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

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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

Abstract
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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.

Keywords: balanced truncation, SODEs, Galerkin method, Levy processes, SPDEs..

Mathematics Subject Classification: Primary: 34K28, 60H10, 60H15, 60J75; Secondary: 78M3.

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).
[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.
[3]	R. F. Curtain, Stability of Stochastic Partial Differential Equation,, J. Math. Anal. Appl., 79 (1981), 352.
[4]	T. Damm, Rational Matrix Equations in Stochastic Control,, Lecture Notes in Control and Information Sciences 297, (2004).
[5]	W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Aust. Math. Soc., 54 (1996), 79.
[6]	E. Hausenblas, Approximation for Semilinear Stochastic Evolution Equations,, Potential Anal., 18 (2003), 141.
[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.
[8]	B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction,, IEEE Trans. Autom. Control, 26 (1981), 17.
[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).
[10]	A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems,, SIAM Rev., 23 (1981), 25.
[11]	M. Redmann and P. Benner, Model Reduction for Stochastic Systems,, Stoch PDE: Anal Comp, 3(3) (2015), 291.

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References:

[1]	A. C. Antoulas, Approximation of large-scale dynamical systems,, Advances in Design and Control 6. Philadelphia, (2005).
[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.
[3]	R. F. Curtain, Stability of Stochastic Partial Differential Equation,, J. Math. Anal. Appl., 79 (1981), 352.
[4]	T. Damm, Rational Matrix Equations in Stochastic Control,, Lecture Notes in Control and Information Sciences 297, (2004).
[5]	W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Aust. Math. Soc., 54 (1996), 79.
[6]	E. Hausenblas, Approximation for Semilinear Stochastic Evolution Equations,, Potential Anal., 18 (2003), 141.
[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.
[8]	B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction,, IEEE Trans. Autom. Control, 26 (1981), 17.
[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).
[10]	A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems,, SIAM Rev., 23 (1981), 25.
[11]	M. Redmann and P. Benner, Model Reduction for Stochastic Systems,, Stoch PDE: Anal Comp, 3(3) (2015), 291.

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