Communications in Mathematical Sciences

Volume 14 (2016)

Number 5

Data-driven stochastic representations of unresolved features in multiscale models

Pages: 1213 – 1236



Nick Verheul (Centrum Wiskunde and Informatica (CWI), Amsterdam, The Netherlands)

Daan Crommelin (Centrum Wiskunde and Informatica (CWI), Amsterdam, The Netherlands; and KdV Institute for Mathematics, University of Amsterdam, The Netherlands)


In this study, we investigate how to use sample data, generated by a fully resolved multiscale model, to construct stochastic representations of unresolved scales in reduced models. We explore three methods to model these stochastic representations. They employ empirical distributions, conditional Markov chains, and conditioned Ornstein–Uhlenbeck processes, respectively. The Kac–Zwanzig heat bath model is used as a prototype model to illustrate the methods. We demonstrate that all tested strategies reproduce the dynamics of the resolved model variables accurately. Furthermore, we show that the computational cost of the reduced model is several orders of magnitude lower than that of the fully resolved model.


multiscale modeling, stochastic model reduction, Kac–Zwanzig heat bath model

2010 Mathematics Subject Classification

37M05, 60H10, 60H35, 65C20

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