Abstract
This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.
| Original language | English |
|---|---|
| Pages (from-to) | 553-589 |
| Number of pages | 37 |
| Journal | Japan Journal of Industrial and Applied Mathematics |
| Volume | 35 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - Jul 1 2018 |
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Keywords
- Dimensionality reduction
- Inverse problem
- Nonlinear dynamical systems
- Predictability
- Stochastic differential equation
ASJC Scopus subject areas
- Engineering(all)
- Applied Mathematics
Cite this
Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates. / Nakano, Naoto; Inatsu, Masaru; Kusuoka, Seiichiro; Saiki, Yoshitaka.
In: Japan Journal of Industrial and Applied Mathematics, Vol. 35, No. 2, 01.07.2018, p. 553-589.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates
AU - Nakano, Naoto
AU - Inatsu, Masaru
AU - Kusuoka, Seiichiro
AU - Saiki, Yoshitaka
PY - 2018/7/1
Y1 - 2018/7/1
N2 - This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.
AB - This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.
KW - Dimensionality reduction
KW - Inverse problem
KW - Nonlinear dynamical systems
KW - Predictability
KW - Stochastic differential equation
UR - http://www.scopus.com/inward/record.url?scp=85045142376&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85045142376&partnerID=8YFLogxK
U2 - 10.1007/s13160-017-0296-2
DO - 10.1007/s13160-017-0296-2
M3 - Article
AN - SCOPUS:85045142376
VL - 35
SP - 553
EP - 589
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
SN - 0916-7005
IS - 2
ER -