Abstract
Model order reduction (MOR) techniques are often used to reduce the order of spatially discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure has already been extended to bilinear equations, an important subclass of nonlinear systems. The choice of Gramians in Al-Baiyat and Bettayeb (In: Proceedings of the 32nd IEEE conference on decision and control, 1993) is the most frequently used approach. A balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on this choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend SPA to stochastic systems with bilinear drift and linear diffusion term. However, we propose a slightly modified reduced order model in comparison to previous work and choose a different reachability Gramian. Based on this new approach, an L^2-error bound is proved for SPA which is the main result of this paper. This bound is new even for deterministic bilinear systems.
Highlights
Many phenomena in real life can be described by partial differential equations (PDEs)
This work on singular perturbation approximation (SPA) for stochastic bilinear systems, see (1), can be interpreted as a generalization of the deterministic bilinear case [18]. This extension builds a bridge between stochastic linear systems and stochastic nonlinear systems such that SPA can possibly be used for many more stochastic equations and applications
We investigated a large-scale stochastic bilinear system
Summary
Many phenomena in real life can be described by partial differential equations (PDEs). Discretizing in space can be considered as a first step This can, for example, be done by spectral Galerkin [17,19,20] or finite element methods [2,21,22]. This usually leads to large-scale SDEs. This usually leads to large-scale SDEs Solving such complex SDE systems causes large computational cost. In this context, model order reduction (MOR) is used to save computational time by replacing high-dimensional systems by systems of low order in which the main information of the original system should be captured
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