## ESAIM: Mathematical Modelling and Numerical Analysis

### Towards effective dynamics in complex systems by Markov kernel approximation

a1 Institut für Mathematik II, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany. schuette@math.fu-berlin.de

a2 Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe (TH), Kaiserstr. 93, 76133 Karlsruhe, Germany. jahnke@math.uni-karlsruhe.de

Abstract

Many complex systems occurring in various application share the property that the underlying Markov process remains in certain regions of the state space for long times, and that transitions between such metastable sets occur only rarely. Often the dynamics within each metastable set is of minor importance, but the transitions between these sets are crucial for the behavior and the understanding of the system. Since simulations of the original process are usually prohibitively expensive, the effective dynamics of the system, i.e. the switching between metastable sets, has to be approximated in a reliable way. This is usually done by computing the dominant eigenvectors and eigenvalues of the transfer operator associated to the Markov process. In many real applications, however, the matrix representing the spatially discretized transfer operator can be extremely large, such that approximating eigenvectors and eigenvalues is a computationally critical problem. In this article we present a novel method to determine the effective dynamics via the transfer operator without computing its dominant spectral elements. The main idea is that a time series of the process allows to approximate the sampling kernel of the process, which is an integral kernel closely related to the transition function of the transfer operator. Metastability is taken into account by representing the approximative sampling kernel by a linear combination of kernels each of which represents the process on one of the metastable sets. The effect of the approximation error on the dynamics of the system is discussed, and the potential of the new approach is illustrated by numerical examples.

(Online publication July 8 2009)

Key Words:

• Effective dynamics;
• complex systems;
• Markov process;
• metastability;
• transfer operators;
• model reduction;
• mixture models.

Mathematics Subject Classification:

• 60J25;
• 60J35;
• 62M05;
• 60J22;
• 65C40