Martin Redmann

Martin Redmann, Melina A. Freitag

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

Nahid Jamshidi, Martin Redmann

In this paper, we investigate large-scale linear systems driven by a fractional Brownian motion (fBm) with Hurst parameter $H\in [1/2, 1)$. We interpret these equations either in the sense of Young ($H>1/2$) or Stratonovich ($H=1/2$). Especially fractional Young differential equations are well suited for modeling real-world phenomena as they capture memory effects. Although it is very complex to solve them in high dimensions, model reduction schemes for Young or Stratonovich settings have not yet been studied much. To address this gap, we analyze important features of fundamental solutions associated to the underlying systems. We prove a weak type of semigroup property which is the foundation of studying system Gramians. From the introduced Gramians, dominant subspace can be identified which is shown in this paper as well. The difficulty for fractional drivers with $H>1/2$ is that there is no link of the corresponding Gramians to algebraic equations making the computation very difficult. Therefore, we further propose empirical Gramians that can be learned from simulation data. Subsequently, we introduce projection-based reduced order models (ROMs) using the dominant subspace information. We point out that such projections are not always optimal for Stratonovich equations as stability might not be preserved and since the error might be larger than expected. Therefore, an improved ROM is proposed for $H=1/2$. We validate our techniques conducting numerical experiments on some large-scale stochastic differential equations driven by fBm resulting from spatial discretizations of fractional stochastic PDEs. Overall, our study provides useful insights into the applicability and effectiveness of reduced order methods for stochastic systems with fractional noise, which can potentially aid in the development of more efficient computational strategies for practical applications.