Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion
arxiv(2024)
摘要
In this paper, we establish new strategies to reduce the dimension of
large-scale controlled stochastic differential equations with non-zero initial
states. The first approach transforms the original setting into a stochastic
system with zero initial states. This transformation naturally leads to
equations with controlled diffusion. A detailed analysis of dominant subspaces
and bounds for the reduction error is provided in this controlled diffusion
framework. Subsequently, we introduce a reduced system for the original
framework and prove an a-priori error bound for the first ansatz. This bound
involves so-called Hankel singular values that are linked to a new pair of
Gramians. A second strategy is presented that is based on the idea of reducing
control and initial state dynamics separately. Here, different Gramians are
used in order to derive a reduced model and their relation to dominant
subspaces are pointed out. We also show an a posteriori error bound for the
second approach involving two types of Hankel singular values.
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