Gramian-based model reduction for unstable stochastic systems

This paper considers large-scale linear stochastic systems representing, e.g., spatially discretized stochastic partial differential equations. Since asymptotic stability can often not be ensured in such a stochastic setting (e.g. due to larger noise), the main focus is on establishing model order reduction (MOR) schemes applicable to unstable systems. MOR is vital to reduce the dimension of the problem in order to lower the enormous computational complexity of for instance sampling methods in high dimensions. In particular, a new type of Gramian-based MOR approach is proposed in this paper that can be used in very general settings. The considered Gramians are constructed to identify dominant subspaces of the stochastic system as pointed out in this work. Moreover, they can be computed via Lyapunov equations. However, covariance information of the underlying systems enters these equations which is not directly available. Therefore, efficient sampling based methods relying on variance reduction techniques are established to derive the required covariances and hence the Gramians. Alternatively, an ansatz to compute the Gramians by deterministic approximations of covariance functions is investigated. An error bound for the studied MOR methods is proved yielding an a-priori criterion for the choice of the reduced system dimension. This bound is new and beneficial even in the deterministic case. The paper is concluded by numerical experiments showing the efficiency of the proposed MOR schemes.