Filters

2016 - 20172

More filters

20192
Reset filters

Settings

Type: Report × Subject: stochastic systems ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Type II singular perturbation approximation for linear systems with Lévy noise
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Redmann, Martin
    When solving linear stochastic 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 singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems.
  • Thumbnail Image
    Item
    Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Redmann, Martin; Freitag, Melina A.
    When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.