Dynamic programming using radial basis functions
Oliver Junge Alex Schreiber
Discrete & Continuous Dynamical Systems - A 2015, 35(9): 4439-4453 doi: 10.3934/dcds.2015.35.4439
We propose a discretization of the optimality principle in dynamic programming based on radial basis functions and Shepard's moving least squares approximation method. We prove convergence of the value iteration scheme, derive a statement about the stability region of the closed loop system using the corresponding approximate optimal feedback law and present several numerical experiments.
keywords: optimal feedback. radial basis function Dynamic programming moving least squares Shepard's method
Computing coherent sets using the Fokker-Planck equation
Andreas Denner Oliver Junge Daniel Matthes
Journal of Computational Dynamics 2016, 3(2): 163-177 doi: 10.3934/jcd.2016008
We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.
keywords: transfer operator Coherent set Fokker-Planck equation.
Global optimal feedbacks for stochastic quantized nonlinear event systems
Stefan Jerg Oliver Junge Marcus Post
Journal of Computational Dynamics 2014, 1(1): 163-176 doi: 10.3934/jcd.2014.1.163
We consider nonlinear control systems for which only quantized and event-triggered state information is available and which are subject to random delays and losses in the transmission of the state to the controller. We present an optimization based approach for computing globally stabilizing controllers for such systems. Our method is based on recently developed set oriented techniques for transforming the problem into a shortest path problem on a weighted hypergraph. We show how to extend this approach to a system subject to a stochastic parameter and propose a corresponding model for dealing with transmission delays.
keywords: global feedback event system. quantized system Dynamic programming set oriented numerics
Topological method for rigorously computing periodic orbits using Fourier modes
Anthony W. Baker Michael Dellnitz Oliver Junge
Discrete & Continuous Dynamical Systems - A 2005, 13(4): 901-920 doi: 10.3934/dcds.2005.13.901
We present a technique for the rigorous computation of periodic orbits in certain ordinary differential equations. The method combines set oriented numerical techniques for the computation of invariant sets in dynamical systems with topological index arguments. It not only allows for the proof of existence of periodic orbits but also for a precise (and rigorous) approximation of these. As an example we compute a periodic orbit for a differential equation introduced in [2].
keywords: periodic orbit set oriented methods Dynamical system computer assisted proof.
Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013)
Gary Froyland Oliver Junge Kathrin Padberg-Gehle
Journal of Computational Dynamics 2015, 2(1): i-ii doi: 10.3934/jcd.2015.2.1i
This issue comprises manuscripts collected on the occasion of the 4th International Workshop on Set-Oriented Numerics which took place at the Technische Universität Dresden in September 2013. The contributions cover a broad spectrum of different subjects in computational dynamics ranging from purely discrete problems on graphs to computer assisted proofs of bifurcations in dissipative PDEs. In many cases, ideas related to set-oriented paradigms turn out to be useful in the computations, for example by quantizing the state space, or by using interval arithmetic to perform rigorous computations.

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Sparse control of alignment models in high dimension
Mattia Bongini Massimo Fornasier Oliver Junge Benjamin Scharf
Networks & Heterogeneous Media 2015, 10(3): 647-697 doi: 10.3934/nhm.2015.10.647
For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.
keywords: Poincaré recurrences Dimension theory multifractal analysis.

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