@inbook{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 = {{Non-Smooth and Complementarity-Based Distributed Parameter Systems}},
editor = {{Michael, Hintermüller and Roland, Herzog and Christian, Kanzow and Michael, Ulbrich and Stefan, Ulbrich}},
isbn = {{978-3-030-79392-0}},
pages = {{43--76}},
publisher = {{Springer}},
title = {{{ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation}}},
doi = {{10.1007/978-3-030-79393-7_3}},
year = {{2022}},
}
@inbook{30294,
abstract = {{With the ever increasing capabilities of sensors and controllers, autonomous driving is quickly becoming a reality. This disruptive change in the automotive industry poses major challenges for manufacturers as well as suppliers as entirely new design and testing strategies have to be developed to remain competitive. Most importantly, the complexity of autonomously driving vehicles in a complex, uncertain, and safety-critical environment requires new testing procedures to cover the almost infinite range of potential scenarios.}},
author = {{Peitz, Sebastian and Dellnitz, Michael and Bannenberg, Sebastian}},
booktitle = {{German Success Stories in Industrial Mathematics}},
editor = {{Bock, H. G. and Küfer, K.-H. and Maas, P. and Milde, A. and Schulz, V.}},
isbn = {{9783030814540}},
issn = {{1612-3956}},
publisher = {{Springer International Publishing}},
title = {{{Efficient Virtual Design and Testing of Autonomous Vehicles}}},
doi = {{10.1007/978-3-030-81455-7_23}},
volume = {{35}},
year = {{2022}},
}
@article{29673,
abstract = {{Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science.}},
author = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian}},
journal = {{Journal of Physics A: Mathematical and Theoretical}},
number = {{31}},
pages = {{314002}},
publisher = {{IOP Publishing Ltd.}},
title = {{{Koopman analysis of quantum systems}}},
doi = {{10.1088/1751-8121/ac7d22}},
volume = {{55}},
year = {{2022}},
}
@unpublished{34618,
abstract = {{In this article, we show how second-order derivative information can be
incorporated into gradient sampling methods for nonsmooth optimization. The
second-order information we consider is essentially the set of coefficients of
all second-order Taylor expansions of the objective in a closed ball around a
given point. Based on this concept, we define a model of the objective as the
maximum of these Taylor expansions. Iteratively minimizing this model
(constrained to the closed ball) results in a simple descent method, for which
we prove convergence to minimal points in case the objective is convex. To
obtain an implementable method, we construct an approximation scheme for the
second-order information based on sampling objective values, gradients and
Hessian matrices at finitely many points. Using a set of test problems, we
compare the resulting method to five other available solvers. Considering the
number of function evaluations, the results suggest that the method we propose
is superior to the standard gradient sampling method, and competitive compared
to other methods.}},
author = {{Gebken, Bennet}},
booktitle = {{arXiv:2210.04579}},
title = {{{Using second-order information in gradient sampling methods for nonsmooth optimization}}},
year = {{2022}},
}
@phdthesis{31556,
abstract = {{Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind.}},
author = {{Gebken, Bennet}},
title = {{{Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization}}},
doi = {{10.17619/UNIPB/1-1327}},
year = {{2022}},
}
@unpublished{33150,
abstract = {{In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem.}},
author = {{Berkemeier, Manuel Bastian and Peitz, Sebastian}},
booktitle = {{arXiv:2208.12094}},
title = {{{Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients}}},
year = {{2022}},
}
@article{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}},
journal = {{IEEE Transactions on Pattern Analysis and Machine Intelligence}},
number = {{11}},
pages = {{7797--7808}},
publisher = {{IEEE}},
title = {{{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}}},
doi = {{10.1109/TPAMI.2021.3114962}},
volume = {{44}},
year = {{2022}},
}
@article{24169,
author = {{Nüske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia}},
issn = {{0167-2789}},
journal = {{Physica D: Nonlinear Phenomena}},
title = {{{Tensor-based computation of metastable and coherent sets}}},
doi = {{10.1016/j.physd.2021.133018}},
year = {{2021}},
}
@article{24170,
author = {{Klus, Stefan and Gelß, Patrick and Nüske, Feliks and Noé, Frank}},
issn = {{2632-2153}},
journal = {{Machine Learning: Science and Technology}},
title = {{{Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry}}},
doi = {{10.1088/2632-2153/ac14ad}},
year = {{2021}},
}
@article{21195,
author = {{Goelz, Christian and Mora, Karin and Stroehlein, Julia Kristin and Haase, Franziska Katharina and Dellnitz, Michael and Reinsberger, Claus and Vieluf, Solveig}},
journal = {{Cognitive Neurodynamics}},
title = {{{Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults}}},
doi = {{10.1007/s11571-020-09656-9}},
year = {{2021}},
}
@article{21337,
abstract = {{We present a flexible trust region descend algorithm for unconstrained and
convexly constrained multiobjective optimization problems. It is targeted at
heterogeneous and expensive problems, i.e., problems that have at least one
objective function that is computationally expensive. The method is
derivative-free in the sense that neither need derivative information be
available for the expensive objectives nor are gradients approximated using
repeated function evaluations as is the case in finite-difference methods.
Instead, a multiobjective trust region approach is used that works similarly to
its well-known scalar pendants. Local surrogate models constructed from
evaluation data of the true objective functions are employed to compute
possible descent directions. In contrast to existing multiobjective trust
region algorithms, these surrogates are not polynomial but carefully
constructed radial basis function networks. This has the important advantage
that the number of data points scales linearly with the parameter space
dimension. The local models qualify as fully linear and the corresponding
general scalar framework is adapted for problems with multiple objectives.
Convergence to Pareto critical points is proven and numerical examples
illustrate our findings.}},
author = {{Berkemeier, Manuel Bastian and Peitz, Sebastian}},
issn = {{2297-8747}},
journal = {{Mathematical and Computational Applications}},
number = {{2}},
title = {{{Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models}}},
doi = {{10.3390/mca26020031}},
volume = {{26}},
year = {{2021}},
}
@article{21820,
abstract = {{The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.}},
author = {{Nüske, Feliks and Koltai, Péter and Boninsegna, Lorenzo and Clementi, Cecilia}},
issn = {{1099-4300}},
journal = {{Entropy}},
title = {{{Spectral Properties of Effective Dynamics from Conditional Expectations}}},
doi = {{10.3390/e23020134}},
year = {{2021}},
}
@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}},
pages = {{696--723}},
title = {{{An efficient descent method for locally Lipschitz multiobjective optimization problems}}},
doi = {{10.1007/s10957-020-01803-w}},
volume = {{188}},
year = {{2021}},
}
@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}},
pages = {{3--29}},
publisher = {{Springer}},
title = {{{Inverse multiobjective optimization: Inferring decision criteria from data}}},
doi = {{10.1007/s10898-020-00983-z}},
volume = {{80}},
year = {{2021}},
}
@phdthesis{32057,
abstract = {{Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung.}},
author = {{Gerlach, Raphael}},
title = {{{The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}}},
doi = {{10.17619/UNIPB/1-1278}},
year = {{2021}},
}
@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}},
pages = {{380--403}},
title = {{{Explicit multiobjective model predictive control for nonlinear systems with symmetries}}},
doi = {{10.1002/rnc.5281}},
volume = {{31(2)}},
year = {{2021}},
}
@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}},
}
@article{21819,
abstract = {{Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.}},
author = {{Klus, Stefan and Nüske, Feliks and Hamzi, Boumediene}},
issn = {{1099-4300}},
journal = {{Entropy}},
title = {{{Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator}}},
doi = {{10.3390/e22070722}},
year = {{2020}},
}
@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}},
}
@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}},
}