Sensitivity analysis and model order reduction for random linear dynamical systems

We consider linear dynamical systems defined by differential algebraic equations. The associated input-output behaviour is given by a transfer function in the frequency domain. Physical parameters of the dynamical system are replaced by random variables to quantify uncertainties. We analyse the sensitivity of the transfer function with respect to the random variables. Total sensitivity coefficients are computed by a nonintrusive and by an intrusive method based on the expansions in series of the polynomial chaos. In addition, a reduction of the state space is applied in the intrusive method. Due to the sensitivities, we perform a model order reduction within the random space by changing unessential random variables back to constants. The error of this reduction is analysed. We present numerical simulations of a test example modelling a linear electric network.

[1]  Luca Daniel,et al.  A Piecewise-Linear Moment-Matching Approach to Parameterized Model-Order Reduction for Highly Nonlinear Systems , 2007, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[2]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[3]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[4]  Roland Pulch.  POLYNOMIAL CHAOS FOR SEMIEXPLICIT DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1 , 2013 .

[5]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[6]  Danny C. Sorensen,et al.  A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction , 2012, SIAM J. Numer. Anal..

[7]  Linda R. Petzold,et al.  The Effect of Problem Perturbations on Nonlinear Dynamical Systems and their Reduced-Order Models , 2005, SIAM J. Sci. Comput..

[8]  H. V. D. Vorst,et al.  Model Order Reduction: Theory, Research Aspects and Applications , 2008 .

[9]  Peter Benner,et al.  A Robust Algorithm for Parametric Model Order Reduction Based on Implicit Moment Matching , 2014 .

[10]  Danny C. Sorensen,et al.  A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems , 2014, SIAM J. Sci. Comput..

[11]  Saifon Chaturantabut,et al.  Structure-preserving model reduction for nonlinear port-Hamiltonian systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[12]  P. Rentrop,et al.  Stochastic Galerkin techniques for random ordinary differential equations , 2012, Numerische Mathematik.

[13]  Roland Pulch,et al.  Polynomial Chaos for the Computation of Failure Probabilities in Periodic Problems , 2010 .

[14]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[15]  Roland Pulch,et al.  POLYNOMIAL CHAOS FOR LINEAR DIFFERENTIAL ALGEBRAIC EQUATIONS WITH RANDOM PARAMETERS , 2011 .

[16]  Adrian Sandu,et al.  Modeling multibody systems with uncertainties. Part II: Numerical applications , 2006 .

[17]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[18]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[19]  Michael Günther,et al.  Modelling and discretization of circuit problems , 2005 .

[20]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[21]  Hung Dinh Nong,et al.  Passivity Preserving Model Reduction via Interpolation of Spectral Zeros: Selection Criteria and Implementation , 2007 .

[22]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[23]  Bernd Simeon,et al.  Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach , 2013 .

[24]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[25]  Serkan Gugercin,et al.  Model Reduction of Descriptor Systems by Interpolatory Projection Methods , 2013, SIAM J. Sci. Comput..

[26]  Wil H. A. Schilders,et al.  Index-Aware Model Order Reduction for Linear Index-2 DAEs with Constant Coefficients , 2013, SIAM J. Sci. Comput..

[27]  Luís Miguel Silveira,et al.  SPARE - a Scalable algorithm for passive, structure preserving, Parameter-Aware model order REduction , 2008, 2008 Design, Automation and Test in Europe.

[28]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[29]  Saltelli Andrea,et al.  Sensitivity Analysis for Nonlinear Mathematical Models. Numerical ExperienceSensitivity Analysis for Nonlinear Mathematical Models. Numerical Experience , 1995 .

[30]  Utz Wever,et al.  Application of the Polynomial Chaos Expansion to the simulation of chemical reactors with uncertainties , 2012, Math. Comput. Simul..

[31]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[32]  Linda R. Petzold,et al.  Approved for public release; further dissemination unlimited Error Estimation for Reduced Order Models of Dynamical Systems ∗ , 2003 .

[33]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[34]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[35]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[36]  Peter Benner,et al.  Interpolatory Projection Methods for Parameterized Model Reduction , 2011, SIAM J. Sci. Comput..

[37]  Dongbin Xiu,et al.  Variance-based global sensitivity analysis via sparse-grid interpolation and cubature , 2011 .

[38]  Peter Benner,et al.  Krylov-Subspace Based Model Reduction of Nonlinear Circuit Models Using Bilinear and Quadratic-Linear Approximations , 2012 .

[39]  Adrian Sandu,et al.  Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects , 2006 .

[40]  Roland Pulch,et al.  Sensitivity analysis of linear dynamical systems in uncertainty quantification , 2013 .

[41]  E. Jan W. ter Maten,et al.  Model Reduction for Circuit Simulation , 2011 .

[42]  R. Freund.  Model reduction methods based on Krylov subspaces , 2003, Acta Numerica.

[43]  A. Antoulas,et al.  H 2 Model Reduction for Large-scale Linear Dynamical Systems * , 2022 .

[44]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[45]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control) , 2005 .

[46]  Roland Pulch.  Modelling and simulation of autonomous oscillators with random parameters , 2011, Math. Comput. Simul..

[47]  Roland Pulch,et al.  Stochastic Galerkin methods and model order reduction for linear dynamical systems , 2013 .

[48]  P. Rentrop,et al.  Polynomial chaos for the approximation of uncertainties: Chances and limits , 2008, European Journal of Applied Mathematics.

[49]  Roland Pulch.  Polynomial Chaos for Analysing Periodic Processes of Differential Algebraic Equations with Random Parameters , 2008 .

[50]  R. D. Berry,et al.  Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System , 2011, SIAM J. Sci. Comput..