@phdthesis{10594,
abstract = {Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute
the set of optimal compromises (the Pareto set) between the conflicting objectives.
Since – in contrast to the solution of a single objective optimization problem – the
Pareto set generally consists of an infinite number of solutions, the computational
effort can quickly become challenging. This is even more the case when many problems have to be solved, when the number of objectives is high, or when the objectives
are costly to evaluate. Consequently, this thesis is devoted to the identification and
exploitation of structure both in the Pareto set and the dynamics of the underlying
model as well as to the development of efficient algorithms for solving problems with
additional parameters, with a high number of objectives or with PDE-constraints.
These three challenges are addressed in three respective parts.
In the first part, predictor-corrector methods are extended to entire Pareto sets.
When certain smoothness assumptions are satisfied, then the set of parameter dependent Pareto sets possesses additional structure, i.e. it is a manifold. The tangent
space can be approximated numerically which yields a direction for the predictor
step. In the corrector step, the predicted set converges to the Pareto set at a new
parameter value. The resulting algorithm is applied to an example from autonomous
driving.
In the second part, the hierarchical structure of Pareto sets is investigated. When
considering a subset of the objectives, the resulting solution is a subset of the Pareto
set of the original problem. Under additional smoothness assumptions, the respective subsets are located on the boundary of the Pareto set of the full problem. This
way, the “skeleton” of a Pareto set can be computed and due to the exponential
increase in computing time with the number of objectives, the computations of
these subsets are significantly faster which is demonstrated using an example from
industrial laundries.
In the third part, PDE-constrained multiobjective optimal control problems are
addressed by reduced order modeling methods. Reduced order models exploit the
structure in the system dynamics, for example by describing the dynamics of only the
most energetic modes. The model reduction introduces an error in both the function values and their gradients, which has to be taken into account in the development of
algorithms. Both scalarization and set-oriented approaches are coupled with reduced
order modeling. Convergence results are presented and the numerical benefit is
investigated. The algorithms are applied to semi-linear heat flow problems as well
as to the Navier-Stokes equations.
},
author = {Peitz, Sebastian},
title = {{ Exploiting structure in multiobjective optimization and optimal control}},
doi = {10.17619/UNIPB/1-176},
year = {2017},
}
@article{16499,
author = {Biasco, Luca and Di Gregorio, Laura},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
pages = {303--362},
title = {{A Birkhoff–Lewis Type Theorem for the Nonlinear Wave Equation}},
doi = {10.1007/s00205-009-0240-y},
year = {2009},
}
@inproceedings{8758,
abstract = {In this contribution we compare two different approaches to the implementation of a Model Predictive Controller in an electric vehicle with respect to the quality of the solution and real-time applicability. The goal is to develop an intelligent cruise control in order to extend the vehicle range, i.e. to minimize energy consumption, by computing the optimal torque profile for a given track. On the one hand, a path-based linear model with strong simplifications regarding the vehicle dynamics is used. On the other hand, a nonlinear model is employed in which the dynamics of the mechanical and electrical subsystem are modeled.},
author = {Eckstein, Julian and Peitz, Sebastian and Schäfer, Kai and Friedel, Patrick and Köhler, Ulrich and Molo, Mirko Hessel-von and Ober-Blöbaum, Sina and Dellnitz, Michael},
booktitle = {Procedia Technology},
issn = {2212-0173},
pages = {465--472},
title = {{A Comparison of two Predictive Approaches to Control the Longitudinal Dynamics of Electric Vehicles}},
doi = {10.1016/j.protcy.2016.08.059},
year = {2016},
}
@inbook{16553,
author = {Dellnitz, Michael and Froyland, Gary and Sertl, Stefan},
booktitle = {Equadiff 99},
isbn = {9789810243593},
title = {{A Conjecture on the Existence of Isolated Eigenvalues of the Perron-Frobenius Operator}},
doi = {10.1142/9789812792617_0199},
year = {2000},
}
@inproceedings{8750,
abstract = {In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.},
author = {Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael},
booktitle = {Numerical and Evolutionary Optimization – NEO 2017},
isbn = {9783319961033},
issn = {1860-949X},
title = {{A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems}},
doi = {10.1007/978-3-319-96104-0_2},
year = {2018},
}
@inproceedings{16666,
author = {Schütze, Oliver and Talbi, El-ghazali and Pulido, Gregorio Toscano and Coello, Carlos Coello and Santana-Quintero, Luis Vicente},
booktitle = {2007 IEEE Swarm Intelligence Symposium},
isbn = {1424407087},
title = {{A Memetic PSO Algorithm for Scalar Optimization Problems}},
doi = {10.1109/sis.2007.368036},
year = {2007},
}
@article{16574,
author = {Dellnitz, Michael and Ober-Blöbaum, Sina and Post, Marcus and Schütze, Oliver and Thiere, Bianca},
issn = {0923-2958},
journal = {Celestial Mechanics and Dynamical Astronomy},
pages = {33--59},
title = {{A multi-objective approach to the design of low thrust space trajectories using optimal control}},
doi = {10.1007/s10569-009-9229-y},
year = {2009},
}
@article{8756,
abstract = {We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.},
author = {Peitz, Sebastian and Schäfer, Kai and Ober-Blöbaum, Sina and Eckstein, Julian and Köhler, Ulrich and Dellnitz, Michael},
issn = {2405-8963},
journal = {IFAC-PapersOnLine},
number = {1},
pages = {8674--8679},
title = {{A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles}},
doi = {10.1016/j.ifacol.2017.08.1526},
volume = {50},
year = {2017},
}
@article{16657,
author = {Peitz, Sebastian and Schäfer, Kai and Ober-Blöbaum, Sina and Eckstein, Julian and Köhler, Ulrich and Dellnitz, Michael},
issn = {2405-8963},
journal = {IFAC-PapersOnLine},
pages = {8674--8679},
title = {{A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles * *This research was funded by the German Federal Ministry of Education and Research (BMBF) within the Leading-Edge Cluster Intelligent Technical Systems OstWestfalenLippe (it’s OWL).}},
doi = {10.1016/j.ifacol.2017.08.1526},
year = {2017},
}
@inproceedings{16643,
author = {Ober-Blöbaum, Sina and Seifried, Albert},
booktitle = {2013 European Control Conference (ECC)},
isbn = {9783033039629},
title = {{A multiobjective optimization approach for optimal control problems of mechanical systems with uncertainties}},
doi = {10.23919/ecc.2013.6669594},
year = {2013},
}
@article{16678,
author = {Witting, Katrin and Schulz, Bernd and Dellnitz, Michael and Böcker, Joachim and Fröhleke, Norbert},
issn = {1433-2779},
journal = {International Journal on Software Tools for Technology Transfer},
pages = {223--231},
title = {{A new approach for online multiobjective optimization of mechatronic systems}},
doi = {10.1007/s10009-008-0066-1},
year = {2008},
}
@inbook{16664,
author = {Schütze, Oliver},
booktitle = {Lecture Notes in Computer Science},
isbn = {9783540018698},
issn = {0302-9743},
title = {{A New Data Structure for the Nondominance Problem in Multi-objective Optimization}},
doi = {10.1007/3-540-36970-8_36},
year = {2003},
}
@article{16542,
author = {Dellnitz, M and Melbourne, I},
issn = {0951-7715},
journal = {Nonlinearity},
pages = {1067--1075},
title = {{A note on the shadowing lemma and symmetric periodic points}},
doi = {10.1088/0951-7715/8/6/010},
year = {1995},
}
@article{16527,
author = {Day, S. and Junge, O. and Mischaikow, K.},
issn = {1536-0040},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {117--160},
title = {{A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems}},
doi = {10.1137/030600210},
year = {2004},
}
@article{16619,
author = {Junge, Oliver and Osinga, Hinke M.},
issn = {1292-8119},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {259--270},
title = {{A set oriented approach to global optimal control}},
doi = {10.1051/cocv:2004006},
year = {2004},
}
@article{16613,
author = {Grüne, Lars and Junge, Oliver},
issn = {0167-6911},
journal = {Systems & Control Letters},
pages = {169--180},
title = {{A set oriented approach to optimal feedback stabilization}},
doi = {10.1016/j.sysconle.2004.08.005},
year = {2005},
}
@article{16581,
author = {Dellnitz, Michael and Klus, Stefan and Ziessler, Adrian},
issn = {1536-0040},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {120--138},
title = {{A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty}},
doi = {10.1137/16m1072735},
year = {2017},
}
@article{16710,
abstract = {In this work we present a set-oriented path following method for the computation of relative global
attractors of parameter-dependent dynamical systems. We start with an initial approximation of the
relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method.
By using previously obtained approximations of the parameter-dependent relative global attractor
we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole
subdivision procedure. We illustrate the feasibility of the set-oriented path following method by
exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence
and of large-scale atmospheric regime changes .
},
author = {Gerlach, Raphael and Ziessler, Adrian and Eckhardt, Bruno and Dellnitz, Michael},
issn = {1536-0040},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {705--723},
title = {{A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors}},
doi = {10.1137/19m1247139},
year = {2020},
}
@article{17015,
author = {Dellnitz, Michael and Hohmann, Andreas},
issn = {0029-599X},
journal = {Numerische Mathematik},
pages = {293--317},
title = {{A subdivision algorithm for the computation of unstable manifolds and global attractors}},
doi = {10.1007/s002110050240},
volume = {75},
year = {1997},
}
@article{16627,
abstract = { The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system. },
author = {Krauskopf, B. and Osinga, H. M. and Doedel, E. J. and Henderson, M. E. and Guckenheimer, J. and Vladimirsky, A. and Dellnitz, M. and Junge, O.},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {763--791},
title = {{A Survey of Methods for Computing (un)stable Manifolds of Vector Fields}},
doi = {10.1142/s0218127405012533},
year = {2005},
}
@article{8751,
abstract = {Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort.},
author = {Peitz, Sebastian and Dellnitz, Michael},
issn = {2297-8747},
journal = {Mathematical and Computational Applications},
number = {2},
title = {{A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction}},
doi = {10.3390/mca23020030},
volume = {23},
year = {2018},
}
@article{16677,
author = {Witting, Katrin and Ober-Blöbaum, Sina and Dellnitz, Michael},
issn = {0925-5001},
journal = {Journal of Global Optimization},
pages = {331--345},
title = {{A variational approach to define robustness for parametric multiobjective optimization problems}},
doi = {10.1007/s10898-012-9972-6},
year = {2013},
}
@article{16532,
author = {Dellnitz, M and Heinrich, C},
issn = {0951-7715},
journal = {Nonlinearity},
pages = {1039--1066},
title = {{Admissible symmetry increasing bifurcations}},
doi = {10.1088/0951-7715/8/6/009},
year = {1995},
}
@article{16714,
author = {Vieluf, Solveig and Mora, Karin and Gölz, Christian and Reuter, Eva-Maria and Godde, Ben and Dellnitz, Michael and Reinsberger, Claus and Voelcker-Rehage, Claudia},
issn = {0306-4522},
journal = {Neuroscience},
pages = {203--213},
title = {{Age- and Expertise-Related Differences of Sensorimotor Network Dynamics during Force Control}},
doi = {10.1016/j.neuroscience.2018.07.025},
year = {2018},
}
@article{16535,
abstract = { Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time. },
author = {Dellnitz, Michael and Junge, Oliver},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {2475--2485},
title = {{Almost Invariant Sets in Chua's Circuit}},
doi = {10.1142/s0218127497001655},
year = {1997},
}
@article{16614,
author = {Guder, Rabbijah and Dellnitz, Michael and Kreuzer, Edwin},
issn = {0960-0779},
journal = {Chaos, Solitons & Fractals},
pages = {525--534},
title = {{An adaptive method for the approximation of the generalized cell mapping}},
doi = {10.1016/s0960-0779(96)00118-x},
year = {1997},
}
@article{16536,
author = {Dellnitz, Michael and Junge, Oliver},
issn = {1432-9360},
journal = {Computing and Visualization in Science},
pages = {63--68},
title = {{An adaptive subdivision technique for the approximation of attractors and invariant measures}},
doi = {10.1007/s007910050006},
year = {1998},
}
@article{16617,
author = {Junge, Oliver},
issn = {1468-9367},
journal = {Dynamical Systems},
pages = {213--222},
title = {{An adaptive subdivision technique for the approximation of attractors and invariant measures: proof of convergence}},
doi = {10.1080/14689360109696233},
year = {2001},
}
@article{16624,
author = {Klus, Stefan and Sahai, Tuhin and Liu, Cong and Dellnitz, Michael},
issn = {0377-0427},
journal = {Journal of Computational and Applied Mathematics},
pages = {3053--3062},
title = {{An efficient algorithm for the parallel solution of high-dimensional differential equations}},
doi = {10.1016/j.cam.2010.12.026},
year = {2011},
}
@article{16867,
abstract = {In this article, we present an efficient descent method for locally Lipschitz
continuous multiobjective optimization problems (MOPs). The method is realized
by combining a theoretical result regarding the computation of descent
directions for nonsmooth MOPs with a practical method to approximate the
subdifferentials of the objective functions. We show convergence to points
which satisfy a necessary condition for Pareto optimality. Using a set of test
problems, we compare our method to the multiobjective proximal bundle method by
M\"akel\"a. The results indicate that our method is competitive while being
easier to implement. While the number of objective function evaluations is
larger, the overall number of subgradient evaluations is lower. Finally, we
show that our method can be combined with a subdivision algorithm to compute
entire Pareto sets of nonsmooth MOPs.},
author = {Gebken, Bennet and Peitz, Sebastian},
journal = {Journal of Optimization Theory and Applications},
title = {{An efficient descent method for locally Lipschitz multiobjective optimization problems}},
doi = {10.1007/s10957-020-01803-w},
year = {2021},
}
@inproceedings{17048,
author = {Timmermann, Robert and Dellnitz, Michael},
booktitle = {Performance Analysis of Sport IX, Part 8, Routledge},
title = {{Analysis of team and player performance using recorded trajectory data}},
year = {2013},
}
@unpublished{16293,
abstract = {Kernel transfer operators, which can be regarded as approximations of
transfer operators such as the Perron-Frobenius or Koopman operator in
reproducing kernel Hilbert spaces, are defined in terms of covariance and
cross-covariance operators and have been shown to be closely related to the
conditional mean embedding framework developed by the machine learning
community. The goal of this paper is to show how the dominant eigenfunctions of
these operators in combination with gradient-based optimization techniques can
be used to detect long-lived coherent patterns in high-dimensional time-series
data. The results will be illustrated using video data and a fluid flow
example.},
author = {Klus, Stefan and Peitz, Sebastian and Schuster, Ingmar},
booktitle = {arXiv:1805.10118},
title = {{Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions}},
year = {2018},
}
@inbook{16658,
author = {Preis, Robert and Monien, Burkhard and Schamberger, Stefan},
booktitle = {Handbook of Approximation Algorithms and Metaheuristics},
isbn = {9781584885504},
issn = {2154-4573},
title = {{Approximation Algorithms for Multilevel Graph Partitioning}},
doi = {10.1201/9781420010749.ch60},
year = {2007},
}
@article{16583,
author = {Demoures, Francois and Gay-Balmaz, Francois and Leitz, Thomas and Leyendecker, Sigrid and Ober-Blöbaum, Sina and Ratiu, Tudor S.},
issn = {1617-7061},
journal = {PAMM},
pages = {45--46},
title = {{Asynchronous variational Lie group integration for geometrically exact beam dynamics}},
doi = {10.1002/pamm.201310018},
year = {2013},
}
@article{17014,
author = {Dellnitz, Michael},
journal = {Schlaglichter der Forschung: Zum 75. Jahrestag der Universität Hamburg},
pages = {411--428},
title = {{Collisions of chaotic attractors}},
year = {1994},
}
@inbook{16584,
author = {Deuflhard, Peter and Dellnitz, Michael and Junge, Oliver and Schütte, Christof},
booktitle = {Computational Molecular Dynamics: Challenges, Methods, Ideas},
isbn = {9783540632429},
issn = {1439-7358},
title = {{Computation of Essential Molecular Dynamics by Subdivision Techniques}},
doi = {10.1007/978-3-642-58360-5_5},
year = {1999},
}
@article{16545,
author = {Dellnitz, M. and Witting, K.},
issn = {1752-5055},
journal = {International Journal of Computing Science and Mathematics},
title = {{Computation of robust Pareto points}},
doi = {10.1504/ijcsm.2009.027876},
year = {2009},
}
@article{16498,
author = {Aston, P. J. and Dellnitz, M.},
issn = {1364-5021},
journal = {Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences},
pages = {2933--2955},
title = {{Computation of the dominant Lyapunov exponent via spatial integration using matrix norms}},
doi = {10.1098/rspa.2003.1143},
year = {2003},
}
@article{17012,
author = {Dellnitz, Michael},
journal = {IMA Journal of Numerical Analysis},
number = {3},
pages = {429--455},
title = {{Computational bifurcation of periodic solutions in systems with symmetry}},
doi = {10.1093/imanum/12.3.429},
volume = {12},
year = {1992},
}
@article{16682,
author = {Dellnitz, Michael and Werner, Bodo},
issn = {0377-0427},
journal = {Journal of Computational and Applied Mathematics},
pages = {97--123},
title = {{Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points}},
doi = {10.1016/0377-0427(89)90150-7},
year = {1989},
}
@article{16608,
author = {Gail, Tobias and Leyendecker, Sigrid and Ober-Blöbaum, Sina},
issn = {1617-7061},
journal = {PAMM},
pages = {43--44},
title = {{Computing time investigations for variational multirate integration}},
doi = {10.1002/pamm.201310017},
year = {2013},
}
@inbook{16543,
author = {Dellnitz, Michael and Preis, Robert},
booktitle = {Lecture Notes in Computer Science},
isbn = {9783540405542},
issn = {0302-9743},
title = {{Congestion and Almost Invariant Sets in Dynamical Systems}},
doi = {10.1007/3-540-45084-x_8},
year = {2003},
}
@inbook{16575,
author = {Dellnitz, Michael and Padberg, Kathrin and Preis, Robert and Thiere, Bianca},
booktitle = {Nonlinear Science and Complexity},
isbn = {9789048198832},
title = {{Continuous and Discrete Concepts for Detecting Transport Barriers in the Planar Circular Restricted Three Body Problem}},
doi = {10.1007/978-90-481-9884-9_12},
year = {2010},
}
@article{16709,
author = {Sahai, Tuhin and Ziessler, Adrian and Klus, Stefan and Dellnitz, Michael},
issn = {0924-090X},
journal = {Nonlinear Dynamics},
title = {{Continuous relaxations for the traveling salesman problem}},
doi = {10.1007/s11071-019-05092-5},
year = {2019},
}
@article{16597,
author = {Flaßkamp, Kathrin and Timmermann, Julia and Ober-Blöbaum, Sina and Trächtler, Ansgar},
issn = {0020-7179},
journal = {International Journal of Control},
pages = {1886--1905},
title = {{Control strategies on stable manifolds for energy-efficient swing-ups of double pendula}},
doi = {10.1080/00207179.2014.893450},
year = {2014},
}
@unpublished{16292,
abstract = {In a recent article, we presented a framework to control nonlinear partial
differential equations (PDEs) by means of Koopman operator based reduced models
and concepts from switched systems. The main idea was to transform a control
system into a set of autonomous systems for which the optimal switching
sequence has to be computed. These individual systems can be approximated very
efficiently by reduced order models obtained from data, and one can guarantee
equality of the full and the reduced objective function under certain
assumptions. In this article, we extend these results to continuous control
inputs using convex combinations of multiple Koopman operators corresponding to
constant controls, which results in a bilinear control system. Although
equality of the objectives can be carried over when the PDE depends linearly on
the control, we show that this approach is also valid in other scenarios using
several flow control examples of varying complexity.},
author = {Peitz, Sebastian},
booktitle = {arXiv:1801.06419},
title = {{Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data}},
year = {2018},
}
@article{16668,
author = {Schütze, Oliver and Laumanns, Marco and Coello Coello, Carlos A. and Dellnitz, Michael and Talbi, El-Ghazali},
issn = {0925-5001},
journal = {Journal of Global Optimization},
pages = {559--577},
title = {{Convergence of stochastic search algorithms to finite size pareto set approximations}},
doi = {10.1007/s10898-007-9265-7},
year = {2008},
}
@inbook{16665,
author = {Schütze, Oliver and Mostaghim, Sanaz and Dellnitz, Michael and Teich, Jürgen},
booktitle = {Lecture Notes in Computer Science},
isbn = {9783540018698},
issn = {0302-9743},
title = {{Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques}},
doi = {10.1007/3-540-36970-8_9},
year = {2003},
}
@article{16684,
author = {Dellnitz, M. and Sch�tze, O. and Hestermeyer, T.},
issn = {0022-3239},
journal = {Journal of Optimization Theory and Applications},
pages = {113--136},
title = {{Covering Pareto Sets by Multilevel Subdivision Techniques}},
doi = {10.1007/s10957-004-6468-7},
year = {2005},
}
@article{16550,
author = {Dellnitz, Michael and Field, Michael and Golubitsky, Martin and Ma, Jun and Hohmann, Andreas},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {1243--1247},
title = {{Cycling Chaos}},
doi = {10.1142/s0218127495000909},
year = {1995},
}
@article{16288,
abstract = {We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.},
author = {Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof},
issn = {0167-2789},
journal = {Physica D: Nonlinear Phenomena},
title = {{Data-driven approximation of the Koopman generator: Model reduction, system identification, and control}},
doi = {10.1016/j.physd.2020.132416},
volume = {406},
year = {2020},
}
@article{16309,
abstract = {In recent years, the success of the Koopman operator in dynamical systems
analysis has also fueled the development of Koopman operator-based control
frameworks. In order to preserve the relatively low data requirements for an
approximation via Dynamic Mode Decomposition, a quantization approach was
recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control
of nonlinear dynamical systems can be realized by means of switched systems
techniques, using only a finite set of autonomous Koopman operator-based
reduced models. These individual systems can be approximated very efficiently
from data. The main idea is to transform a control system into a set of
autonomous systems for which the optimal switching sequence has to be computed.
In this article, we extend these results to continuous control inputs using
relaxation. This way, we combine the advantages of the data efficiency of
approximating a finite set of autonomous systems with continuous controls. We
show that when using the Koopman generator, this relaxation --- realized by
linear interpolation between two operators --- does not introduce any error for
control affine systems. This allows us to control high-dimensional nonlinear
systems using bilinear, low-dimensional surrogate models. The efficiency of the
proposed approach is demonstrated using several examples with increasing
complexity, from the Duffing oscillator to the chaotic fluidic pinball.},
author = {Peitz, Sebastian and Otto, Samuel E. and Rowley, Clarence W.},
journal = {SIAM Journal on Applied Dynamical Systems},
number = {3},
pages = {2162--2193},
title = {{Data-Driven Model Predictive Control using Interpolated Koopman Generators}},
doi = {10.1137/20M1325678},
volume = {19},
year = {2020},
}
@article{16290,
abstract = {The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.},
author = {Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael},
issn = {0935-4964},
journal = {Theoretical and Computational Fluid Dynamics},
pages = {577–591},
title = {{Deep model predictive flow control with limited sensor data and online learning}},
doi = {10.1007/s00162-020-00520-4},
volume = {34},
year = {2020},
}
@inproceedings{16631,
author = {Li, R. and Pottharst, A. and Frohieke, N. and Becker, J. and Witting, K. and Dellnitz, M. and Znamenshchykov, O. and Feldmann, R.},
booktitle = {Twentieth Annual IEEE Applied Power Electronics Conference and Exposition, 2005. APEC 2005.},
isbn = {0780389751},
title = {{Design and implementation of a hybrid energy supply system for railway vehicles}},
doi = {10.1109/apec.2005.1452979},
year = {2005},
}
@article{16669,
author = {Schütze, Oliver and Vasile, Massimiliano and Junge, Oliver and Dellnitz, Michael and Izzo, Dario},
issn = {0305-215X},
journal = {Engineering Optimization},
pages = {155--181},
title = {{Designing optimal low-thrust gravity-assist trajectories using space pruning and a multi-objective approach}},
doi = {10.1080/03052150802391734},
year = {2009},
}
@article{16600,
author = {Froyland, Gary and Dellnitz, Michael},
issn = {1064-8275},
journal = {SIAM Journal on Scientific Computing},
pages = {1839--1863},
title = {{Detecting and Locating Near-Optimal Almost-Invariant Sets and Cycles}},
doi = {10.1137/s106482750238911x},
year = {2003},
}
@inproceedings{16675,
author = {Thiere, Bianca and Ober-Blöbaum, Sina and Pergola, Pierpaolo},
booktitle = {AIAA/AAS Astrodynamics Specialist Conference},
isbn = {9781624101502},
title = {{Detecting Initial Guesses for Trajectories in the (P)CRTBP}},
doi = {10.2514/6.2010-7962},
year = {2010},
}
@article{16518,
author = {Barany, Ernest and Dellnitz, Michael and Golubitsky, Martin},
issn = {0167-2789},
journal = {Physica D: Nonlinear Phenomena},
pages = {66--87},
title = {{Detecting the symmetry of attractors}},
doi = {10.1016/0167-2789(93)90198-a},
year = {1993},
}
@article{16602,
author = {Froyland, Gary and Padberg, Kathrin and England, Matthew H. and Treguier, Anne Marie},
issn = {0031-9007},
journal = {Physical Review Letters},
title = {{Detection of Coherent Oceanic Structures via Transfer Operators}},
doi = {10.1103/physrevlett.98.224503},
year = {2007},
}
@inproceedings{8761,
abstract = {In this contribution, the range extension problem of electric vehicles is addressed. To this aim, an intelligent cruise control is developed based on the formulation of an optimal control problem. Solutions of this optimal control problem are energy efficient accelerator pedal position profiles. They can be computed numerically by a direct optimal control method using sequential quadratic programming. The approach is applied to two different driving scenarios. The results show that the energy efficiency is increased by using optimal control for both an artificial and a realistic scenario.},
author = {Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian},
booktitle = {Procedia Technology},
issn = {2212-0173},
pages = {285--294},
title = {{Development of an Intelligent Cruise Control Using Optimal Control Methods}},
doi = {10.1016/j.protcy.2014.09.082},
volume = {15},
year = {2014},
}
@article{16586,
author = {Elsässer, Robert and Monien, Burkhard and Preis, Robert},
issn = {1432-4350},
journal = {Theory of Computing Systems},
pages = {305--320},
title = {{Diffusion Schemes for Load Balancing on Heterogeneous Networks}},
doi = {10.1007/s00224-002-1056-4},
year = {2002},
}
@article{16635,
author = {Meyer, A.},
issn = {1474-6670},
journal = {IFAC Proceedings Volumes},
pages = {28--33},
title = {{Discontinuity Induced Bifurcations in Timed Continuous Petri Nets}},
doi = {10.3182/20121003-3-mx-4033.00008},
year = {2012},
}
@inproceedings{16630,
abstract = {This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.},
author = {Leyendecker, Sigrid and Ober-Blöbaum, Sina and Marsden, Jerrold E. and Ortiz, Michael},
booktitle = {Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C},
isbn = {079184806X},
title = {{Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics}},
doi = {10.1115/detc2007-34574},
year = {2007},
}
@article{16582,
author = {Demoures, F. and Gay-Balmaz, F. and Leyendecker, S. and Ober-Blöbaum, S. and Ratiu, T. S. and Weinand, Y.},
issn = {0029-599X},
journal = {Numerische Mathematik},
pages = {73--123},
title = {{Discrete variational Lie group formulation of geometrically exact beam dynamics}},
doi = {10.1007/s00211-014-0659-4},
year = {2015},
}
@inproceedings{16672,
author = {Specht, Andreas and Ober-Blobaum, Sina and Wallscheid, Oliver and Romaus, Christoph and Bocker, Joachim},
booktitle = {2013 International Electric Machines & Drives Conference},
isbn = {9781467349741},
title = {{Discrete-time model of an IPMSM based on variational integrators}},
doi = {10.1109/iemdc.2013.6556322},
year = {2013},
}
@inproceedings{16594,
author = {Flasskamp, Kathrin and Murphey, Todd and Ober-Blobaum, Sina},
booktitle = {2013 European Control Conference (ECC)},
isbn = {9783033039629},
title = {{Discretized switching time optimization problems}},
doi = {10.23919/ecc.2013.6669577},
year = {2013},
}
@inproceedings{16623,
author = {Klöpper, Benjamin and Podlogar, Herbert and Gausemeier, Jürgen and Witting, Katrin},
booktitle = {2008 19th International Conference on Database and Expert Systems Applications},
isbn = {9780769532998},
title = {{Domain Spanning Search for the Identification of Solution Patterns for the Conceptual Design of Self-Optimizing Systems}},
doi = {10.1109/dexa.2008.111},
year = {2008},
}
@inbook{16544,
author = {Dellnitz, Michael and Scheurle, Jürgen},
booktitle = {Dynamics, Bifurcation and Symmetry},
isbn = {9789401044134},
title = {{Eigenvalue Movement for a Class of Reversible Hamiltonian Systems with Three Degrees of Freedom}},
doi = {10.1007/978-94-011-0956-7_9},
year = {1994},
}
@inproceedings{16589,
author = {Flaskamp, K. and Ober-Blobaum, S.},
booktitle = {2012 American Control Conference (ACC)},
isbn = {9781457710964},
title = {{Energy efficient control for mechanical systems based on inherent dynamical structures}},
doi = {10.1109/acc.2012.6314820},
year = {2012},
}
@inproceedings{16626,
author = {Knoke, Tobias and Romaus, Christoph and Bocker, Joachim and Dell'Aere, Alessandro and Witting, Katrin},
booktitle = {IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics},
isbn = {1424403901},
issn = {1553-572X},
title = {{Energy Management for an Onboard Storage System Based on Multi-Objective Optimization}},
doi = {10.1109/iecon.2006.347705},
year = {2006},
}
@inproceedings{16663,
author = {Schneider, T. and Schulz, B. and Henke, C. and Witting, K. and Steenken, D. and Bocker, J.},
booktitle = {2009 IEEE International Electric Machines and Drives Conference},
isbn = {9781424442515},
title = {{Energy transfer via linear doubly-fed motor in different operating modes}},
doi = {10.1109/iemdc.2009.5075267},
year = {2009},
}
@article{16297,
abstract = {In real-world problems, uncertainties (e.g., errors in the measurement,
precision errors) often lead to poor performance of numerical algorithms when
not explicitly taken into account. This is also the case for control problems,
where optimal solutions can degrade in quality or even become infeasible. Thus,
there is the need to design methods that can handle uncertainty. In this work,
we consider nonlinear multi-objective optimal control problems with uncertainty
on the initial conditions, and in particular their incorporation into a
feedback loop via model predictive control (MPC). In multi-objective optimal
control, an optimal compromise between multiple conflicting criteria has to be
found. For such problems, not much has been reported in terms of uncertainties.
To address this problem class, we design an offline/online framework to compute
an approximation of efficient control strategies. This approach is closely
related to explicit MPC for nonlinear systems, where the potentially expensive
optimization problem is solved in an offline phase in order to enable fast
solutions in the online phase. In order to reduce the numerical cost of the
offline phase, we exploit symmetries in the control problems. Furthermore, in
order to ensure optimality of the solutions, we include an additional online
optimization step, which is considerably cheaper than the original
multi-objective optimization problem. We test our framework on a car
maneuvering problem where safety and speed are the objectives. The
multi-objective framework allows for online adaptations of the desired
objective. Alternatively, an automatic scalarizing procedure yields very
efficient feedback controls. Our results show that the method is capable of
designing driving strategies that deal better with uncertainties in the initial
conditions, which translates into potentially safer and faster driving
strategies.},
author = {Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian},
journal = {International Journal of Robust and Nonlinear Control},
number = {17},
pages = {7593--7618},
title = {{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}},
doi = {10.1002/rnc.5197},
volume = {30},
year = {2020},
}
@article{16294,
abstract = {Model predictive control is a prominent approach to construct a feedback
control loop for dynamical systems. Due to real-time constraints, the major
challenge in MPC is to solve model-based optimal control problems in a very
short amount of time. For linear-quadratic problems, Bemporad et al. have
proposed an explicit formulation where the underlying optimization problems are
solved a priori in an offline phase. In this article, we present an extension
of this concept in two significant ways. We consider nonlinear problems and -
more importantly - problems with multiple conflicting objective functions. In
the offline phase, we build a library of Pareto optimal solutions from which we
then obtain a valid compromise solution in the online phase according to a
decision maker's preference. Since the standard multi-parametric programming
approach is no longer valid in this situation, we instead use interpolation
between different entries of the library. To reduce the number of problems that
have to be solved in the offline phase, we exploit symmetries in the dynamical
system and the corresponding multiobjective optimal control problem. The
results are verified using two different examples from autonomous driving.},
author = {Ober-Blöbaum, Sina and Peitz, Sebastian},
journal = {International Journal of Robust and Nonlinear Control},
number = {2},
pages = {380--403},
title = {{Explicit multiobjective model predictive control for nonlinear systems with symmetries}},
doi = {10.1002/rnc.5281},
volume = {31},
year = {2021},
}
@article{16552,
author = {Dellnitz, Michael and Hohmann, Andreas and Junge, Oliver and Rumpf, Martin},
issn = {1054-1500},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
pages = {221--228},
title = {{Exploring invariant sets and invariant measures}},
doi = {10.1063/1.166223},
year = {1997},
}
@inbook{16598,
author = {Froyland, Gary},
booktitle = {Nonlinear Dynamics and Statistics},
isbn = {9781461266488},
title = {{Extracting Dynamical Behavior via Markov Models}},
doi = {10.1007/978-1-4612-0177-9_12},
year = {2001},
}
@inbook{16289,
abstract = {In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.},
author = {Peitz, Sebastian and Klus, Stefan},
booktitle = {Lecture Notes in Control and Information Sciences},
isbn = {9783030357122},
issn = {0170-8643},
pages = {257--282},
publisher = {Springer},
title = {{Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator}},
doi = {10.1007/978-3-030-35713-9_10},
volume = {484},
year = {2020},
}
@article{16556,
author = {Dellnitz, M.},
issn = {0272-4979},
journal = {IMA Journal of Numerical Analysis},
pages = {167--185},
title = {{Finding zeros by multilevel subdivision techniques}},
doi = {10.1093/imanum/22.2.167},
year = {2002},
}
@inproceedings{10597,
abstract = {In comparison to classical control approaches in the field of electrical drives like the field-oriented control (FOC), model predictive control (MPC) approaches are able to provide a higher control performance. This refers to shorter settling times, lower overshoots, and a better decoupling of control variables in case of multi-variable controls. However, this can only be achieved if the used prediction model covers the actual behavior of the plant sufficiently well. In case of model deviations, the performance utilizing MPC remains below its potential. This results in effects like increased current ripple or steady state setpoint deviations. In order to achieve a high control performance, it is therefore necessary to adapt the model to the real plant behavior. When using an online system identification, a less accurate model is sufficient for commissioning of the drive system. In this paper, the combination of a finite-control-set MPC (FCS-MPC) with a system identification is proposed. The method does not require high-frequency signal injection, but uses the measured values already required for the FCS-MPC. An evaluation of the least squares-based identification on a laboratory test bench showed that the model accuracy and thus the control performance could be improved by an online update of the prediction models.},
author = {Hanke, Soren and Peitz, Sebastian and Wallscheid, Oliver and Böcker, Joachim and Dellnitz, Michael},
booktitle = {2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)},
isbn = {9781538694145},
title = {{Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification}},
doi = {10.1109/precede.2019.8753313},
year = {2019},
}
@article{16548,
author = {Dellnitz, M and Melbourne, I and Marsden, J E},
issn = {0951-7715},
journal = {Nonlinearity},
pages = {979--996},
title = {{Generic bifurcation of Hamiltonian vector fields with symmetry}},
doi = {10.1088/0951-7715/5/4/008},
year = {1992},
}
@inbook{16547,
author = {Dellnitz, Michael and Marsden, Jerrold E. and Melbourne, Ian and Scheurle, Jürgen},
booktitle = {Bifurcation and Symmetry},
isbn = {9783034875387},
title = {{Generic Bifurcations of Pendula}},
doi = {10.1007/978-3-0348-7536-3_10},
year = {1992},
}
@article{16541,
author = {Dellnitz, Michael and Melbourne, Ian},
issn = {0377-0427},
journal = {Journal of Computational and Applied Mathematics},
pages = {249--259},
title = {{Generic movement of eigenvalues for equivariant self-adjoint matrices}},
doi = {10.1016/0377-0427(94)90032-9},
year = {1994},
}
@article{16612,
author = {Grüne, L. and Junge, O.},
issn = {0022-3239},
journal = {Journal of Optimization Theory and Applications},
pages = {411--429},
title = {{Global Optimal Control of Perturbed Systems}},
doi = {10.1007/s10957-007-9312-z},
year = {2007},
}
@article{16500,
author = {Chaudhuri, I. and Sertl, S. and Hajnal, Z. and Dellnitz, M. and Frauenheim, Th.},
issn = {0169-4332},
journal = {Applied Surface Science},
pages = {108--113},
title = {{Global optimization of silicon nanoclusters}},
doi = {10.1016/j.apsusc.2003.11.007},
year = {2004},
}
@article{16671,
author = {Sertl, Stefan and Dellnitz, Michael},
issn = {0925-5001},
journal = {Journal of Global Optimization},
pages = {569--587},
title = {{Global Optimization using a Dynamical Systems Approach}},
doi = {10.1007/s10898-005-4384-5},
year = {2006},
}
@inproceedings{8752,
abstract = {In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated into a subdivision algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.},
author = {Peitz, Sebastian and Dellnitz, Michael},
booktitle = {NEO 2016},
isbn = {9783319640624},
issn = {1860-949X},
pages = {159--182},
title = {{Gradient-Based Multiobjective Optimization with Uncertainties}},
doi = {10.1007/978-3-319-64063-1_7},
year = {2017},
}
@inbook{16559,
author = {Dellnitz, Michael and Molo, Mirko Hessel-von and Metzner, Philipp and Preis, Robert and Schütte, Christof},
booktitle = {Analysis, Modeling and Simulation of Multiscale Problems},
isbn = {9783540356561},
title = {{Graph Algorithms for Dynamical Systems}},
doi = {10.1007/3-540-35657-6_23},
year = {2006},
}
@article{16659,
author = {Ringkamp, Maik and Ober-Blöbaum, Sina and Dellnitz, Michael and Schütze, Oliver},
issn = {0305-215X},
journal = {Engineering Optimization},
pages = {1117--1146},
title = {{Handling high-dimensional problems with multi-objective continuation methods via successive approximation of the tangent space}},
doi = {10.1080/0305215x.2011.634407},
year = {2012},
}
@article{17041,
author = {Campos, Cédric M. and Ober-Blöbaum, Sina and Trélat, Emmanuel},
journal = {Discrete & Continuous Dynamical Systems - A},
number = {9},
pages = {4193--4223},
title = {{High order variational integrators in the optimal control of mechanical systems}},
doi = {10.3934/dcds.2015.35.4193},
volume = {35},
year = {2015},
}
@article{16667,
author = {Schütze, Oliver and Coello Coello, Carlos A. and Mostaghim, Sanaz and Talbi, El-Ghazali and Dellnitz, Michael},
issn = {0305-215X},
journal = {Engineering Optimization},
pages = {383--402},
title = {{Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems}},
doi = {10.1080/03052150701821328},
year = {2008},
}
@article{16713,
author = {Gölz, Christian and Voelcker-Rehage, Claudia and Mora, Karin and Reuter, Eva-Maria and Godde, Ben and Dellnitz, Michael and Reinsberger, Claus and Vieluf, Solveig},
issn = {1664-042X},
journal = {Frontiers in Physiology},
title = {{Improved Neural Control of Movements Manifests in Expertise-Related Differences in Force Output and Brain Network Dynamics}},
doi = {10.3389/fphys.2018.01540},
year = {2018},
}
@article{16696,
author = {Zanzottera, A. and Mingotti, G. and Castelli, R. and Dellnitz, M.},
issn = {1007-5704},
journal = {Communications in Nonlinear Science and Numerical Simulation},
pages = {832--843},
title = {{Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of Earth-to-halo transfers in the Sun–Earth–Moon scenario}},
doi = {10.1016/j.cnsns.2011.06.032},
year = {2012},
}
@article{16295,
abstract = {It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.},
author = {Gebken, Bennet and Peitz, Sebastian},
journal = {Journal of Global Optimization},
publisher = {Springer},
title = {{Inverse multiobjective optimization: Inferring decision criteria from data}},
doi = {10.1007/s10898-020-00983-z},
year = {2020},
}
@article{10593,
abstract = {We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.},
author = {Peitz, Sebastian and Klus, Stefan},
issn = {0005-1098},
journal = {Automatica},
pages = {184--191},
title = {{Koopman operator-based model reduction for switched-system control of PDEs}},
doi = {10.1016/j.automatica.2019.05.016},
volume = {106},
year = {2019},
}
@article{16648,
author = {Padberg, Kathrin and Hauff, Thilo and Jenko, Frank and Junge, Oliver},
issn = {1367-2630},
journal = {New Journal of Physics},
pages = {400--400},
title = {{Lagrangian structures and transport in turbulent magnetized plasmas}},
doi = {10.1088/1367-2630/9/11/400},
year = {2007},
}
@article{16649,
author = {Padberg, Kathrin and Thiere, Bianca and Preis, Robert and Dellnitz, Michael},
issn = {1007-5704},
journal = {Communications in Nonlinear Science and Numerical Simulation},
pages = {4176--4190},
title = {{Local expansion concepts for detecting transport barriers in dynamical systems}},
doi = {10.1016/j.cnsns.2009.03.018},
year = {2009},
}
@inbook{16599,
author = {Froyland, Gary},
booktitle = {Equadiff 99},
isbn = {9789810243593},
title = {{Markov modelling for random dynamical systems}},
doi = {10.1142/9789812792617_0200},
year = {2013},
}
@inbook{16546,
author = {Dellnitz, Michael and Golubitsky, Martin and Melbourne, Ian},
booktitle = {Bifurcation and Symmetry},
isbn = {9783034875387},
title = {{Mechanisms of Symmetry Creation}},
doi = {10.1007/978-3-0348-7536-3_9},
year = {1992},
}
@inbook{16679,
author = {Anacker, Harald and Dellnitz, Michael and Flaßkamp, Kathrin and Groesbrink, Stefan and Hartmann, Philip and Heinzemann, Christian and Horenkamp, Christian and Kleinjohann, Bernd and Kleinjohann, Lisa and Korf, Sebastian and Krüger, Martin and Müller, Wolfgang and Ober-Blöbaum, Sina and Oberthür, Simon and Porrmann, Mario and Priesterjahn, Claudia and Radkowski, Rafael and Rasche, Christoph and Rieke, Jan and Ringkamp, Maik and Stahl, Katharina and Steenken, Dominik and Stöcklein, Jörg and Timmermann, Robert and Trächtler, Ansgar and Witting, Katrin and Xie, Tao and Ziegert, Steffen},
booktitle = {Lecture Notes in Mechanical Engineering},
isbn = {9783642454349},
issn = {2195-4356},
title = {{Methods for the Design and Development}},
doi = {10.1007/978-3-642-45435-6_5},
year = {2014},
}
@inbook{17035,
author = {Dellnitz, Michael and Dignath, Florian and Flaßkamp, Kathrin and Molo, Mirko Hessel-von and Krüger, Martin and Timmermann, Robert and Zheng, Qinghua},
booktitle = {Mathematics in Industry},
isbn = {9783642250996},
issn = {1612-3956},
title = {{Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway}},
doi = {10.1007/978-3-642-25100-9_14},
year = {2012},
}
@inbook{16576,
author = {Dellnitz, Michael and Dignath, Florian and Flaßkamp, Kathrin and Molo, Mirko Hessel-von and Krüger, Martin and Timmermann, Robert and Zheng, Qinghua},
booktitle = {Mathematics in Industry},
isbn = {9783642250996},
issn = {1612-3956},
title = {{Modelling and Analysis of the Nonlinear Dynamics of the Transrapid and Its Guideway}},
doi = {10.1007/978-3-642-25100-9_14},
year = {2012},
}
@inproceedings{16528,
author = {Dell'Aere, Alessandro},
booktitle = {IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics},
isbn = {1424403901},
issn = {1553-572X},
title = {{Multi-Objective Optimization in Self-Optimizing Systems}},
doi = {10.1109/iecon.2006.348080},
year = {2006},
}
@inproceedings{16695,
author = {Wang, Fang and Dellnitz, Michael},
booktitle = {2008 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications},
isbn = {9781424428915},
title = {{Multi-objective shape optimization for piezoceramics}},
doi = {10.1109/spawda.2008.4775823},
year = {2009},
}
@inbook{16681,
author = {Dellnitz, Michael and Schütze, Oliver},
booktitle = {Global Analysis of Nonlinear Dynamics},
isbn = {9781461431275},
title = {{Multilevel Subdivision Techniques for Scalar Optimization Problems}},
doi = {10.1007/978-1-4614-3128-2_10},
year = {2012},
}
@inproceedings{8759,
abstract = {In a wide range of applications, it is desirable to optimally control a system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When it is not possible to compute the entire control trajectory in advance, for instance due to uncertainties or unforeseeable events, model predictive control methods can be applied to control the system during operation in real time. In this article, we present an algorithm for the solution of multiobjective model predictive control problems. In an offline scenario, it can be used to compute the entire set of optimal compromises whereas in a real time scenario, one optimal compromise is computed according to an operator's preference. The results are illustrated using the example of an industrial laundry. A logistics model of the laundry is developed and then utilized in the optimization routine. Results are presented for an offline as well as an online scenario.},
author = {Peitz, Sebastian and Gräler, Manuel and Henke, Christian and Molo, Mirko Hessel-von and Dellnitz, Michael and Trächtler, Ansgar},
booktitle = {Procedia Technology},
issn = {2212-0173},
pages = {483--490},
title = {{Multiobjective Model Predictive Control of an Industrial Laundry}},
doi = {10.1016/j.protcy.2016.08.061},
year = {2016},
}
@inbook{16579,
author = {Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian},
booktitle = {Mathematics in Industry},
isbn = {9783319234120},
issn = {1612-3956},
title = {{Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control}},
doi = {10.1007/978-3-319-23413-7_87},
year = {2016},
}
@inproceedings{34,
author = {Dellnitz, Michael and Eckstein, Julian and Flaßkamp, Kathrin and Friedel, Patrick and Horenkamp, Christian and Köhler, Ulrich and Ober-Blöbaum, Sina and Peitz, Sebastian and Tiemeyer, Sebastian},
booktitle = {Progress in Industrial Mathematics at ECMI},
issn = {2212-0173},
pages = {633--641},
publisher = {Springer International Publishing},
title = {{Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control}},
doi = {10.1007/978-3-319-23413-7_87},
volume = {22},
year = {2016},
}
@article{8753,
abstract = {In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient-based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the flow around a cylinder and a backward-facing-step channel flow.},
author = {Peitz, Sebastian and Ober-Blöbaum, Sina and Dellnitz, Michael},
issn = {0167-8019},
journal = {Acta Applicandae Mathematicae},
number = {1},
pages = {171–199},
title = {{Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling}},
doi = {10.1007/s10440-018-0209-7},
volume = {161},
year = {2019},
}
@article{16642,
author = {Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
issn = {1617-7061},
journal = {PAMM},
pages = {639--640},
title = {{Multiobjective optimal control of fluid mixing}},
doi = {10.1002/pamm.201510309},
year = {2015},
}
@inproceedings{16524,
author = {Blesken, Matthias and Ruckert, Ulrich and Steenken, Dominik and Witting, Katrin and Dellnitz, Michael},
booktitle = {2009 NORCHIP},
isbn = {9781424443109},
title = {{Multiobjective optimization for transistor sizing of CMOS logic standard cells using set-oriented numerical techniques}},
doi = {10.1109/norchp.2009.5397800},
year = {2009},
}
@article{16610,
author = {Geisler, M.Sc. Jens and Witting, Dipl.-Math. Katrin and Trächtler, Ansgar and Dellnitz, Michael},
issn = {1474-6670},
journal = {IFAC Proceedings Volumes},
pages = {4380--4386},
title = {{Multiobjective Optimization of Control Trajectories for the Guidance of a Rail-bound Vehicle}},
doi = {10.3182/20080706-5-kr-1001.00738},
year = {2008},
}
@article{1774,
abstract = {In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100.},
author = {Peitz, Sebastian and Dellnitz, Michael},
issn = {1617-7061},
journal = {PAMM},
number = {1},
pages = {613--614},
publisher = {WILEY-VCH Verlag},
title = {{Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction}},
doi = {10.1002/pamm.201510296},
volume = {15},
year = {2015},
}
@inproceedings{8760,
abstract = {n this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100.},
author = {Peitz, Sebastian and Dellnitz, Michael},
booktitle = {PAMM},
issn = {1617-7061},
pages = {613--614},
title = {{Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction}},
doi = {10.1002/pamm.201510296},
year = {2015},
}
@article{16521,
author = {Bezrukov, S. and Elsässer, R. and Monien, B. and Preis, R. and Tillich, J.-P.},
issn = {0304-3975},
journal = {Theoretical Computer Science},
pages = {155--174},
title = {{New spectral lower bounds on the bisection width of graphs}},
doi = {10.1016/j.tcs.2004.03.059},
year = {2004},
}
@inbook{16516,
author = {Baier, Robert and Molo, Mirko Hessel-von},
booktitle = {Large-Scale Scientific Computing},
isbn = {9783642298424},
issn = {0302-9743},
title = {{Newton’s Method and Secant Method for Set-Valued Mappings}},
doi = {10.1007/978-3-642-29843-1_9},
year = {2012},
}
@article{16633,
abstract = {AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.},
author = {Melbourne, Ian and Dellnitz, Michael},
issn = {0305-0041},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
pages = {235--268},
title = {{Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group}},
doi = {10.1017/s0305004100071577},
year = {1993},
}
@article{16560,
author = {Dellnitz, Michael and Junge, Oliver and Post, Marcus and Thiere, Bianca},
issn = {0923-2958},
journal = {Celestial Mechanics and Dynamical Astronomy},
pages = {357--370},
title = {{On target for Venus – set oriented computation of energy efficient low thrust trajectories}},
doi = {10.1007/s10569-006-9008-y},
year = {2006},
}
@article{16537,
author = {Dellnitz, Michael and Junge, Oliver},
issn = {0036-1429},
journal = {SIAM Journal on Numerical Analysis},
pages = {491--515},
title = {{On the Approximation of Complicated Dynamical Behavior}},
doi = {10.1137/s0036142996313002},
year = {1999},
}
@article{16572,
author = {Dellnitz, Michael and Froyland, Gary and Horenkamp, Christian and Padberg, Kathrin},
issn = {0936-7195},
journal = {GAMM-Mitteilungen},
pages = {47--60},
title = {{On the Approximation of Transport Phenomena - a Dynamical Systems Approach}},
doi = {10.1002/gamm.200910004},
year = {2009},
}
@article{16580,
author = {Ziessler, Adrian and Molo, Mirko Hessel-Von and Dellnitz, Michael},
issn = {2158-2491},
journal = {Journal of Computational Dynamics},
pages = {5--5},
title = {{On the computation of attractors for delay differential equations}},
doi = {10.3934/jcd.2016005},
year = {2016},
}
@article{16712,
author = {Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan},
issn = {1468-9367},
journal = {Dynamical Systems},
pages = {1--19},
title = {{On the equivariance properties of self-adjoint matrices}},
doi = {10.1080/14689367.2019.1661355},
year = {2019},
}
@article{10595,
abstract = {In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.},
author = {Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael},
issn = {0925-5001},
journal = {Journal of Global Optimization},
number = {4},
pages = {891--913},
title = {{On the hierarchical structure of Pareto critical sets}},
doi = {10.1007/s10898-019-00737-6},
volume = {73},
year = {2019},
}
@article{16554,
author = {Dellnitz, Michael and Froyland, Gary and Sertl, Stefan},
issn = {0951-7715},
journal = {Nonlinearity},
pages = {1171--1188},
title = {{On the isolated spectrum of the Perron-Frobenius operator}},
doi = {10.1088/0951-7715/13/4/310},
year = {2000},
}
@inproceedings{17045,
author = {Gail, Tobias and Leyendecker, Sigrid and Ober-Blöbaum, Sina},
booktitle = {The 3rdJoint International Conference on Multibody System Dynamics},
title = {{On the role of quadrature rules and system dimensions in variational multirateintegrators}},
year = {2014},
}
@unpublished{20731,
abstract = {We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.},
author = {Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian},
booktitle = {arXiv:2012.07483},
title = {{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}},
year = {2020},
}
@inproceedings{16596,
author = {Flasskamp, Kathrin and Ober-Blobaum, Sina and Schneider, Tobias and Bocker, Joachim},
booktitle = {52nd IEEE Conference on Decision and Control},
isbn = {9781467357173},
title = {{Optimal control of a switched reluctance drive by a direct method using a discrete variational principle}},
doi = {10.1109/cdc.2013.6761075},
year = {2014},
}
@article{16593,
author = {Flaßkamp, Kathrin and Timmermann, Julia and Ober-Blöbaum, Sina and Dellnitz, Michael and Trächtler, Ansgar},
issn = {1617-7061},
journal = {PAMM},
pages = {723--724},
title = {{Optimal Control on Stable Manifolds for a Double Pendulum}},
doi = {10.1002/pamm.201210350},
year = {2012},
}
@article{16587,
abstract = { We discuss nearest neighbor load balancing schemes on processor networks which are represented by a cartesian product of graphs and present a new optimal diffusion scheme for general graphs. In the first part of the paper, we introduce the Alternating-Direction load balancing scheme, which reduces the number of load balance iterations by a factor of 2 for cartesian products of graphs. The resulting flow is theoretically analyzed and can be very high for certain cases. Therefore, we further present the Mixed-Direction scheme which needs the same number of iterations but computes in most cases a much smaller flow. In the second part of the paper, we present a simple optimal diffusion scheme for general graphs, calculating a balancing flow which is minimal in the l2 norm. It is based on the spectra of the graph representing the network and needs only m-1 iterations to balance the load with m being the number of distinct eigenvalues. Known optimal diffusion schemes have the same performance, however the optimal scheme presented in this paper can be implemented in a very simple manner. The number of iterations of optimal diffusion schemes is independent of the load scenario and, thus, they are practical for networks which represent graphs with known spectra. Finally, our experiments exhibit that the new optimal scheme can successfully be combined with the Alternating-Direction and Mixed-Direction schemes for efficient load balancing on product graphs. },
author = {Elsässer, Robert and Monien, Burkhard and Preis, Robert and Frommer, Andreas},
issn = {0129-6264},
journal = {Parallel Processing Letters},
pages = {61--73},
title = {{Optimal Diffusion Schemes and Load Balancing on Product Graphs}},
doi = {10.1142/s0129626404001714},
year = {2004},
}
@inproceedings{16661,
author = {Romaus, C. and Bocker, J. and Witting, K. and Seifried, A. and Znamenshchykov, O.},
booktitle = {2009 IEEE Energy Conversion Congress and Exposition},
isbn = {9781424428939},
title = {{Optimal energy management for a hybrid energy storage system combining batteries and double layer capacitors}},
doi = {10.1109/ecce.2009.5316428},
year = {2009},
}
@inproceedings{16618,
author = {Junge, O. and Ober-Blobaum, S.},
booktitle = {Proceedings of the 44th IEEE Conference on Decision and Control},
isbn = {0780395670},
title = {{Optimal Reconfiguration of Formation Flying Satellites}},
doi = {10.1109/cdc.2005.1582132},
year = {2006},
}
@inproceedings{16621,
author = {Junge, O. and Marsden, J.E. and Ober-Blobaum, S.},
booktitle = {Proceedings of the 45th IEEE Conference on Decision and Control},
isbn = {1424401712},
title = {{Optimal Reconfiguration of Formation Flying Spacecraft ---a Decentralized Approach}},
doi = {10.1109/cdc.2006.376977},
year = {2006},
}
@article{16595,
author = {Flaßkamp, Kathrin and Murphey, Todd and Ober-Blöbaum, Sina},
issn = {1617-7061},
journal = {PAMM},
pages = {401--402},
title = {{Optimization for discretized switched systems}},
doi = {10.1002/pamm.201310196},
year = {2013},
}
@article{16629,
author = {Krüger, Martin and Witting, Katrin and Trächtler, Ansgar and Dellnitz, Michael},
issn = {1474-6670},
journal = {IFAC Proceedings Volumes},
pages = {12611--12619},
title = {{Parametric Model-Order Reduction in Hierarchical Multiobjective Optimization of Mechatronic Systems*}},
doi = {10.3182/20110828-6-it-1002.02487},
year = {2011},
}
@article{10596,
abstract = {Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.},
author = {Schütze, Oliver and Cuate, Oliver and Martín, Adanay and Peitz, Sebastian and Dellnitz, Michael},
issn = {0305-215X},
journal = {Engineering Optimization},
number = {5},
pages = {832--855},
title = {{Pareto Explorer: a global/local exploration tool for many-objective optimization problems}},
doi = {10.1080/0305215x.2019.1617286},
volume = {52},
year = {2020},
}
@inproceedings{16622,
author = {Keuck, L. and Frohleke, N. and Bocker, J. and Ziessler, A.},
booktitle = {2015 17th European Conference on Power Electronics and Applications (EPE'15 ECCE-Europe)},
isbn = {9789075815221},
title = {{PFC-control for improved inductor utilization}},
doi = {10.1109/epe.2015.7309373},
year = {2015},
}
@inproceedings{8757,
abstract = {A framework for set‐oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non‐smooth objective functionals. Using an academic example, we show that the extension to non‐smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.},
author = {Beermann, Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan},
booktitle = {PAMM},
issn = {1617-7061},
pages = {51--54},
title = {{POD-based multiobjective optimal control of PDEs with non-smooth objectives}},
doi = {10.1002/pamm.201710015},
year = {2018},
}
@article{16660,
author = {Ringkamp, Maik and Ober-Blöbaum, Sina and Leyendecker, Sigrid},
issn = {1617-7061},
journal = {PAMM},
pages = {27--30},
title = {{Relaxing mixed integer optimal control problems using a time transformation}},
doi = {10.1002/pamm.201510008},
year = {2015},
}
@article{16674,
author = {Thiere, B.},
issn = {0077-8923},
journal = {Annals of the New York Academy of Sciences},
pages = {44--54},
title = {{Return Time Dynamics as a Tool for Finding Almost Invariant Sets}},
doi = {10.1196/annals.1370.027},
year = {2006},
}
@unpublished{16711,
abstract = {Embedding techniques allow the approximations of finite dimensional
attractors and manifolds of infinite dimensional dynamical systems via
subdivision and continuation methods. These approximations give a topological
one-to-one image of the original set. In order to additionally reveal their
geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our
results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky
equation, as well as for the attractor of the Mackey-Glass delay differential
equation.},
author = {Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael},
booktitle = {arXiv:1902.08824},
title = {{Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems}},
year = {2019},
}
@article{16601,
author = {Froyland, Gary and Junge, Oliver and Ochs, Gunter},
issn = {0167-2789},
journal = {Physica D: Nonlinear Phenomena},
pages = {68--84},
title = {{Rigorous computation of topological entropy with respect to a finite partition}},
doi = {10.1016/s0167-2789(01)00216-0},
year = {2001},
}
@inbook{16616,
author = {Junge, Oliver},
booktitle = {Equadiff 99},
isbn = {9789810243593},
title = {{Rigorous discretization of subdivision techniques}},
doi = {10.1142/9789812792617_0178},
year = {2000},
}
@unpublished{16296,
abstract = {Multiobjective optimization plays an increasingly important role in modern
applications, where several objectives are often of equal importance. The task
in multiobjective optimization and multiobjective optimal control is therefore
to compute the set of optimal compromises (the Pareto set) between the
conflicting objectives. Since the Pareto set generally consists of an infinite
number of solutions, the computational effort can quickly become challenging
which is particularly problematic when the objectives are costly to evaluate as
is the case for models governed by partial differential equations (PDEs). To
decrease the numerical effort to an affordable amount, surrogate models can be
used to replace the expensive PDE evaluations. Existing multiobjective
optimization methods using model reduction are limited either to low parameter
dimensions or to few (ideally two) objectives. In this article, we present a
combination of the reduced basis model reduction method with a continuation
approach using inexact gradients. The resulting approach can handle an
arbitrary number of objectives while yielding a significant reduction in
computing time.},
author = {Banholzer, Stefan and Gebken, Bennet and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan},
booktitle = {arXiv:1906.09075},
title = {{ROM-based multiobjective optimization of elliptic PDEs via numerical continuation}},
year = {2019},
}
@inproceedings{17049,
author = {Jakobsmeyer, R. and Schnittker, R. and Timmermann, R. and Zorn, R. and Rückert, U. and Baumeister, J.},
booktitle = {Performance Analysis of Sport IX, Part 8, Routledge},
title = {{Running performance analysis in basketball using recorded trajectory data}},
year = {2013},
}
@inproceedings{16628,
author = {Krishnamurthy, A. and Preis, R.},
booktitle = {19th IEEE International Parallel and Distributed Processing Symposium},
isbn = {0769523129},
title = {{Satellite Formation, a Mobile Sensor Network in Space}},
doi = {10.1109/ipdps.2005.387},
year = {2005},
}
@article{16573,
author = {Dellnitz, M. and Froyland, G. and Horenkamp, C. and Padberg-Gehle, K. and Sen Gupta, A.},
issn = {1607-7946},
journal = {Nonlinear Processes in Geophysics},
pages = {655--663},
title = {{Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators}},
doi = {10.5194/npg-16-655-2009},
year = {2009},
}
@inproceedings{17031,
author = {Geisler, Jens and Witting, Katrin and Trächtler, Ansgar and Dellnitz, Michael},
booktitle = {7th International Heinz Nixdorf Symposium: Self-optimizing Mechatronic Systems: Designing the Future},
title = {{Self-Optimization of the Guidance Module of a Rail-bound Vehicle}},
year = {2008},
}
@article{16540,
author = {Dellnitz, Michael and Klus, Stefan},
issn = {1468-9367},
journal = {Dynamical Systems},
pages = {61--79},
title = {{Sensing and control in symmetric networks}},
doi = {10.1080/14689367.2016.1215410},
year = {2017},
}
@inproceedings{16561,
author = {Dellnitz, Michael and Padberg, Kathrin and Post, Marcus and Thiere, Bianca},
booktitle = {AIP Conference Proceedings},
issn = {0094-243X},
title = {{Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems}},
doi = {10.1063/1.2710046},
year = {2007},
}
@inbook{16670,
author = {Schütze, Oliver and Witting, Katrin and Ober-Blöbaum, Sina and Dellnitz, Michael},
booktitle = {EVOLVE- A Bridge between Probability, Set Oriented Numerics and Evolutionary Computation},
isbn = {9783642327254},
issn = {1860-949X},
title = {{Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems}},
doi = {10.1007/978-3-642-32726-1_5},
year = {2013},
}
@inbook{16538,
author = {Dellnitz, Michael and Junge, Oliver},
booktitle = {Handbook of Dynamical Systems},
isbn = {9780444501684},
issn = {1874-575X},
title = {{Set Oriented Numerical Methods for Dynamical Systems}},
doi = {10.1016/s1874-575x(02)80026-1},
year = {2002},
}
@inbook{16539,
author = {Dellnitz, Michael and Junge, Oliver},
booktitle = {Elsevier Astrodynamics Series},
isbn = {9780123735621},
issn = {1874-9305},
title = {{Set Oriented Numerical Methods in Space Mission Design}},
doi = {10.1016/s1874-9305(07)80007-0},
year = {2006},
}
@inbook{8754,
abstract = {In this chapter, we combine a global, derivative-free subdivision algorithm for multiobjective optimization problems with a posteriori error estimates for reduced-order models based on Proper Orthogonal Decomposition in order to efficiently solve multiobjective optimization problems governed by partial differential equations. An error bound for a semilinear heat equation is developed in such a way that the errors in the conflicting objectives can be estimated individually. The resulting algorithm constructs a library of locally valid reduced-order models online using a Greedy (worst-first) search. Using this approach, the number of evaluations of the full-order model can be reduced by a factor of more than 1000.},
author = {Beermann, Dennis and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan},
booktitle = {Reduced-Order Modeling (ROM) for Simulation and Optimization},
isbn = {9783319753188},
pages = {47--72},
title = {{Set-Oriented Multiobjective Optimal Control of PDEs Using Proper Orthogonal Decomposition}},
doi = {10.1007/978-3-319-75319-5_3},
year = {2018},
}
@article{16592,
author = {Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Kobilarov, Marin},
issn = {0938-8974},
journal = {Journal of Nonlinear Science},
pages = {599--629},
title = {{Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures}},
doi = {10.1007/s00332-012-9140-7},
year = {2012},
}
@article{16590,
author = {Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Kobilarov, Marin},
issn = {1617-7061},
journal = {PAMM},
pages = {577--578},
title = {{Solving optimal control problems by using inherent dynamical properties}},
doi = {10.1002/pamm.201010281},
year = {2010},
}
@article{16673,
author = {Tantardini, Marco and Fantino, Elena and Ren, Yuan and Pergola, Pierpaolo and Gómez, Gerard and Masdemont, Josep J.},
issn = {0923-2958},
journal = {Celestial Mechanics and Dynamical Astronomy},
pages = {215--232},
title = {{Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem}},
doi = {10.1007/s10569-010-9299-x},
year = {2010},
}
@techreport{17028,
author = {Vasile, Massimiliano and Schütze, Oliver and Junge, Oliver and Radice, Gimbardo and Dellnitz, Michael},
publisher = {European Space Agency},
title = {{Spiral Trajectories in Global Optimisation of Interplanetary and Orbital Transfers}},
year = {2007},
}
@article{16551,
abstract = { Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (ur = muθ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral. },
author = {Dellnitz, Michael and Golubitsky, Martin and Hohmann, Andreas and Stewart, Ian},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {1487--1501},
title = {{Spirals in Scalar Reaction–Diffusion Equations}},
doi = {10.1142/s0218127495001149},
year = {1995},
}
@article{17034,
author = {Hage-Packhäuser, Sebastian and Dellnitz, Michael},
journal = {Discrete & Continuous Dynamical Systems - B},
number = {1},
pages = {239--263},
title = {{Stabilization via symmetry switching in hybrid dynamical systems}},
doi = {10.3934/dcdsb.2011.16.239},
volume = {16},
year = {2011},
}
@inproceedings{16591,
author = {Flasskamp, Kathrin and Murphey, Todd and Ober-Blobaum, Sina},
booktitle = {2012 IEEE 51st IEEE Conference on Decision and Control (CDC)},
isbn = {9781467320665},
title = {{Switching time optimization in discretized hybrid dynamical systems}},
doi = {10.1109/cdc.2012.6426414},
year = {2012},
}
@article{16639,
author = {Meyer, A. and Dellnitz, M. and Hessel-von Molo, M.},
issn = {1751-570X},
journal = {Nonlinear Analysis: Hybrid Systems},
pages = {125--135},
title = {{Symmetries in timed continuous Petri nets}},
doi = {10.1016/j.nahs.2010.05.005},
year = {2011},
}
@article{16636,
author = {Meyer, A. and Dellnitz, M.},
issn = {1474-6670},
journal = {IFAC Proceedings Volumes},
pages = {68--73},
title = {{Symmetries in Timed Continuous Petri Nets}},
doi = {10.3182/20090916-3-es-3003.00013},
year = {2009},
}
@article{16510,
abstract = { In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators. },
author = {Aston, Philip J. and Dellnitz, Michael},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {1643--1676},
title = {{Symmetry Breaking Bifurcations of Chaotic Attractors}},
doi = {10.1142/s021812749500123x},
year = {1995},
}
@article{17039,
author = {Flasskamp, Kathrin and Hage-Packhäuser, Sebastian and Ober-Blöbaum, Sina},
journal = {Journal of Computational Dynamics},
number = {1},
pages = {25--50},
title = {{Symmetry exploiting control of hybrid mechanical systems}},
doi = {10.3934/jcd.2015.2.25},
volume = {2},
year = {2015},
}
@inbook{17411,
abstract = {Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example.},
author = {Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian},
booktitle = {Advances in Dynamics, Optimization and Computation},
editor = {Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
isbn = {9783030512637},
issn = {2198-4182},
publisher = {Springer},
title = {{Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach}},
doi = {10.1007/978-3-030-51264-4_9},
year = {2020},
}
@inbook{16549,
author = {Dellnitz, Michael and Golubitsky, Martin and Nicol, Matthew},
booktitle = {Trends and Perspectives in Applied Mathematics},
isbn = {9781461269243},
issn = {0066-5452},
title = {{Symmetry of Attractors and the Karhunen-Loève Decomposition}},
doi = {10.1007/978-1-4612-0859-4_4},
year = {1994},
}
@article{16632,
author = {Mehta, Prashant G. and Hessel-von Molo, Mirko and Dellnitz, Michael},
issn = {1023-6198},
journal = {Journal of Difference Equations and Applications},
pages = {1147--1178},
title = {{Symmetry of attractors and the Perron-Frobenius operator}},
doi = {10.1080/10236190601045788},
year = {2006},
}
@article{16638,
author = {Meyer, A. and Silva, M.},
issn = {1474-6670},
journal = {IFAC Proceedings Volumes},
pages = {153--159},
title = {{Symmetry Reductions in Timed Continuous Petri Nets Under Infinite Server Semantics}},
doi = {10.3182/20120606-3-nl-3011.00067},
year = {2012},
}
@article{8755,
abstract = {Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von Kármán vortex street and the simulation of two merging vortices.},
author = {Klus, Stefan and Gelß, Patrick and Peitz, Sebastian and Schütte, Christof},
issn = {0951-7715},
journal = {Nonlinearity},
number = {7},
pages = {3359--3380},
title = {{Tensor-based dynamic mode decomposition}},
doi = {10.1088/1361-6544/aabc8f},
volume = {31},
year = {2018},
}
@inbook{16555,
author = {Dellnitz, Michael and Froyland, Gary and Junge, Oliver},
booktitle = {Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems},
isbn = {9783642625244},
title = {{The Algorithms Behind GAIO — Set Oriented Numerical Methods for Dynamical Systems}},
doi = {10.1007/978-3-642-56589-2_7},
year = {2001},
}
@inbook{17994,
abstract = {In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation.},
author = {Gerlach, Raphael and Ziessler, Adrian},
booktitle = {Advances in Dynamics, Optimization and Computation},
editor = {Junge, Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
isbn = {9783030512637},
issn = {2198-4182},
pages = {55--85},
publisher = {Springer},
title = {{The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems}},
doi = {10.1007/978-3-030-51264-4_3},
volume = {304},
year = {2020},
}
@inbook{17018,
author = {Dellnitz, Michael and Junge, Oliver and Rumpf, Martin and Strzodka, Robert},
booktitle = {Equadiff 99},
isbn = {9789810243593},
pages = {1053----1059},
title = {{The computation of an unstable invariant set inside a cylinder containing a knotted flow}},
doi = {10.1142/9789812792617_0204},
year = {2000},
}
@article{17043,
author = {Baier, Robert and Dellnitz, Michael and Hessel-von Molo, Mirko and Sertl, Stefan and Kevrekidis, Ioannis G.},
journal = {Journal of Computational Dynamics},
number = {1},
pages = {39--69},
title = {{The computation of convex invariant sets via Newton's method}},
doi = {10.3934/jcd.2014.1.39},
volume = {1},
year = {2014},
}
@inbook{16513,
author = {Aston, P. J. and Dellnitz, M.},
booktitle = {Equadiff 99},
isbn = {9789810243593},
title = {{The Computation of Lyapunov Exponents via Spatial Integration Using Vector Norms}},
doi = {10.1142/9789812792617_0196},
year = {2000},
}
@article{16511,
author = {Aston, Philip J. and Dellnitz, Michael},
issn = {0045-7825},
journal = {Computer Methods in Applied Mechanics and Engineering},
pages = {223--237},
title = {{The computation of lyapunov exponents via spatial integration with application to blowout bifurcations}},
doi = {10.1016/s0045-7825(98)00196-0},
year = {1999},
}
@inbook{16533,
author = {Dellnitz, Michael and Hohmann, Andreas},
booktitle = {Nonlinear Dynamical Systems and Chaos},
isbn = {9783034875202},
title = {{The Computation of Unstable Manifolds Using Subdivision and Continuation}},
doi = {10.1007/978-3-0348-7518-9_21},
year = {1996},
}
@inbook{16611,
author = {Golubitsky, Martin and Marsden, Jerrold and Stewart, Ian and Dellnitz, Michael},
booktitle = {Normal Forms and Homoclinic Chaos},
isbn = {9780821803264},
title = {{The constrained Liapunov-Schmidt procedure and periodic orbits}},
doi = {10.1090/fic/004/05},
year = {1995},
}
@article{17013,
author = {Dellnitz, Michael},
journal = {Lectures in Applied Mathematics},
pages = {163--169},
title = {{The equivariant Darboux theorem}},
volume = {29},
year = {1993},
}
@article{16708,
abstract = { In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to
the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems.
To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for
the computation of such objects of finite dimensional systems with the results obtained in the work
of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis
of partial differential equations and illustrate its feasibility by computing unstable manifolds of the
one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential
equation.
},
author = {Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael},
issn = {1536-0040},
journal = {SIAM Journal on Applied Dynamical Systems},
pages = {1265--1292},
title = {{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}},
doi = {10.1137/18m1204395},
year = {2019},
}
@inbook{16577,
author = {Dellnitz, Michael and Dumitrescu, Roman and Flaßkamp, Kathrin and Gausemeier, Jürgen and Hartmann, Philip and Iwanek, Peter and Korf, Sebastian and Krüger, Martin and Ober-Blöbaum, Sina and Porrmann, Mario and Priesterjahn, Claudia and Stahl, Katharina and Trächtler, Ansgar and Vaßholz, Mareen},
booktitle = {Lecture Notes in Mechanical Engineering},
isbn = {9783642454349},
issn = {2195-4356},
title = {{The Paradigm of Self-optimization}},
doi = {10.1007/978-3-642-45435-6_1},
year = {2014},
}
@article{16634,
author = {Melbourne, Ian and Dellnitz, Michael and Golubitsky, Martin},
issn = {0003-9527},
journal = {Archive for Rational Mechanics and Analysis},
pages = {75--98},
title = {{The structure of symmetric attractors}},
doi = {10.1007/bf00386369},
year = {1993},
}
@article{16603,
author = {Froyland, Gary and Horenkamp, Christian and Rossi, Vincent and Santitissadeekorn, Naratip and Gupta, Alex Sen},
issn = {1463-5003},
journal = {Ocean Modelling},
pages = {69--75},
title = {{Three-dimensional characterization and tracking of an Agulhas Ring}},
doi = {10.1016/j.ocemod.2012.05.001},
year = {2012},
}
@inproceedings{16525,
author = {Day, Sarah and Junge, Oliver and Mischaikow, Konstantin},
booktitle = {EQUADIFF 2003},
isbn = {9789812561695},
title = {{Towards Automated Chaos Verification}},
doi = {10.1142/9789812702067_0014},
year = {2005},
}
@article{16715,
author = {Bittracher, Andreas and Koltai, Péter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Schütte, Christof},
issn = {0938-8974},
journal = {Journal of Nonlinear Science},
pages = {471--512},
title = {{Transition Manifolds of Complex Metastable Systems}},
doi = {10.1007/s00332-017-9415-0},
volume = {28},
year = {2018},
}
@article{16557,
abstract = { We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids. },
author = {Dellnitz, Michael and Junge, Oliver and Koon, Wang Sang and Lekien, Francois and Lo, Martin W. and Marsden, Jerrold E. and Padberg, Kathrin and Preis, Robert and Ross, Shane D. and Thiere, Bianca},
issn = {0218-1274},
journal = {International Journal of Bifurcation and Chaos},
pages = {699--727},
title = {{Transport in Dynamical Astronomy and Multibody Problems}},
doi = {10.1142/s0218127405012545},
year = {2005},
}
@article{16558,
author = {Dellnitz, Michael and Junge, Oliver and Lo, Martin W. and Marsden, Jerrold E. and Padberg, Kathrin and Preis, Robert and Ross, Shane D. and Thiere, Bianca},
issn = {0031-9007},
journal = {Physical Review Letters},
title = {{Transport of Mars-Crossing Asteroids from the Quasi-Hilda Region}},
doi = {10.1103/physrevlett.94.231102},
year = {2005},
}
@inproceedings{16620,
author = {Junge, O. and Marsden, J.E. and Mezic, I.},
booktitle = {2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)},
isbn = {0780386825},
title = {{Uncertainty in the dynamics of conservative maps}},
doi = {10.1109/cdc.2004.1430379},
year = {2004},
}
@inbook{16640,
author = {Monien, Burkhard and Preis, Robert},
booktitle = {Mathematical Foundations of Computer Science 2001},
isbn = {9783540424963},
issn = {0302-9743},
title = {{Upper Bounds on the Bisection Width of 3- and 4-Regular Graphs}},
doi = {10.1007/3-540-44683-4_46},
year = {2007},
}
@article{16641,
author = {Monien, Burkhard and Preis, Robert},
issn = {1570-8667},
journal = {Journal of Discrete Algorithms},
pages = {475--498},
title = {{Upper bounds on the Bisection Width of 3- and 4-regular Graphs}},
doi = {10.1016/j.jda.2005.12.009},
year = {2006},
}
@inproceedings{16588,
author = {Flaßkamp, Kathrin and Ober-Blöbaum, Sina},
booktitle = {Proceedings of the 14th international conference on Hybrid systems: computation and control - HSCC '11},
isbn = {9781450306294},
title = {{Variational formulation and optimal control of hybrid lagrangian systems}},
doi = {10.1145/1967701.1967736},
year = {2011},
}
@inproceedings{17046,
author = {Ober-Blöbaum, Sina and Lindhorst, Henning},
booktitle = {21st International Symposium on Mathematical Theory of Networks and Systems},
title = {{Variational formulation and structure-preserving discretization ofnonlinear electric circuits}},
year = {2014},
}
@article{17050,
author = {Ober-Blöbaum, Sina and Tao, Molei and Cheng, Mulin and Owhadi, Houman and Marsden, Jerrold E.},
issn = {0021-9991},
journal = {Journal of Computational Physics},
pages = {498--530},
title = {{Variational integrators for electric circuits}},
doi = {10.1016/j.jcp.2013.02.006},
volume = {242},
year = {2013},
}
@inbook{17044,
author = {Leitz, Thomas and Ober-Blöbaum, Sina and Leyendecker, Sigrid},
booktitle = {Multibody Dynamics},
editor = {Terze, Zdravko},
isbn = {9783319072593},
issn = {1871-3033},
publisher = {Springer},
title = {{Variational Lie Group Formulation of Geometrically Exact Beam Dynamics: Synchronous and Asynchronous Integration}},
doi = {10.1007/978-3-319-07260-9},
year = {2014},
}
@article{17017,
author = {Bürkle, David and Dellnitz, Michael and Junge, Oliver and Rumpf, Martin and Spielberg, Michael},
journal = {Proceedings of Visualization 99},
title = {{Visualizing Complicated Dynamics}},
year = {1999},
}