Sampling-based model order reduction for stochastic differential equations driven by fractional Brownian motion

Abstract

In this paper, we study large-scale linear fractional stochastic systems representing, e.g., spatially discretized stochastic partial differential equations (SPDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H > ½. Such equations are more realistic in modeling real-world phenomena in comparison to frameworks not capturing memory effects. To the best of our knowledge, dimension reduction schemes for such settings have not been studied so far.

In this work, we investigate empirical reduced order methods that are either based on snapshots (e.g. proper orthogonal decomposition) or on approximated Gramians. In each case, dominant subspaces are learned from data. Such model reduction techniques are introduced and analyzed for stochastic systems with fractional noise and later applied to spatially discretized SPDEs driven by fBm in order to reduce the computational cost arising from both the high dimension of the considered stochastic system and a large number of required Monte Carlo runs.

We validate our proposed techniques with numerical experiments for some large-scale stochastic differential equations driven by fBm.