24 research outputs found
Type II balanced truncation for deterministic bilinear control systems
When solving partial differential equations numerically, usually a high order
spatial discretisation is needed. Model order reduction (MOR) techniques are
often used to reduce the order of spatially-discretised systems and hence
reduce computational complexity. A particular MOR technique to obtain a reduced
order model (ROM) is balanced truncation (BT), a method which has been
extensively studied for deterministic linear systems. As so-called type I BT it
has already been extended to bilinear equations, an important subclass of
nonlinear systems. We provide an alternative generalisation of the linear
setting to bilinear systems which is called type II BT. The Gramians that we
propose in this context contain information about the control. It turns out
that the new approach delivers energy bounds which are not just valid in a
small neighbourhood of zero. Furthermore, we provide an -error bound
which so far is not known when applying type I BT to bilinear systems
Energy estimates and model order reduction for stochastic bilinear systems
In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far
Energy estimates and model order reduction for stochastic bilinear systems
In this paper, we investigate a large-scale stochastic system with bilinear
drift and linear diffusion term. Such high dimensional systems appear for
example when discretizing a stochastic partial differential equations in space.
We study a particular model order reduction technique called balanced
truncation (BT) to reduce the order of spatially-discretized systems and hence
reduce computational complexity. We introduce suitable Gramians to the system
and prove energy estimates that can be used to identify states which contribute
only very little to the system dynamics. When BT is applied the reduced system
is obtained by removing these states from the original system. The main
contribution of this paper is an -error bound for BT for stochastic
bilinear systems. This result is new even for deterministic bilinear equations.
In order to achieve it, we develop a new technique which is not available in
the literature so far