An -type error bound for balancing-related model order reduction of linear systems with Lévy noise
Introduction
Model order reduction (MOR) is of major importance in the field of deterministic control theory. It is used to save computational time by replacing large scale systems by systems of low order in which the main information of the original system should be captured. Such high dimensional problems occur for example after the spatial discretization of a partial differential equation (PDE) which can be used to model chemical, physical or biological phenomena. A particular ansatz to obtain a reduced order model is to balance a system such that the dominant reachable and observable states are the same. Afterwards, the difficult to observe and difficult to reach states are neglected. One way to do that is to use balanced truncation (BT) which was introduced by Moore [1] and a thorough treatment of the topic can be found in Antoulas [2] or Obinata, Anderson [3].
Since many phenomena in computational sciences and engineering contain uncertainties, it is natural to extend PDE models by adding a noise term. This leads to stochastic PDEs (SPDEs) which are studied in Da Prato, Zabczyk [4] and in Prévôt and Röckner [5] for the Wiener case. Peszat, Zabczyk consider more general equations with Lévy noise in [6], where the solutions may have jumps. To solve SPDEs numerically, one can reduce them to large scale ordinary SDEs by using the Galerkin method. For that reason, generalizing MOR techniques to stochastic systems can be motivated. The mentioned Galerkin approximation is for example investigated in Grecksch, Kloeden [7], Hausenblas [8], Jentzen, Kloeden [9] and Redmann, Benner [10].
To reduce large scale SDEs, balancing related methods are generalized. BT is considered for SDEs with Wiener noise in Benner, Damm [11] and for systems with Lévy noise it is done by Benner, Redmann in [12]. Benner and Redmann provide an -type error bound and the preservation of mean square asymptotic stability is shown in Benner et al. [13]. In Benner et al. [14] and Damm, Benner [15] an example is presented which clarifies that the -error bound from the deterministic case does not hold for stochastic systems. Recently, a new ansatz to extend BT to SDEs is considered by Benner et al. [14] or Damm, Benner [15] in which a new reachability Gramian is used. This alternative Gramian so far has no integral representation involving the fundamental solution of the system which is in contrast to the first approach. The advantage of the new ansatz is the existence of an -error bound and the preservation of mean square asymptotic stability. It only remains to prove an -error bound to have a closed theory. This -error bound analysis is present in this paper.
In this paper, we focus on BT for SDEs with Lévy noise. We start with giving an overview about the two ways to generalize the deterministic framework and state the most important results that are already proven. In Section 2, we briefly discuss the procedure and emphasize results on error bounds and the stability analysis of the methods. In Section 3, we contribute an -type error bound for the new ansatz in [14] and [15] to close the gap in the error bound analysis. The non-negative number bounds the worst case mean error between the original and the reduced order output and as follows: As a first step, we provide a representation of which can be taken for practical computations and hence be used for finding a suitable reduced order dimension. For this representation, we need to solve three matrix equations which are much cheaper than computing the expected value . Furthermore, we prove that can be rewritten as an expression depending on the truncated Hankel singular values (HSVs) of the system similar to the error bound. This second representation can be used to find a suitable reduced order dimension based on the HSVs and it shows that the error bound is small if the truncated states are unimportant (states corresponding to the small HSVs).
Section snippets
Balancing of stochastic systems with Lévy noise
Let , , and . For and we consider the following linear stochastic system: where are scalar uncorrelated and square integrable Lévy processes with mean zero defined on a filtered probability space .1 In addition, we assume () to be -adapted and the increments to be independent of
-type error bound for type balanced truncation
For simplicity of notation, we assume to have a balanced realization of system (1) in terms of the type approach. This balanced realization we denote by in order to distinguish between the coefficients of the type and the type ansatz. Since we are in a balanced situation, such that where . Below, we use the following suitable partitions
Conclusions
In this paper, we have described two ways to generalize BT for linear controlled SDEs with Lévy noise, the type and the type ansatz. We discussed the procedures to obtain the ROMs and summarized all known facts in that field including an -type error bound, a stability result for type BT and an -type error bound, a stability result for type BT. As our contribution we established an -type error bound for the type ansatz and proved that there is a realization which only depends on
Acknowledgments
We thank the reviewers for their comments and suggestions for the manuscript. These comments definitely helped to improve the paper.
References (15)
- et al.
Positive operators and stable truncation
Linear Algebra Appl.
(2016) Principal component analysis in linear systems: Controllability, observability, and model reduction
IEEE Trans. Automat. Control
(1981)Approximation of Large-scale Dynamical Systems
- et al.
Model Reduction for Control System Design
- et al.
Stochastic Equations in Infinite Dimensions
- et al.
A Concise Course on Stochastic Partial Differential Equations
- et al.
Stochastic Partial Differential Equations with Lévy Noise. An evolution equation approach
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