10,301 research outputs found
A new estimation of the lower error bound in balanced truncation method
For a single-input/single-output (SISO) linear time-invariant dynamical system, the standard H-infinity-norm lower error bound of balanced truncation method is parallel to G(s) - G(r)(s)parallel to(H infinity) >= sigma(r+1), where sigma(i), i = 1,..., n, are the Hankel singular values of system in decreasing order. In this paper we provide a new estimation of the lower error, namely; parallel to G(s) - G(r)(s)parallel to(H infinity) >= max{sigma(d), 2 vertical bar Sigma(i is not an element of g) s(i)sigma(i)vertical bar},; where s(i) is the sign associated with the Hankel singular value sigma(i) in Ober's canonical form. The subset g and the index d in the above inequality will be introduced in the paper. We show by means of an example that the new bound may be relevant in deciding which states need to be kept in the balanced truncation method, and that using the standard result does not always yield the best approximationPreprin
Early stopping for statistical inverse problems via truncated SVD estimation
We consider truncated SVD (or spectral cut-off, projection) estimators for a
prototypical statistical inverse problem in dimension . Since calculating
the singular value decomposition (SVD) only for the largest singular values is
much less costly than the full SVD, our aim is to select a data-driven
truncation level only based on the knowledge of
the first singular values and vectors. We analyse in detail
whether sequential {\it early stopping} rules of this type can preserve
statistical optimality. Information-constrained lower bounds and matching upper
bounds for a residual based stopping rule are provided, which give a clear
picture in which situation optimal sequential adaptation is feasible. Finally,
a hybrid two-step approach is proposed which allows for classical oracle
inequalities while considerably reducing numerical complexity.Comment: slightly modified version. arXiv admin note: text overlap with
arXiv:1606.0770
Balanced Truncation of Networked Linear Passive Systems
This paper studies model order reduction of multi-agent systems consisting of
identical linear passive subsystems, where the interconnection topology is
characterized by an undirected weighted graph. Balanced truncation based on a
pair of specifically selected generalized Gramians is implemented on the
asymptotically stable part of the full-order network model, which leads to a
reduced-order system preserving the passivity of each subsystem. Moreover, it
is proven that there exists a coordinate transformation to convert the
resulting reduced-order model to a state-space model of Laplacian dynamics.
Thus, the proposed method simultaneously reduces the complexity of the network
structure and individual agent dynamics, and it preserves the passivity of the
subsystems and the synchronization of the network. Moreover, it allows for the
a priori computation of a bound on the approximation error. Finally, the
feasibility of the method is demonstrated by an example
Order Reduction of the Chemical Master Equation via Balanced Realisation
We consider a Markov process in continuous time with a finite number of
discrete states. The time-dependent probabilities of being in any state of the
Markov chain are governed by a set of ordinary differential equations, whose
dimension might be large even for trivial systems. Here, we derive a reduced
ODE set that accurately approximates the probabilities of subspaces of interest
with a known error bound. Our methodology is based on model reduction by
balanced truncation and can be considerably more computationally efficient than
the Finite State Projection Algorithm (FSP) when used for obtaining transient
responses. We show the applicability of our method by analysing stochastic
chemical reactions. First, we obtain a reduced order model for the
infinitesimal generator of a Markov chain that models a reversible,
monomolecular reaction. In such an example, we obtain an approximation of the
output of a model with 301 states by a reduced model with 10 states. Later, we
obtain a reduced order model for a catalytic conversion of substrate to a
product; and compare its dynamics with a stochastic Michaelis-Menten
representation. For this example, we highlight the savings on the computational
load obtained by means of the reduced-order model. Finally, we revisit the
substrate catalytic conversion by obtaining a lower-order model that
approximates the probability of having predefined ranges of product molecules.Comment: 12 pages, 6 figure
Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition
We study reduced-order models of three-dimensional perturbations in
linearized channel flow using balanced proper orthogonal decomposition (BPOD).
The models are obtained from three-dimensional simulations in physical space as
opposed to the traditional single-wavenumber approach, and are therefore better
able to capture the effects of localized disturbances or localized actuators.
In order to assess the performance of the models, we consider the impulse
response and frequency response, and variation of the Reynolds number as a
model parameter. We show that the BPOD procedure yields models that capture the
transient growth well at a low order, whereas standard POD does not capture the
growth unless a considerably larger number of modes is included, and even then
can be inaccurate. In the case of a localized actuator, we show that POD modes
which are not energetically significant can be very important for capturing the
energy growth. In addition, a comparison of the subspaces resulting from the
two methods suggests that the use of a non-orthogonal projection with adjoint
modes is most likely the main reason for the superior performance of BPOD. We
also demonstrate that for single-wavenumber perturbations, low-order BPOD
models reproduce the dominant eigenvalues of the full system better than POD
models of the same order. These features indicate that the simple, yet accurate
BPOD models are a good candidate for developing model-based controllers for
channel flow.Comment: 35 pages, 20 figure
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