@unpublished{29236,
abstract = {{The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.}},
author = {{McLachlan, Robert and Offen, Christian}},
keywords = {{variational integrators, backward error analysis, Euler--Lagrange equations, multistep methods, conjugate symplectic methods}},
pages = {{17}},
title = {{{Backward error analysis for conjugate symplectic methods}}},
year = {{2022}},
}
@article{23382,
abstract = {{Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.}},
author = {{Offen, Christian and Ober-Blöbaum, Sina}},
journal = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}},
publisher = {{AIP}},
title = {{{Symplectic integration of learned Hamiltonian systems}}},
doi = {{10.1063/5.0065913}},
volume = {{32(1)}},
year = {{2022}},
}
@unpublished{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}},
booktitle = {{arXiv:2201.12062}},
title = {{{Koopman analysis of quantum systems}}},
year = {{2022}},
}
@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{30490,
author = {{Cresson, Jacky and Jiménez, Fernando and Ober-Blöbaum, Sina}},
journal = {{AIMS}},
pages = {{57--89}},
title = {{{Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations}}},
volume = {{14(1)}},
year = {{2022}},
}
@inproceedings{30733,
abstract = {{Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.}},
author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}},
location = {{London}},
title = {{{Verification of safety critical control policies using kernel methods}}},
year = {{2022}},
}
@article{30861,
abstract = {{AbstractWe consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). According to Pontryagin’s maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or Fuller phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the turnpike property. This property is further used in order to design simple and realistic suboptimal control strategies.}},
author = {{Caillau, Jean-Baptiste and Djema, Walid and Gouzé, Jean-Luc and Maslovskaya, Sofya and Pomet, Jean-Baptiste}},
issn = {{0022-3239}},
journal = {{Journal of Optimization Theory and Applications}},
keywords = {{Applied Mathematics, Management Science and Operations Research, Control and Optimization}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Turnpike Property in Optimal Microbial Metabolite Production}}},
doi = {{10.1007/s10957-022-02023-0}},
year = {{2022}},
}
@unpublished{31057,
abstract = {{In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.}},
author = {{Schütte, Philipp and Barkhofen, Sonja and Weich, Tobias}},
title = {{{Semiclassical Formulae For Wigner Distributions}}},
year = {{2022}},
}
@unpublished{19941,
abstract = {{In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized
partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.}},
author = {{McLachlan, Robert I and Offen, Christian}},
booktitle = {{Journal of Geometric Mechanics}},
publisher = {{AIMS}},
title = {{{Backward error analysis for variational discretisations of partial differential equations}}},
year = {{2022}},
}