Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

Int. J. UncertaintyQuantification

2018

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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.

Section: Test Examplementioning

confidence: 85%

“…for each t. The estimate (29) converges for k, m → ∞ in probability (provided that the orthonormal basis is complete). The error measures (28) and (25) are given point-wise in time. Global measures can be obtained by taking (integral) mean values on a finite time interval.…”

Section: Error Measures In Time Domainmentioning

confidence: 99%

Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems

Jakeman

Int. J. UncertaintyQuantification

2018

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“…Often it holds that rank(Ĉ) = m provided that m ≤n. More details on the stochastic Galerkin method for linear dynamical systems are given in [22,28].…”

Section: Stochastic Galerkin Methodsmentioning

confidence: 99%

Stability-preserving model order reduction for linear stochastic Galerkin systems

2019

J.Math.Industry

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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 stochastic 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.

“…The stochastic Galerkin systems generate approximations of the partial variances (15). Assuming thatV j − V j is sufficiently small for j = q ′ + 1, .…”

Section: Proofmentioning

confidence: 99%

“…Our contribution is application of model order reduction (MOR) strategies to the high-dimensional stochastic Galerkin system, which results in a much smaller reduced-order model that is much easier to simulate. General information on MOR can be found in [1,22]; in [15], the stochastic Galerkin method was reduced by a Krylov subspace method. In this paper, we use the technique of balanced truncation, where a-priori error bounds are available for the MOR.…”

Section: Introductionmentioning

confidence: 99%

Sensitivity analysis of random linear dynamical systems using quadratic outputs

Journal of Computational and Applied Mathematics

Narayan

2021

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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.