### Abstract

A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): First, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems. More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp. The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.

Original language | English |
---|---|

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 17 |

Issue number | 4 |

Pages (from-to) | 2574-2614 |

ISSN | 1536-0040 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Keywords

- Dimension reduction
- Implicit equation-free methods
- Michaelis- Menten kinetics
- Slow-fast systems
- Stochastic differential equations

### Cite this

*SIAM Journal on Applied Dynamical Systems*,

*17*(4), 2574-2614. https://doi.org/10.1137/17M1126084

}

*SIAM Journal on Applied Dynamical Systems*, vol. 17, no. 4, pp. 2574-2614. https://doi.org/10.1137/17M1126084

**Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems.** / Sieber, Jan; Marschler, Christian; Starke, Jens.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems

AU - Sieber, Jan

AU - Marschler, Christian

AU - Starke, Jens

PY - 2018/1/1

Y1 - 2018/1/1

N2 - A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): First, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems. More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp. The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.

AB - A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): First, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for some time, and finally a user-defined restriction operator maps down into a low-dimensional space again. We prove convergence of equation-free methods for finite time scale separation with respect to a method parameter, the so-called healing time. Our convergence result justifies equation-free methods as a tool for performing high-level tasks such as bifurcation analysis on high-dimensional systems. More precisely, if the high-dimensional system has an attracting invariant manifold with smaller expansion and attraction rates in the tangential direction than in the transversal direction (normal hyperbolicity), and restriction and lifting satisfy some generic transversality conditions, then an implicit formulation of the lift-evolve-restrict procedure generates an approximate map that converges to the flow on the invariant manifold for healing time going to infinity. In contrast to all previous results, our result does not require the time scale separation to be large. A demonstration with Michaelis-Menten kinetics shows that the error estimates of our theorem are sharp. The ability to achieve convergence even for finite time scale separation is especially important for applications involving stochastic systems, where the evolution occurs at the level of distributions, governed by the Fokker-Planck equation. In these applications the spectral gap is typically finite. We investigate a low-dimensional stochastic differential equation where the ratio between the decay rates of fast and slow variables is 2.

KW - Dimension reduction

KW - Implicit equation-free methods

KW - Michaelis- Menten kinetics

KW - Slow-fast systems

KW - Stochastic differential equations

U2 - 10.1137/17M1126084

DO - 10.1137/17M1126084

M3 - Journal article

VL - 17

SP - 2574

EP - 2614

JO - S I A M Journal on Applied Dynamical Systems

JF - S I A M Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 4

ER -