Balanced model order reduction for linear random dynamical systems driven by Lévy noise

Martin Redmann; Melina A. Freitag

doi:10.3934/jcd.2018002

Article Contents

2018, Volume 5: 33-59. Doi: 10.3934/jcd.2018002

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Balanced model order reduction for linear random dynamical systems driven by Lévy noise

Martin Redmann^1, and
Melina A. Freitag^2, ,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
2.
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

* Corresponding author: Melina A. Freitag
* Corresponding author: Melina A. Freitag

Published: November 2018

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93A15, 93B40, 93E03, 65C30, 60J75; Secondary: 93A30, 15A24.

Citation:

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Figure 1. Galerkin solution to the stochastic damped wave equation in (6).

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Figure 2. Components of the output (7) (position and velocity in the middle of the string) of the stochastic damped wave equation in (6).

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Figure 3. Output of stochastic damped wave equation in (6)-(7) in the phase plane.

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Figure 4. Logarithmic errors of BT for position $y^1$ and velocity $y^2$ with $r = 6$.

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Figure 5. Logarithmic errors of SPA for position $y^1$ and velocity $y^2$ with $r = 24$.

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Figure 6. Logarithmic errors of SPA for position $y^1$ and velocity $y^2$ with $r = 6$.

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Figure 7. Logarithmic errors of BT for position $y^1$ and velocity $y^2$ with $r = 24$.

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Table 1. Error and error bounds for both BT and SPA and several dimensions of the reduced order model (ROM).

Dim. ROM	Error BT	Error bound BT	Error SPA	Error bound SPA
2	7.6387e-02	9.3245e-02	1.0852e-01	1.2293e-01
4	8.5160e-03	1.2180e-02	8.6050e-03	1.2185e-02
8	5.1560e-03	9.6638e-03	5.6720e-03	9.7072e-03
16	1.8570e-03	6.6764e-03	2.4970e-03	6.7382e-03
32	6.7050e-04	4.3849e-03	1.4410e-03	4.9106e-03
64	9.9130e-05	2.3491e-03	3.1440e-04	2.6354e-03

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