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arXiv · 2408.00581

Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion

Martin Redmann
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Abstract

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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Martin Redmann. 2024-08-01. Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion. https://arxiv.org/abs/2408.00581

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