Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

Ansmann, Gerrit

doi:10.1063/1.5019320

Cited by 62 publications

(36 citation statements)

References 42 publications

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“…We employ a Euler-Maruyama integrator, for simplicity. There are more reliable and faster integrators, for example JiTCSDE (Ansmann, 2018). For the present case, with an integration over 500 time units and with a timestep of 0.001, which can be seen in Fig.…”

Section: Exemplary One-dimensional Ornstein-uhlenbeck Processmentioning

confidence: 83%

kramersmoyal: Kramers--Moyal coefficients for stochastic processes

Gorjão¹,

Meirinhos²

2019

JOSS

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A general problem for evaluating Markovian stochastic processes is the retrieval of the moments or the Kramers-Moyal coefficients M from data or time-series. The Kramers-Moyal coefficients are derived from an Taylor expansion of the master equation that describes the probability evolution of a Markovian stochastic process.Given a set of stochastic data, ergodic or quasi-stationary, the extensive literature of stochastic processes awards a set of measures, such as the Kramers-Moyal coefficients or its moments, which link stochastic processes to a probabilistic description of the process or of the family of processes (Risken, 1996). Most commonly known is the Fokker-Planck equation or truncated forward Kolmogorov equation, partial differential equations, obtained from the Taylor expansion of the master equation.Of particular relevance is the growing evidence that real-world data displays higher-order (n > 2) Kramers-Moyal coefficients, which has a two-fold consequence: The common truncation at third order of the forward Kolmogorov equation, giving rise to the Fokker-Planck equation, is no longer valid. The existence of higher-order (n > 2) Kramers-Moyal coefficients in recorded data thus invalidates the aforementioned common argument for truncation, thus rendering the Fokker-Planck description insufficient (Tabar, 2019). A clear and common example is the presence of discontinuous jumps in data (Aït-Sahalia, 2002;Anvari, Tabar, Peinke, & Lehnertz, 2016), which can give rise to higher-order Kramers-Moyal coefficients, as are evidenced in Gorjão, Heysel, Lehnertz, & Tabar (2019) and references within.Calculating the moments or Kramers-Moyal coefficients strictly from data can be computationally heavy for long data series and is prone to innaccuracy especially where the density of data points is scarce, for example, usually at the boundaries on the domain of the process. The most straightforward approach is to perform a histogram-based estimation to evaluate the moments of the system at hand. This has two main drawbacks: it requires a discrete space of examination of the process and is shown to be less accurate than using kernel-based estimators (Lamouroux & Lehnertz, 2009). This library is based on a kernel-based estimation, i.e., the Nadaraya-Watson kernel estimator (Nadaraya, 1964;Watson, 1964), which allows for more robust results given both a wider range of possible kernel shapes to perform the calculation, as well as retrieving the results in a nonbinned coordinate space, unlike histogram regressions (Silverman, 2018). It further employs a convolution of the time series with the selected kernel, circumventing the computational issue of sequential array summation, the most common bottleneck in integration time and computer memory.

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Section: Exemplary One-dimensional Ornstein-uhlenbeck Processmentioning

confidence: 83%

kramersmoyal: Kramers--Moyal coefficients for stochastic processes

Gorjão¹,

Meirinhos²

2019

JOSS

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“…All the numerical integrations presented below are performed using a Dormand-Prince method with adaptive time step and a spin time of 100 time units, with runs of 1000 time units. We make use of the Python module JiTCODE [81], an extension of SciPy's ODE that allows to numerically simulate ordinary differential equations, computing quantities of interest as Lyapunov exponents as well. All results have been double checked and confirmed using the MATLAB function ode45 where integrations are performed using the 4th order Runge-Kutta integrator with adaptive time step [82].…”

Section: Resultsmentioning

confidence: 99%

Mechanics and thermodynamics of a new minimal model of the atmosphere

Vissio

Lucarini

2020

Eur. Phys. J. Plus

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The understanding of the fundamental properties of the climate system has long benefitted from the use of simple numerical models able to parsimoniously represent the essential ingredients of its processes. Here, we introduce a new model for the atmosphere that is constructed by supplementing the now-classic Lorenz ’96 one-dimensional lattice model with temperature-like variables. The model features an energy cycle that allows for energy to be converted between the kinetic form and the potential form and for introducing a notion of efficiency. The model’s evolution is controlled by two contributions—a quasi-symplectic and a gradient one, which resemble (yet not conforming to) a metriplectic structure. After investigating the linear stability of the symmetric fixed point, we perform a systematic parametric investigation that allows us to define regions in the parameters space where at steady-state stationary, quasi-periodic, and chaotic motions are realised, and study how the terms responsible for defining the energy budget of the system depend on the external forcing injecting energy in the kinetic and in the potential energy reservoirs. Finally, we find preliminary evidence that the model features extensive chaos. We also introduce a more complex version of the model that is able to accommodate for multiscale dynamics and that features an energy cycle that more closely mimics the one of the Earth’s atmosphere.

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“…This can be done by solving Eq. (6) numerically using a solver for delay differential equations 50 . For the simulation the system was initialized at the fixed point and a disturbance is introduced to one of the two areas.…”

Section: Homogeneous Inertiamentioning

confidence: 99%

Time delay effects in the control of synchronous electricity grids

Böttcher

Otto

Kettemann

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The expansion of inverter-connected generation facilities (i.e. wind and photovoltaics) and the removal of conventional power plants is necessary to mitigate the impacts of climate change. Whereas conventional generation with large rotating generator masses provides stabilizing inertia, inverter-connected generation does not. Since the underlying power system and the control mechanisms that keep it close to a desired reference state, were not

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Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

Cited by 62 publications

References 42 publications

kramersmoyal: Kramers--Moyal coefficients for stochastic processes

kramersmoyal: Kramers--Moyal coefficients for stochastic processes

Mechanics and thermodynamics of a new minimal model of the atmosphere

Time delay effects in the control of synchronous electricity grids

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