G. Mathematics of Computing
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Reconstruction of photonic crystal geometries using a reduced basis method for nonlinear outputs
(2016)
Maxwell solvers based on the hp-adaptive finite element method allow for accurate geometrical modeling and high numerical accuracy. These features are indispensable for the optimization of optical properties or reconstruction of parameters through inverse processes. High computational complexity prohibits the evaluation of the solution for many parameters. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem allowing to compute solutions for a parameter configuration orders of magnitude faster. The RBM allows to evaluate linear and nonlinear outputs of interest like Fourier transform or the enhancement of the electromagnetic field in milliseconds. We apply the RBM to compute light-scattering off two dimensional photonic crystal structures made of silicon and reconstruct geometrical parameters.
Model order reduction for the time-harmonic Maxwell equation applied to complex nanostructures
(2016)
Fields such as optical metrology and computational lithography require fast and efficient methods for solving
the time-harmonic Maxwell’s equation. Highly accurate geometrical modeling and numerical accuracy atcomputational costs are a prerequisite for any simulation study of complex nano-structured photonic devices.
We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based
on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter
dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scatteringoptical wavelengths off periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such
as the width and height of the fin and gate can vary in a large range.
Nano-optical scattering problems play an important role in our modern, technologically driven society. Computers, smartphones and all kinds of electronic devices are manufactured by the semiconductor industry which relies on production using photomasks as well as optical process control. The digital world, e.g. the world wide web, is based on optical interconnects and so-called quantum computers based on optics are supposed to be next generation computers. Moreover, global economic progress demands new and sustainable energy resources and one option is to make use of the power stored in optical radiation from the sun. Additionally, understanding fundamental physics such as the optical properties of asymmetric, or chiral, structures could promote future innovations in engineering. In order to understand and manipulate these kinds of processes, physics provides a well established model: the so-called Maxwell’s equations. Stated by James Clerk Maxwell in 1862, this description of the interaction of light and matter still provides a profound basis for the analysis of electromagnetic phenomena. However, real world problems cannot be calculated using simple mathematics. Rather, computer simulations are needed to obtain solutions of the physical model. Finding suitable methods to solve these problems opens up a wide variety of possibilities. On the one hand, there are methods which require long computing times. On the other hand, some algorithms depend on high memory usage. That is why the field of numerics deals with the question which method is optimally suited for specific problems. The aim of this work is to investigate the applicability of the so-called Fourier Modal Method (FMM) to nano-optical scattering problems in general. Since simple analytical solutions are non-existent for most recent physical problems, we use the Finite Element Method (FEM) to double-check performance of the FMM. Mathematics provide reliable procedures to control the errors of numerics using the FEM. Yet up to now it has not been possible to rigorously classify the quality of the Fourier Modal Method’s results. It is not fully understood whether the process of investing more and more computing resources yields more accurate results. So, we have to ask ourselves: does the numerical method invariably converge? In spite of this uncertainty when using the FMM, it is a well established method dating back to the 1980s. This numerical method has recently been used to optimize performance of solar cells [19] as well as to improve the optical properties of so-called single-photon sources [41] which are essential for quantum cryptography. The latter is a promising candidate to increase digital security and revolutionise cryptography techniques. Furthermore, with the help of the Fourier Modal Method an important issue in optics has been partly resolved: angular filtering of light was made possible by using a mirror which becomes transparent at a certain viewing angle [77]. In addition, an improved numerical technique to design so-called Photonic Crystal waveguides based on the FMM was developed recently [15]. Photonic Crystals are used in the fields of optical bio-sensing and for the construction of novel semiconductor devices. Moreover, approaches to link the FMM and the FEM try to combine advantages of both methods to obtain fast and accurate results [81]. These ideas are closely linked to the well-known concept of Domain Decomposition within the FEM [88]. Here, one possibility to couple domains is to use the scattering matrix formalism as it is done in the FMM. In the scope of this convergence study, we state Maxwell’s equations, particularly for periodic geometries. We describe two physical phenomena of nano-optics, namely chirality and opto-electrical coupling, and define the errors of our simulations. Afterwards, the two investigated methods are analysed with respect to their general properties and a way to unify modelling physics when using both algorithms is presented. With the help of various numerical experiments, we explore convergence characteristics of the FMM and draw conclusions about the ability of this approach to provide accurate results and, consequently, its potential for research on technological innovations.
We propose a new coarse-to-fine approach to solve certain linear programs by column generation. The problems that we address contain layers corresponding to different levels of detail, i.e., coarse layers as well as fine layers. These layers are utilized to design
efficient pricing rules. In a nutshell, the method shifts the pricing of a fine linear program to a coarse counterpart. In this way, major decisions are taken in the coarse layer, while minor
details are tackled within the fine layer. We elucidate our methodology by an application to a complex railway rolling stock rotation problem. We provide comprehensive computational results that demonstrate the benefit of this new technique for the solution of large scale problems.
Bei der numerischen Lösung von Optimalsteuerungsproblemen mit
elliptischen partiellen Differentialgleichungen als Nebenbedingung
treten unvermeidlich Diskretisierungs- und Iterationsfehler auf.
Man ist aus Aufwandsgründen daran interessiert die dabei entstehenden
Fehler nicht sehr klein wählen zu müssen. In der Folge werden die linearisierten Nebenbedingungen in einem Composite-Step-Verfahren nicht exakt erfüllt. In dieser Arbeit wird der Einfluss dieser Ungenauigkeit auf das Konvergenzverhalten von Newton-Lagrange-Verfahren untersucht.
Dabei sollen mehrere einschlägige lokale Konvergenzresultate diskutiert werden.
Anschließend wird ein konkretes Composite-Step-Verfahren formuliert, in dem die Genauigkeit der
inneren Iterationsverfahren adaptiv gesteuert werden kann.
Am Ende der Arbeit wird an zwei Musterproblemen die hohe Übereinstimmung der analytischen
Voraussagen und der tatsächlichen Performanz der dargestellten Methoden demonstriert.
We consider a system dynamics model that describes the effect of human activity on natural resources. The central stocks are the accumulated profit, the industry structures, and the water resources. The model can be controlled through two time-dependent parameters. The goal in this paper is to find a parameter setting that leads to a maximization of a performance index, which reflects both environmental and economic aspects. Thus, the goal is to identify the most sustainable stock of industry structures within the model's constraints and assumptions. In order to find a proven global optimal parameter set, we formulate the System Dynamics Optimization model as a mixed-integer nonlinear problem that is accessible for numerical solvers. Due to the dynamic structure of the model, certain steps of the solution process must be handled with greater care, compared to standard non-dynamic problems. We describe our approach of solving the industry structure model and present computational results. In addition, we discuss the limitations of the approach and next steps.
Contemporary supercomputers can easily provide years of
CPU time per wall-clock hour. One challenge of today's software
development is how to harness this wast computing power in order to solve
really hard mixed integer programming instances. In 2010, two out of
six open MIPLIB2003 instances could be solved by ParaSCIP in more than
ten consecutive runs, restarting from checkpointing files.
The contribution of this paper is threefold:
For the first time, we present computational results of single runs for
those two instances. Secondly, we provide new improved upper and lower
bounds for all of the remaining four open MIPLIB2003 instances.
Finally, we explain which new developments led to these results and
discuss the current progress of ParaSCIP. Experiments were conducted on
HLRNII, on HLRN III, and on the Titan supercomputer, using up to 35,200 cores.
This thesis represents a game-theoretic investigation of the allocation of inspectors in a transportation network, comparing Nash and Stackelberg equilibrium strategies to a strategy in which inspections are conducted proportionally to the traffic volume. It contains specifications for the integration of space and time dependencies and extensive experimental tests for the application on the transportation network of German motorways using real data. Main results are that - although the formulated spot-checking game is not zero-sum - we are able to compute a Nash equilibrium using linear programming and secondly, that experimental results yield that a Nash equilibrium strategy represents a good trade-off for the Stackelberg equilibrium strategy between efficiency of controls and computation time.
Trajectory- or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality - their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge.
Bei der numerischen Lösung von Optimalsteuerungsproblemen mit
elliptischen partiellen Differentialgleichungen als Nebenbedingung
treten unvermeidlich Diskretisierungs- und Iterationsfehler auf.
Man ist aus Aufwandsgründen daran interessiert die dabei entstehenden
Fehler nicht sehr klein wählen zu müssen. In der Folge werden die linearisierten Nebenbedingungen in einem Composite-Step-Verfahren nicht exakt erfüllt. In dieser Arbeit wird der Einfluss dieser Ungenauigkeit auf das Konvergenzverhalten von Newton-Lagrange-Verfahren untersucht.
Dabei sollen mehrere einschlägige lokale Konvergenzresultate diskutiert werden.
Anschließend wird ein konkretes Composite-Step-Verfahren formuliert, in dem die Genauigkeit der
inneren Iterationsverfahren adaptiv gesteuert werden kann.
Am Ende der Arbeit wird an zwei Musterproblemen die hohe Übereinstimmung der analytischen
Voraussagen und der tatsächlichen Performanz der dargestellten Methoden demonstriert.
Sports rankings are obtained by applying a system of rules to evaluate the
performance of the participants in a competition.
We consider rankings that result from assigning an ordinal rank to each
competitor according to their performance.
We develop an integer programming model for rankings that allows us
to calculate the number of points needed to guarantee
a team the ith position, as well as the minimum number of points
that could yield the ith place.
The model is very general and can thus be applied to many types of sports.
We discuss examples coming from football (soccer), ice hockey, and
Formula~1. We answer various questions and debunk a few myths along the way.
Are 40 points enough to avoid relegation in the Bundesliga?
Do 95 points guarantee the participation of a team in the NHL playoffs?
Moreover, in the season restructuration currently under consideration in the NHL,
will it be easier or harder to access the playoffs?
Is it possible to win the Formula~1 World Championship without winning at least one race
or without even climbing once on the podium?
Finally, we observe that the optimal solutions of the aforementioned model
are associated to extreme situations which are unlikely to happen. Thus,
to get closer to realistic scenarios, we enhance the model by adding some
constraints inferred from the results of the previous years.
Two fundamental mathematical formulations for railway timetabling are compared on a common set of sample problems, representing both multiple track high density services in Europe and single track bidirectional operations in North America. One formulation, ACP, enforces against conflicts by constraining time intervals between trains, while the other formulation, HGF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and HGF return comparable solutions in the aggregate, with some significant differences in select instances, and a pattern of significant differences in performance and constraint enforcement overall.
We present a new semidefinite representation for the trace of
a real function f applied to symmetric matrices, when a
semidefinite representation of the convex function f is known. Our construction
is intuitive, and yields a representation that is more compact than the previously known one.
We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices
that for a rational number p in the interval (0,1],
the matrix X raised to the exponent p is the largest element
of a set represented by linear matrix inequalities.
We give numerical results for a problem inspired from the theory
of experimental designs, which show that the new semidefinite programming formulation
yields a speed-up factor in the order of 10.
This paper provides a highly integrated solution approach for rolling stock
planning problems in the context of intercity passenger traffic. The main
contributions are a generic hypergraph based mixed integer programming
model and an integrated algorithm for the considered rolling stock rotation
planning problem. The new developed approach is able to handle a very large
set of industrial railway requirements, such as vehicle composition,
maintenance constraints, infrastructure capacity, and regularity aspects.
By the integration of this large bundle of technical railway aspects, we show
that our approach has the power to produce implementable rolling stock
rotations for our industrial cooperation partner DB Fernverkehr.
This is the first time that the rolling stock rotations at DB Fernverkehr
could be optimized by an automated system utilizing advanced mathematical
programming techniques.
Railway Track Allocation
(2012)
This article gives an overview of the results of the author's PhD thesis. The thesis deals with the
mathematical optimization for the efficient use of
railway infrastructure. We address the optimal allocation of the available
railway track capacity - the track allocation problem. This track allocation
problem is a major challenge for a railway company, independent of whether
a free market, a private monopoly, or a public monopoly is given. 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 sizes of the problem instances.
Mathematical models and optimization techniques can result in huge gains
for both railway customers and operators, e.g., in terms of cost reductions or
service quality improvements. We tackle this challenge by developing novel
mathematical models and associated innovative algorithmic solution methods
for large scale instances. We made considerable progress on solving track
allocation problems by two main features - a novel modeling approach for the
macroscopic track allocation problem and algorithmic improvements based on the
utilization of the bundle method. This allows us to produce for the first time reliable solutions for a real world instance, i.e., the Simplon corridor in Switzerland.
This work aims to develop a new algorithm to calculate the free energy of water molecules by using a deterministic way.
For this purpose, we assume a closed system confined to a physical volume, having water molecules in gas phase.
To calculate the free energy of this sytem we utilized Mayer cluster expansion and the fluctuation free integration
method.
It is clear that a transformation to sustainable value creation is needed, because business as usual is not an option for preserving competitive advantages of leading industries. What does that mean? This contribution proposes possible approaches for a shift in existing manufacturing paradigms. In a first step, sustainability aspects from the German Sustainability Strategy and from the tools of life cycle sustainability assessment are chosen to match areas of a value creation process. Within these aspects are indicators, which can be measured within a manufacturing process. Once these data are obtained they can be used to set up a mathematical linear pulse model of manufacturing in order to analyse the evolution of the system over time, that is the transition process, by using a system dynamics approach. An increase of technology development by a factor of 2 leads to an increase of manufacturing but also to an increase of climate change. Compensation measures need to be taken. This can be done by e.g. taking money from the GDP (as an indicator of the aspect ``macroeconomic performance''). The value of the arc from that building block towards climate change must then be increased by a factor of 10. The choice of independent and representative indicators or aspects shall be validated and double-checked for their significance with the help of multi-criteria mixed-integer programming optimisation methods.
In this paper we give an analytical description on the structure of
solutions to the gas nomination validation problem in gas
transportation networks. These networks are assumed to contain no
active devices, only certain hypothetical pipelines, where the flow
of gas is modeled by a generalized version of the quadratic
Weymouth's equation. The purpose of considering generalized flow
formulas is to be able to adapt our results to various gas network
optimization problems involving gas flow formulas beyond Weymouth's
equation. Such formulas can appear in leaves of branch and bound
trees, or they can stem from discretization and linearization
carried out at active devices. We call a balanced supply-demand
vector a nomination, and the passive nomination validation problem
is to decide whether there exist pressures at the nodes generating a
given nomination. We prove that in our setup the pressure square
vectors generating a given nomination form a one-dimensional
connected and continuous curve in the pressure square space, and
this curve is a line for the classical Weymouth's equation. We also
present a visual approach for the easy comprehension of how this
solution curve arises; we give a short investigation of the set of
feasible nominations; and finally we give a proof that the
nomination validation problem in gas networks with active devices is
NP-complete.
In the last 20 years competitive analysis has become the main tool for
analyzing the quality of online algorithms. Despite of this,
competitive analysis has also been criticized: It sometimes cannot
discriminate between algorithms that exhibit significantly different
empirical behavior, or it even favors an algorithm that is worse from
an empirical point of view. Therefore, there have been several
approaches to circumvent these drawbacks. In this survey, we discuss
probabilistic alternatives for competitive analysis.
We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite
matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP.
The optimization problems of this class are semidefinite packing problems,
which are the SDP analogs to vector packing problems.
Of particular interest is the case in which our result guarantees the existence of a solution
of rank one: we show that the computation of this solution actually reduces to a
Second Order Cone Program (SOCP).
We point out an application in statistics, in the optimal design of experiments.
In this paper, we study the hop constrained chain polytope, that is, the convex hull of the incidence vectors of (s,t)-chains using at most k arcs of a given digraph, and its dominant. We use extended formulations (implied by the inherent structure of the Moore-Bellman-Ford algorithm) to derive facet defining inequalities for these polyhedra via projection. Our findings result into characterizations of all facet defining {0,+1,-1}-inequalities for the hop constrained chain polytope and all facet defining {0,1}-inequalities
for its dominant. Although the derived inequalities are already known, such classifications were not previously given to the best of our knowledge. Moreover, we use this approach to generalize so called jump inequalities, which have been introduced in a paper of Dahl and Gouveia in 2004.
This thesis is about mathematical optimization for the efficient use of railway infrastructure. We address the
optimal allocation of the available railway track capacity - the track allocation problem.
This track allocation problem is a major challenge for a railway company, independent of
whether a free market, a private monopoly, or a public monopoly is given.
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 sizes of the problem instances. Mathematical models and optimization
techniques can result in huge gains for both railway customers and operators, e.g.,
in terms of cost reductions or service quality improvements.
We tackle this challenge by developing novel mathematical models and associated innovative algorithmic
solution methods for large scale instances. This allows us to produce for the first time reliable
solutions for a real world instance, i.e., the Simplon corridor in Switzerland.
The opening chapter gives a comprehensive overview on railway planning problems.
This provides insights into the regulatory and technical framework,
it discusses the interaction of several planning steps, and identifies optimization potentials in
railway transportation. The remainder of the thesis is comprised of two major parts.
The first part is concerned with modeling railway systems to allow for resource and capacity analysis.
Railway capacity has basically two dimensions, a space dimension which are the physical
infrastructure elements as well as a time dimension that refers to the train movements, i.e.,
occupation or blocking times, on the physical infrastructure. Railway safety systems operate
on the same principle all over the world. A train has to reserve infrastructure blocks for some time to pass through.
Two trains reserving the same block of the infrastructure within the same point in time is called block conflict.
Therefore, models for railway capacity involve the definition
and calculation of reasonable running and associated reservation and
blocking times to allow for a conflict free allocation.
In the second and main part of the thesis, the optimal track
allocation problem for macroscopic models of the railway system is considered.
The literature for related problems is surveyed.
A graph-theoretic model for the track allocation problem is
developed. In that model optimal track allocations correspond to
conflict-free paths in special time-expanded graphs.
Furthermore, we made considerable progress on solving track allocation problems by two
main features - a novel modeling approach for the macroscopic track
allocation problem and algorithmic improvements based on the
utilization of the bundle method.
Finally, we go back to practice and present in the last chapter several case
studies using the tools netcast and tsopt.
We provide a computational comparison of
our new models and standard packing models used in the literature.
Our computational experience indicates that our approach, i.e.,
``configuration models'', outperforms other models. Moreover, the rapid branching
heuristic and the bundle method enable us to produce high quality solutions for very large scale
instances, which has not been possible before.
In addition, we present results for a theoretical and rather visionary auction framework
for track allocation. We discuss several auction design questions and analyze experiments of
various auction simulations.
The highlights are results for the Simplon corridor in Switzerland.
We optimized the train traffic through this tunnel using our models and
software tools.
To the best knowledge of the author and confirmed by several railway
practitioners this was the first time that fully automatically produced
track allocations on a macroscopic scale fulfill the requirements
of the originating microscopic model, withstand the evaluation in the
microscopic simulation tool OpenTrack, and exploit the infrastructure capacity.
This documents the success of our approach in practice and the usefulness
and applicability of mathematical optimization to railway track allocation.
„Diskrete Mathematik, was ist das?“, ist eine typische Frage von Lehrern mit traditioneller Mathematikausbildung, denn dort kam und kommt diskrete Mathematik kaum vor. Die etwas Aufgeschlosseneren fragen: „Wenn (schon wieder) etwas Neues unterrichtet werden soll, was soll denn dann im Lehrplan gestrichen werden?“ Auf die zweite Frage wird hier nicht eingegangen. Das Ziel dieses Aufsatzes ist es, in diskrete Mathematik einzuführen, Interesse an diesem Fachgebiet zu wecken und dazu anzuregen, dieses auch im Schulunterricht (ein wenig) zu berücksichtigen. Die Schüler und Schülerinnen werden dafür dankbar sein – eine Erfahrung, die in vielen Unterrichtsreihen gemacht wurde.
Particle methods have become indispensible in conformation dynamics to
compute transition rates in protein folding, binding processes and
molecular design, to mention a few.
Conformation dynamics requires at a decomposition of a molecule's position
space into metastable conformations.
In this paper, we show how this decomposition
can be obtained via the design of either ``soft'' or ``hard''
molecular conformations.
We show, that the soft approach results in a larger metastabilitiy of
the decomposition and is thus more advantegous. This is illustrated
by a simulation of Alanine Dipeptide.