To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. If one uses the well-known Galerkin scheme, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations we consider balanced truncation as a model order reduction technique. This approach is well-known from deterministic control theory and successfully employed in practice for decades. So, we generalize balanced truncation for controlled linear systems with Levy noise, discuss properties of the reduced order model, provide an error bound, and give some examples.

To solve a stochastic linear evolution equation numerically, nite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations, we consider the singular perturbation approximation as a model order reduction technique in this paper. This approach is well-known from deterministic control theory and here we generalize it for controlled linear systems with Lévy noise. Additionally, we discuss properties of the reduced order model, provide an error bound, and give some examples to demonstrate the quality of this model order reduction technique.

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When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatiallydiscretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an H 2 error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments.

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