36 research outputs found
Stability-preserving model order reduction for linear stochastic Galerkin systems
Mathematical modeling often yields linear dynamical systems in science and
engineering. We change physical parameters of the system into random variables
to perform an uncertainty quantification. The stochastic Galerkin method yields
a larger linear dynamical system, whose solution represents an approximation of
random processes. A model order reduction (MOR) of the Galerkin system is
advantageous due to the high dimensionality. However, asymptotic stability may
be lost in some MOR techniques. In Galerkin-type MOR methods, the stability can
be guaranteed by a transformation to a dissipative form. Either the original
dynamical system or the stochastic Galerkin system can be transformed. We
investigate the two variants of this stability-preserving approach. Both
techniques are feasible, while featuring different properties in numerical
methods. Results of numerical computations are demonstrated for two test
examples modeling a mechanical application and an electric circuit,
respectively.Comment: 34 pages, 16 figure
Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest
We examine nonlinear dynamical systems of ordinary differential equations or
differential algebraic equations. In an uncertainty quantification, physical
parameters are replaced by random variables. The inner variables as well as a
quantity of interest are expanded into series with orthogonal basis functions
like the polynomial chaos expansions, for example. On the one hand, the
stochastic Galerkin method yields a large coupled dynamical system. On the
other hand, a stochastic collocation method, which uses a quadrature rule or a
sampling scheme, can be written in the form of a large weakly coupled dynamical
system. We apply projection-based methods of nonlinear model order reduction to
the large systems. A reduced-order model implies a low-dimensional
representation of the quantity of interest. We focus on model order reduction
by proper orthogonal decomposition. The error of a best approximation located
in a low-dimensional subspace is analysed. We illustrate results of numerical
computations for test examples.Comment: 28 pages, 18 figure
Model order reduction and sparse orthogonal expansions for random linear dynamical systems
We consider linear dynamical systems of ordinary differential equations or
differential algebraic equations. Physical parameters are substituted by random
variables for an uncertainty quantification. We expand the state variables as
well as a quantity of interest into an orthogonal system of basis functions,
which depend on the random variables. For example, polynomial chaos expansions
are applicable. The stochastic Galerkin method yields a larger linear dynamical
system, whose solution approximates the unknown coefficients in the expansions.
The Hardy norms of the transfer function provide information about the
input-output behaviour of the Galerkin system. We investigate two approaches to
construct a sparse representation of the quantity of interest, where just a low
number of coefficients is non-zero. Firstly, a standard basis is reduced by the
omission of basis functions, whose accompanying Hardy norms are relatively
small. Secondly, a projection-based model order reduction is applied to the
Galerkin system and allows for the definition of new basis functions as a
sparse representation. In both cases, we prove error bounds on the sparse
approximation with respect to Hardy norms. Numerical experiments are
demonstrated for a test example modelling a linear electric circuit.Comment: 24 pages, 11 figure
Stability preservation in Galerkin-type projection-based model order reduction
We consider linear dynamical systems consisting of ordinary differential
equations with high dimensionality. The aim of model order reduction is to
construct an approximating system of a much lower dimension. Therein, the
reduced system may be unstable, even though the original system is
asymptotically stable. We focus on projection-based model order reduction of
Galerkin-type. A transformation of the original system guarantees an
asymptotically stable reduced system. This transformation requires the
numerical solution of a high-dimensional Lyapunov equation. We specify an
approximation of the solution, which allows for an efficient iterative
treatment of the Lyapunov equation under a certain assumption. Furthermore, we
generalise this strategy to preserve the asymptotic stability of stationary
solutions in model order reduction of nonlinear dynamical systems. Numerical
results for high-dimensional examples confirm the computational feasibility of
the stability-preserving approach.Comment: 23 pages, 11 figure
Balanced truncation for model order reduction of linear dynamical systems with quadratic outputs
We investigate model order reduction (MOR) for linear dynamical systems,
where a quadratic output is defined as a quantity of interest. The system can
be transformed into a linear dynamical system with many linear outputs. MOR is
feasible by the method of balanced truncation, but suffers from the large
number of outputs in approximate methods. To ameliorate this shortcoming we
derive an equivalent quadratic-bilinear system with a single linear output and
analyze the properties of this system. We examine MOR for this system via the
technique of balanced truncation, which requires a stabilization of the system.
Therein, the solution of two quadratic Lyapunov equations is traced back to the
solution of just two linear Lyapunov equations. We present numerical results
for several test examples comparing the two MOR approaches.Comment: 35 pages, 20 figure
Poly-Sinc Solution of Stochastic Elliptic Differential Equations
In this paper, we introduce a numerical solution of a stochastic partial
differential equation (SPDE) of elliptic type using polynomial chaos along side
with polynomial approximation at Sinc points. These Sinc points are defined by
a conformal map and when mixed with the polynomial interpolation, it yields an
accurate approximation. The first step to solve SPDE is to use stochastic
Galerkin method in conjunction with polynomial chaos, which implies a system of
deterministic partial differential equations to be solved. The main difficulty
is the higher dimensionality of the resulting system of partial differential
equations. The idea here is to solve this system using a small number of
collocation points. Two examples are presented, mainly using Legendre
polynomials for stochastic variables. These examples illustrate that we require
to sample at few points to get a representation of a model that is sufficiently
accurate
Stability preservation in stochastic Galerkin projections of dynamical systems
In uncertainty quantification, critical parameters of mathematical models are
substituted by random variables. We consider dynamical systems composed of
ordinary differential equations. The unknown solution is expanded into an
orthogonal basis of the random space, e.g., the polynomial chaos expansions. A
Galerkin method yields a numerical solution of the stochastic model. In the
linear case, the Galerkin-projected system may be unstable, even though all
realizations of the original system are asymptotically stable. We derive a
basis transformation for the state variables in the original system, which
guarantees a stable Galerkin-projected system. The transformation matrix is
obtained from a symmetric decomposition of a solution of a Lyapunov equation.
In the nonlinear case, we examine stationary solutions of the original system.
Again the basis transformation preserves the asymptotic stability of the
stationary solutions in the stochastic Galerkin projection. We present results
of numerical computations for both a linear and a nonlinear test example.Comment: 24 pages, 12 figure
Sensitivity analysis of random linear dynamical systems using quadratic outputs
In uncertainty quantification, a stochastic modelling is often applied, where
parameters are substituted by random variables. We investigate linear dynamical
systems of ordinary differential equations with a quantity of interest as
output. Our objective is to analyse the sensitivity of the output with respect
to the random variables. A variance-based approach generates partial variances
and sensitivity indices. We expand the random output using the generalised
polynomial chaos. The stochastic Galerkin method yields a larger system of
ordinary differential equations. The partial variances represent quadratic
outputs of this system. We examine system norms of the stochastic Galerkin
formulation to obtain sensitivity measures. Furthermore, we apply model order
reduction by balanced truncation, which allows for an efficient computation of
the system norms with guaranteed error bounds. Numerical results are shown for
a test example.Comment: 26 pages, 14 figure
Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems
Polynomial chaos methods have been extensively used to analyze systems in
uncertainty quantification. Furthermore, several approaches exist to determine
a low-dimensional approximation (or sparse approximation) for some quantity of
interest in a model, where just a few orthogonal basis polynomials are
required. We consider linear dynamical systems consisting of ordinary
differential equations with random variables. The aim of this paper is to
explore methods for producing low-dimensional approximations of the quantity of
interest further. We investigate two numerical techniques to compute a
low-dimensional representation, which both fit the approximation to a set of
samples in the time domain. On the one hand, a frequency domain analysis of a
stochastic Galerkin system yields the selection of the basis polynomials. It
follows a linear least squares problem. On the other hand, a sparse
minimization yields the choice of the basis polynomials by information from the
time domain only. An orthogonal matching pursuit produces an approximate
solution of the minimization problem. We compare the two approaches using a
test example from a mechanical application
Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction
We investigate linear dynamical systems consisting of ordinary differential
equations with high dimensionality. Model order reduction yields alternative
systems of much lower dimensions. However, a reduced system may be unstable,
although the original system is asymptotically stable. We consider
projection-based model order reduction of Galerkin-type. A transformation of
the original system ensures that any reduced system is asymptotically stable.
This transformation requires the solution of a high-dimensional Lyapunov
inequality. We solve this problem using a specific Lyapunov equation. Its
solution can be represented as a matrix-valued integral in the frequency
domain. Consequently, quadrature rules yield numerical approximations, where
large sparse linear systems of algebraic equations have to be solved. We
analyse this approach and show a sufficient condition on the error to meet the
Lyapunov inequality. Furthermore, this technique is extended to systems of
differential-algebraic equations with strictly proper transfer functions by a
regularisation. Finally, we present results of numerical computations for
high-dimensional examples, which indicate the efficiency of this
stability-preserving method.Comment: 32 pages, 16 figure