# Dimension Reduction for Systems with Slow Relaxation

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## Abstract

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize ‘optimal’ model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.

## Keywords

Dimension reduction Slow relaxation Sloppy models Mori–Zwanzig projection Multi-scale Aging Weathering Glassy systems Oil spills## Notes

### Acknowledgements

S.V. would like to acknowledge the many, very illuminating discussions with Kevin Lin who was very generous with his time and his ideas. We are grateful to an anonymous referee for pointing out the potential connections between our work and the sloppy models universality class. This viewpoint turns out to be particularly fruitful. This work was funded in part by a Grant from GoMRI. We also received support from NSF-DMS-1109856 and NSF-OCE-1434198.

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