@misc{Winkelmann2016, author = {Winkelmann, Stefanie}, title = {Markov Control with Rare State Observation: Average Optimality}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60981}, year = {2016}, abstract = {This paper investigates the criterion of long-term average costs for a Markov decision process (MDP) which is not permanently observable. Each observation of the process produces a fixed amount of \textit{information costs} which enter the considered performance criterion and preclude from arbitrarily frequent state testing. Choosing the \textit{rare} observation times is part of the control procedure. In contrast to the theory of partially observable Markov decision processes, we consider an arbitrary continuous-time Markov process on a finite state space without further restrictions on the dynamics or the type of interaction. Based on the original Markov control theory, we redefine the control model and the average cost criterion for the setting of information costs. We analyze the constant of average costs for the case of ergodic dynamics and present an optimality equation which characterizes the optimal choice of control actions and observation times. For this purpose, we construct an equivalent freely observable MDP and translate the well-known results from the original theory to the new setting.}, language = {en} } @misc{WinkelmannSchuette2016, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60999}, year = {2016}, abstract = {Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models (MSM) as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.}, language = {en} } @article{NiemannWinkelmannWolfetal.2021, author = {Niemann, Jan-Hendrik and Winkelmann, Stefanie and Wolf, Sarah and Sch{\"u}tte, Christof}, title = {Agent-based modeling: Population limits and large timescales}, volume = {31}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, number = {3}, issn = {1438-0064}, doi = {10.1063/5.0031373}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-77309}, year = {2021}, abstract = {Modeling, simulation and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from informal descriptions of interaction dynamics to more formal, mathematical models. In this paper, we study agent-based models (ABMs) given as continuous-time stochastic processes and their pathwise approximation by ordinary and stochastic differential equations (ODEs and SDEs, respectively) for medium to large populations. By means of an appropriately adapted transfer operator approach we study the behavior of the ABM process on long time scales. We show that, under certain conditions, the transfer operator approach allows to bridge the gap between the pathwise results for large populations on finite timescales, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales like large deviation theory. The latter provides a rigorous analysis of rare events including the associated asymptotic rates on timescales that scale exponentially with the population size. We demonstrate that it is possible to reveal metastable structures and timescales of rare events of the ABM process by finite-length trajectories of the SDE process for large enough populations. This approach has the potential to drastically reduce computational effort for the analysis of ABMs.}, language = {en} } @misc{GelssMateraSchuette2015, author = {Gelß, Patrick and Matera, Sebastian and Sch{\"u}tte, Christof}, title = {Solving the master equation without kinetic Monte Carlo: tensor train approximations for a CO oxidation model}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55743}, year = {2015}, abstract = {In multiscale models of heterogeneous catalysis, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. This usually is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO_2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.}, language = {en} } @misc{WinkelmannSchuette2017, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {Hybrid Models for Chemical Reaction Networks: Multiscale Theory and Application to Gene Regulatory Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64264}, year = {2017}, abstract = {Well-mixed stochastic chemical kinetics are properly modelled by the chemical master equation (CME) and associated Markov jump processes in molecule number space. If the reactants are present in large amounts, however, corresponding simulations of the stochastic dynamics become computationally expensive and model reductions are demanded. The classical model reduction approach uniformly rescales the overall dynamics to obtain deterministic systems characterized by ordinary differential equations, the well-known mass action reaction rate equations. For systems with multiple scales there exist hybrid approaches that keep parts of the system discrete while another part is approximated either using Langevin dynamics or deterministically. This paper aims at giving a coherent overview of the different hybrid approaches, focusing on their basic concepts and the relation between them. We derive a novel general description of such hybrid models that allows to express various forms by one type of equation. We also check in how far the approaches apply to model extensions of the CME for dynamics which do not comply with the central well-mixed condition and require some spatial resolution. A simple but meaningful gene expression system with negative self-regulation is analysed to illustrate the different approximation qualities of some of the hybrid approaches discussed.}, language = {en} } @misc{BittracherKoltaiKlusetal.2017, author = {Bittracher, Andreas and Koltai, P{\´e}ter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Sch{\"u}tte, Christof}, title = {Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63822}, year = {2017}, abstract = {We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.}, language = {en} } @misc{StraubeWinkelmannHoefling2022, author = {Straube, Arthur and Winkelmann, Stefanie and H{\"o}fling, Felix}, title = {Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles}, issn = {1438-0064}, doi = {10.12752/8817}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88179}, year = {2022}, abstract = {Our theoretical study concerns an urea-urease-based pH oscillator confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the limit cycle, which controls the dynamics for giant vesicles and dominates the strongly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, amenable to analytic treatments, and show that the accuracy of predictions, including the period of oscillations, is highly sensitive to the choice of the reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The accurate description of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.}, language = {en} }