An H2-type error bound for balancing-related model order reduction of linear systems with Lévy noise

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Abstract

To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear differential equations of high order. To reduce the high dimension for practical computations, model order reduction is frequently used. Balanced truncation (BT) is a well-known technique from deterministic control theory and it was already extended for controlled linear systems with Lévy noise. Recently, a new ansatz was investigated which provides an alternative way to generalize BT for stochastic systems. There, the question of the existence of an H2-error bound was asked which we answer in this paper. We discuss how this bound can be computed practically and how to use it to find a suitable reduced order dimension.

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 H2-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 H-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 H-error bound and the preservation of mean square asymptotic stability. It only remains to prove an H2-error bound to have a closed theory. This H2-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 H2-type error bound ϵ̃ for the new ansatz in [14] and [15] to close the gap in the error bound analysis. The non-negative number ϵ̃ bounds the worst case mean error between the original and the reduced order output Y and YR as follows: supt[0,T]EY(t)ỸR(t)2ϵ̃uLT2.As a first step, we provide a representation of ϵ̃ 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 EY(t)ỸR(t)2. Furthermore, we prove that ϵ̃ can be rewritten as an expression depending on the truncated Hankel singular values (HSVs) of the system similar to the H 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 A, NkRn×n, BRn×m and CRp×n. For t0 and X(0)=x0 we consider the following linear stochastic system: dX(t)=[AX(t)+Bu(t)]dt+k=1qNkX(t)dMk(t),Y(t)=CX(t),where M1,,Mq are scalar uncorrelated and square integrable Lévy processes with mean zero defined on a filtered probability space Ω,F,(Ft)t0,P.1 In addition, we assume Mk (k=1,,q) to be (Ft)t0-adapted and the increments Mk(t+h)Mk(t) to be independent of Ft

H2-type error bound for type 2 balanced truncation

For simplicity of notation, we assume to have a balanced realization of system (1) in terms of the type 2 approach. This balanced realization we denote by (Ã,B̃,C̃,Ñ) in order to distinguish between the coefficients of the type 1 and the type 2 ansatz. Since we are in a balanced situation, P̃=Q=Σ̃ such that ÃTΣ̃+Σ̃Ã+ÑTΣ̃Ñc=C̃TC̃,ÃTΣ̃1+Σ̃1Ã+ÑTΣ̃1ÑcΣ̃1B̃B̃TΣ̃1,where cEM2(1). Below, we use the following suitable partitions Ã=Ã11Ã12Ã21Ã22,B̃=B̃1B̃2,C̃=C̃1C̃2,Ñ=Ñ11Ñ12Ñ21Ñ

Conclusions

In this paper, we have described two ways to generalize BT for linear controlled SDEs with Lévy noise, the type 1 and the type 2 ansatz. We discussed the procedures to obtain the ROMs and summarized all known facts in that field including an H2-type error bound, a stability result for type 1 BT and an H-type error bound, a stability result for type 2 BT. As our contribution we established an H2-type error bound for the type 2 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.

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