2015, 2015(special): 945-953. doi: 10.3934/proc.2015.0945

Approximation and model order reduction for second order systems with Levy-noise

1. 

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany, Germany

Received  September 2014 Revised  September 2015 Published  November 2015

We consider a controlled second order stochastic partial differential equation (SPDE) with Levy noise. To solve this system numerically, we apply a Galerkin scheme leading to a sequence of ordinary SDEs of large order. To reduce the high dimension we use balanced truncation.
Citation: Martin Redmann, Peter Benner. Approximation and model order reduction for second order systems with Levy-noise. Conference Publications, 2015, 2015 (special) : 945-953. doi: 10.3934/proc.2015.0945
References:
[1]

A. C. Antoulas, Approximation of large-scale dynamical systems,, Advances in Design and Control 6. Philadelphia, (2005).   Google Scholar

[2]

P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems,, SIAM J. Control Optim., 49 (2011), 686.   Google Scholar

[3]

R. F. Curtain, Stability of Stochastic Partial Differential Equation,, J. Math. Anal. Appl., 79 (1981), 352.   Google Scholar

[4]

T. Damm, Rational Matrix Equations in Stochastic Control,, Lecture Notes in Control and Information Sciences 297, (2004).   Google Scholar

[5]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Aust. Math. Soc., 54 (1996), 79.   Google Scholar

[6]

E. Hausenblas, Approximation for Semilinear Stochastic Evolution Equations,, Potential Anal., 18 (2003), 141.   Google Scholar

[7]

A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise,, Proc. R. Soc. A 2009, 465 (2009), 649.   Google Scholar

[8]

B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction,, IEEE Trans. Autom. Control, 26 (1981), 17.   Google Scholar

[9]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An evolution equation approach,, Encyclopedia of Mathematics and Its Applications 113, (2007).   Google Scholar

[10]

A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems,, SIAM Rev., 23 (1981), 25.   Google Scholar

[11]

M. Redmann and P. Benner, Model Reduction for Stochastic Systems,, Stoch PDE: Anal Comp, 3(3) (2015), 291.   Google Scholar

show all references

References:
[1]

A. C. Antoulas, Approximation of large-scale dynamical systems,, Advances in Design and Control 6. Philadelphia, (2005).   Google Scholar

[2]

P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems,, SIAM J. Control Optim., 49 (2011), 686.   Google Scholar

[3]

R. F. Curtain, Stability of Stochastic Partial Differential Equation,, J. Math. Anal. Appl., 79 (1981), 352.   Google Scholar

[4]

T. Damm, Rational Matrix Equations in Stochastic Control,, Lecture Notes in Control and Information Sciences 297, (2004).   Google Scholar

[5]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Aust. Math. Soc., 54 (1996), 79.   Google Scholar

[6]

E. Hausenblas, Approximation for Semilinear Stochastic Evolution Equations,, Potential Anal., 18 (2003), 141.   Google Scholar

[7]

A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise,, Proc. R. Soc. A 2009, 465 (2009), 649.   Google Scholar

[8]

B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction,, IEEE Trans. Autom. Control, 26 (1981), 17.   Google Scholar

[9]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An evolution equation approach,, Encyclopedia of Mathematics and Its Applications 113, (2007).   Google Scholar

[10]

A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems,, SIAM Rev., 23 (1981), 25.   Google Scholar

[11]

M. Redmann and P. Benner, Model Reduction for Stochastic Systems,, Stoch PDE: Anal Comp, 3(3) (2015), 291.   Google Scholar

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