective martingale is constructed via a backward regression
procedure starting at the last exercise date. The thus constructed martingales are
nally used to compute an upper price bound. At the same time, the algorithm
also provides approximate continuation functions which may be used to construct
a price lower bound. The algorithm is applied to the pricing of
exible caps
in a Hull and White (1990) model setup. The simple model choice allows for
comparison of the computed price bounds with the exact price which is obtained
by means of a trinomial tree implementation. As a result, we obtain tight price
bounds for the considered application. Moreover, the algorithm is generically
designed for multi-dimensional problems and is tractable to implement.
Primal heuristics are an important component of state-of-the-art codes for mixed integer nonlinear programming (MINLP). In this article we give a compact overview of primal heuristics for MINLP that have been suggested in the literature of recent years. We sketch the fundamental concepts of different classes of heuristics and discuss specific implementations. A brief computational experiment shows that primal heuristics play a key role in achieving feasibility and finding good primal bounds within a global MINLP solver.
We present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.
We show that pricing a big class of relevant options by hedging
and no-arbitrage can be extended beyond semimartingale models. To
this end we construct a subclass of self-financing portfolios that
contains hedges for these options, but does not contain arbitrage
opportunities, even if the stock price process is a
non-semimartingale of some special type.
Moreover, we show that the option prices depend
essentially only on a path property of the stock price process,
viz. on the quadratic variation. As a consequence, we can
incorporate many stylized facts to a pricing model without
changing the option prices.
This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical ‘delta hedge’ in complete markets.
Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in
fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any
potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal
representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and
Meziou on the max-plus decomposition of supermartingales, and they provide a singular analogue to
the non-linear Riesz representation in El Karoui and Föllmer.
We propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Positivity and stability are preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
We propose a model reduction method for positive systems that ensures the positivity of the reduced model. For both, continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Stability is preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
We introduce an algorithm for
diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both space and diffusion direction.
The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models.
It works by embedding the measurement space into a space with defined metric and group operations, in this case the Lie group of three-dimensional Euclidean motion SE(3).
Subsequently, pairwise comparisons of the values of the diffusion
weighted signal are used for adaptation.
The position-orientation adaptive smoothing preserves the edges of the observed fine and anisotropic structures.
The POAS-algorithm is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and
variability of quantities derived from the data for specific models.
We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and
hence acquisition time while achieving similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.
Our main result is that every n-dimensional polytope can be
described by at most 2n ? 1 polynomial inequalities and, moreover, these
polynomials can explicitly be constructed. For an n-dimensional pointed
polyhedral cone we prove the bound 2n ? 2 and for arbitrary polyhedra we
get a constructible representation by 2n polynomial inequalities.
We consider stochastic programs with risk measures in the objective and study
stability properties as well as decomposition structures. Thereby we place emphasis on dynamic
models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we
define the class of polyhedral risk measures such that stochastic programs with risk measures taken
from this class have favorable properties. Polyhedral risk measures are defined as optimal values of
certain linear stochastic programs where the arguments of the risk measure appear on the right-hand
side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and
used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we
propose multiperiod extensions of the Conditional-Value-at-Risk.
We compare different multiperiod risk measures taken from the class of polyhedral risk measures with respect to the effect
they show when used in the objective of a stochastic program. For this purpose, simulation results of a stochastic programming
model for optimizing the electricity portfolio of a German municipal power utility are presented and analyzed. This model
aims to minimize risk and expected overall cost simultaneously.
We prove global convergence of an inexact polyhedral Gau\ss--Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of
their domains of definition. While being known to be very slow by themselves, such methods are a cornerstone for fast, globally convergent multigrid methods. Our result generalizes the proof of Kornhuber and Krause [2006] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach.
The randomized k-number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k-partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k = 2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k > 2 in the restricted problem and show that the vector of differences between the k sums converges to a k - 1-dimensional Poisson point process.
The main focus of this paper is on an a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by
parabolic and elliptic PDEs. Based on a perturbation method it is deduced how far the suboptimal
control, computed on the basis of the POD model, is from the (unknown)
exact one. Numerical examples illustrate the realization of the proposed approach for linear-quadratic problems governed by parabolic and elliptic partial differential equations.
Millionen von Menschen werden allein in Deutschland täglich von
Bussen, Bahnen und Flugzeugen transportiert. Der öffentliche
Personenverkehr (ÖV) ist von großer Bedeutung für die
Lebensqualität einzelner aber auch für die Leistungsfähigkeit ganzer
Regionen. Qualität und Effizienz von ÖV-Systemen hängen ab von
politischen Rahmenbedingungen (staatlich geplant,
wettbewerblich organisiert) und der Eignung der Infrastruktur
(Schienensysteme, Flughafenstandorte), vom vorhandenen
Verkehrsangebot (Fahr- und Flugplan), von der Verwendung
angemessener Technologien (Informations-, Kontroll- und
Buchungssysteme) und dem bestmöglichen Einsatz der
Betriebsmittel (Energie, Fahrzeuge und Personal). Die hierbei
auftretenden Entscheidungs-, Planungs- und
Optimierungsprobleme sind z.T. gigantisch und "schreien"
aufgrund ihrer hohen Komplexität nach Unterstützung durch Mathematik.
Dieser Artikel skizziert den Stand und die Bedeutung des Einsatzes von
Mathematik bei der Planung und Durchführung von öffentlichem
Personenverkehr, beschreibt die bestehenden Herausforderungen und
regt zukunftsweisende Maßnahmen an.
Every day, millions of people are transported by buses, trains, and airplanes
in Germany. Public transit (PT) is of major importance for the quality of
life of individuals as well as the productivity of entire regions. Quality and
efficiency of PT systems depend on the political framework (state-run, market
oriented) and the suitability of the infrastructure (railway tracks, airport
locations), the existing level of service (timetable, flight schedule), the use
of adequate technologies (information, control, and booking systems), and
the best possible deployment of equipment and resources (energy, vehicles,
crews). The decision, planning, and optimization problems arising in this
context are often gigantic and “scream” for mathematical support because of
their complexity.
This article sketches the state and the relevance of mathematics in planning
and operating public transit, describes today’s challenges, and suggests a
number of innovative actions.
The current contribution of mathematics to public transit is — depending
on the transportation mode — of varying depth. Air traffic is already well
supported by mathematics. Bus traffic made significant advances in recent
years, while rail traffic still bears significant opportunities for improvements.
In all areas of public transit, the existing potentials are far from being exhausted.
For some PT problems, such as vehicle and crew scheduling in bus and
air traffic, excellent mathematical tools are not only available, but used in
many places. In other areas, such as rolling stock rostering in rail traffic,
the performance of the existing mathematical algorithms is not yet sufficient.
Some topics are essentially untouched from a mathematical point
of view; e.g., there are (except for air traffic) no network design or fare
planning models of practical relevance. PT infrastructure construction is
essentially devoid of mathematics, even though enormous capital investments
are made in this area. These problems lead to questions that can only be
tackled by engineers, economists, politicians, and mathematicians in a joint
effort.
Among other things, the authors propose to investigate two specific topics,
which can be addressed at short notice, are of fundamental importance not
only for the area of traffic planning, should lead to a significant improvement
in the collaboration of all involved parties, and, if successful, will be of real
value for companies and customers:
• discrete optimal control: real-time re-planning of traffic systems in case
of disruptions,
• model integration: service design in bus and rail traffic.
Work on these topics in interdisciplinary research projects could be funded
by the German ministry of research and education (BMBF), the German
ministry of economics (BMWi), or the German science foundation (DFG).
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations
(2007)
We discuss the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations. We
show that there is a substantial difference in the behavior under these perturbations. We also discuss the perturbation of real eigenvalues of real
skew-Hamiltonian matrices under structured perturbations and use these results to analyze the properties of the URV method of computing the
eigenvalues of Hamiltonian matrices.
Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle.
We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability
as well as the index of stability for these subspaces.
We discuss the perturbation analysis for
eigenvalues and eigenvectors of structured homogeneous matrix polynomials with
Hermitian, skew-Hermitian, H-even and H-odd structure.
We construct minimal structured perturbations (structured backward errors) such that an
approximate eigenpair is an exact eigenpair of an appropriate perturbed structured matrix
polynomial. We present various comparisons with unstructured backward
errors and previous error bounds derived for the non-homogeneous case
and show that our bounds present a significant improvement.
In this work we propose a general framework for the structured perturbation
analysis of several classes of structured matrix polynomials in homogeneous
form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate eigenpair is an exact eigenpair of an appropriately perturbed matrix polynomial. This work extends previous work for the non-homogeneous case (we include infinite eigenvalues) and we show that the structured backward errors improve the known unstructured backward errors.
Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case
(2012)
We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
(2013)
We consider boundary value problems for semilinear hyperbolic systems of the type
$$
\partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n
$$
with smooth coefficient functions $a_j$
and $b_j$
such that
$b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$.
We state conditions for Hopf bifurcation, i.e.,
for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence
on $\la$
of time-periodic solutions bifurcating from the zero stationary solution. Furthermore,
we derive a formula which determines the bifurcation direction.
The proof is done by means of a Liapunov-Schmidt reduction procedure.
For this purpose, Fredholm properties of the linearized
system and implicit function
theorem techniques are used.
There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs:
First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters
is much more delicate. And second,
a sufficient amount of dissipativity is needed in the system, and a priori
it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
(2013)
This paper concerns linear first-order hyperbolic systems in one space dimension of the type
$$
\partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space. We state a kind of dissipativity condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem, i.e., either there is a nontrivial solution to the homogeneous problem (in this case the space of such solutions has finite dimension) or the nonhomogeneous problem is uniquely solvable for any right-hand side (in this case the solution depends continuously on the right-hand side). In particular, under those conditions no small denominator effects occur.
Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
Finally, in the case that all coefficients $a_j$ are $t$-independent, we show that the solutions are $C^\infty$-smooth if the data are $C^\infty$-smooth.
We describe the appearance and stability of spatio-temporal periodic
patterns (rotating waves) in unidirectional rings of coupled oscillators
with delayed couplings. We show how delays in the coupling lead
to a splitting of each rotating wave into several new ones. The appearance
of rotating waves is mediated by Hopf bifurcations of the symmetric
equilibrium.
We also conclude that the coupling delays can be effectively
replaced by increasing the number of oscillators in the chain.
The phenomena are shown for Stuart-Landau
oscillators as well as for coupled FitzHugh-Nagumo systems interacting
via excitatory chemical synapses.