June & December  2018, 5(1&2): 33-59. doi: 10.3934/jcd.2018002

Balanced model order reduction for linear random dynamical systems driven by Lévy noise

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany

2. 

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

* Corresponding author: Melina A. Freitag

Published  October 2018

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

Citation: Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002
References:
[1]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control 6. Philadelphia, PA: SIAM, 2005. doi: 10.1137/1.9780898718713.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781.

[3]

K. S. Arun and S. Y. Kung, Balanced approximation of stochastic systems, SIAM J. Matrix Anal. Appl., 11 (1990), 42-68.  doi: 10.1137/0611003.

[4]

O. E. Barndorff-NielsenJ. L. Jensen and M. Sørensen, Some stationary processes in discrete and continuous time, Adv. in Appl. Probab., 30 (1998), 989-1007.  doi: 10.1239/aap/1035228204.

[5]

C. BeattieS. Gugercin and V. Mehrmann, Model reduction for systems with inhomogeneous initial conditions, Systems Control Lett., 99 (2017), 99-106.  doi: 10.1016/j.sysconle.2016.11.007.

[6]

P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems., SIAM J. Control Optim., 49 (2011), 686-711.  doi: 10.1137/09075041X.

[7]

P. Benner and M. Redmann, Model reduction for stochastic systems, Stoch PDE: Anal Comp, 3 (2015), 291-338.  doi: 10.1007/s40072-015-0050-1.

[8]

R. F. Curtain, Stability of Stochastic Partial Differential Equation, J. Math. Anal. Appl., 79 (1981), 352-369.  doi: 10.1016/0022-247X(81)90031-7.

[9]

T. Damm and P. Benner, Balanced truncation for stochastic linear systems with guaranteed error bound, Proceedings of MTNS-2014, Groningen, The Netherlands, 2014, 1492-1497.

[10]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Aust. Math. Soc., 54 (1996), 79-85.  doi: 10.1017/S0004972700015094.

[11]

C. Hartmann, Balanced model reduction of partially observed langevin equations: An averaging principle, Math. Comput. Model. Dyn. Syst., 17 (2011), 463-490.  doi: 10.1080/13873954.2011.576517.

[12]

C. HartmannB. Schafer-Bung and A. Thons-Zueva, Balanced averaging of bilinear systems with applications to stochastic control, SIAM Journal on Control and Optimization, 51 (2013), 2356-2378.  doi: 10.1137/100796844.

[13]

C. Hartmann and C. Schütte, Balancing of partially-observed stochastic differential equations., in Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, IEEE, 2008, 4867-4872.

[14]

E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.

[15]

M. HeinkenschlossT. Reis and A. C. Antoulas, Balanced truncation model reduction for systems with inhomogeneous initial conditions, Automatica J. IFAC, 47 (2011), 559-564.  doi: 10.1016/j.automatica.2010.12.002.

[16]

D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.

[17]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.  doi: 10.1016/j.cam.2006.03.039.

[18]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes. 2nd ed., Grundlehren der Mathematischen Wissenschaften. 288. Berlin: Springer, 2003. doi: 10.1007/978-3-662-05265-5.

[19]

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), 649-667.  doi: 10.1098/rspa.2008.0325.

[20]

H.-H. Kuo, Introduction to Stochastic Integration, Universitext. New York, NJ: Springer, 2006.

[21]

Y. Liu and B. D. Anderson, Singular perturbation approximation of balanced systems, Int. J. Control, 50 (1989), 1379-1405.  doi: 10.1080/00207178908953437.

[22]

M. Metivier, Semimartingales: A Course on Stochastic Processes, De Gruyter Studies in Mathematics, 2. Berlin - New York: de Gruyter, 1982.

[23]

B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction., IEEE Trans. Autom. Control, 26 (1981), 17-32.  doi: 10.1109/TAC.1981.1102568.

[24]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications 113. Cambridge: Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373.

[25]

A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems, SIAM Rev., 23 (1981), 25-52.  doi: 10.1137/1023003.

[26]

M. Redmann, Balancing Related Model Order Reduction Applied to Linear Controlled Evolution Equations with Lévy Noise, Ph.D. thesis, Otto-von-Guericke-Universität Magdeburg, 2016.

[27]

M. Redmann and P. Benner, Approximation and model order reduction for second order systems with Lévy-noise, AIMS Proceedings, 2015, 945-953. doi: 10.3934/proc.2015.0945.

[28]

M. Redmann and P. Benner, Singular perturbation approximation for linear systems with Lévy noise, Stochastics and Dynamics, 18 (2018), 1850033, 23pp. doi: 10.1142/S0219493718500338.

[29]

K.-I. Sato and M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, Probability Theory and Mathematical Statistics, 1021 (2006), 541-551.  doi: 10.1007/BFb0072949.

[30]

W. H. Schilders, H. A. Van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications, vol. 13, Springer, 2008. doi: 10.1007/978-3-540-78841-6.

[31]

J. Zabczyk, Controllability of stochastic linear systems, Systems Control Lett., 1 (1981), 25-31.  doi: 10.1016/S0167-6911(81)80008-4.

show all references

References:
[1]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control 6. Philadelphia, PA: SIAM, 2005. doi: 10.1137/1.9780898718713.

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University Press, 2009. doi: 10.1017/CBO9780511809781.

[3]

K. S. Arun and S. Y. Kung, Balanced approximation of stochastic systems, SIAM J. Matrix Anal. Appl., 11 (1990), 42-68.  doi: 10.1137/0611003.

[4]

O. E. Barndorff-NielsenJ. L. Jensen and M. Sørensen, Some stationary processes in discrete and continuous time, Adv. in Appl. Probab., 30 (1998), 989-1007.  doi: 10.1239/aap/1035228204.

[5]

C. BeattieS. Gugercin and V. Mehrmann, Model reduction for systems with inhomogeneous initial conditions, Systems Control Lett., 99 (2017), 99-106.  doi: 10.1016/j.sysconle.2016.11.007.

[6]

P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems., SIAM J. Control Optim., 49 (2011), 686-711.  doi: 10.1137/09075041X.

[7]

P. Benner and M. Redmann, Model reduction for stochastic systems, Stoch PDE: Anal Comp, 3 (2015), 291-338.  doi: 10.1007/s40072-015-0050-1.

[8]

R. F. Curtain, Stability of Stochastic Partial Differential Equation, J. Math. Anal. Appl., 79 (1981), 352-369.  doi: 10.1016/0022-247X(81)90031-7.

[9]

T. Damm and P. Benner, Balanced truncation for stochastic linear systems with guaranteed error bound, Proceedings of MTNS-2014, Groningen, The Netherlands, 2014, 1492-1497.

[10]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Aust. Math. Soc., 54 (1996), 79-85.  doi: 10.1017/S0004972700015094.

[11]

C. Hartmann, Balanced model reduction of partially observed langevin equations: An averaging principle, Math. Comput. Model. Dyn. Syst., 17 (2011), 463-490.  doi: 10.1080/13873954.2011.576517.

[12]

C. HartmannB. Schafer-Bung and A. Thons-Zueva, Balanced averaging of bilinear systems with applications to stochastic control, SIAM Journal on Control and Optimization, 51 (2013), 2356-2378.  doi: 10.1137/100796844.

[13]

C. Hartmann and C. Schütte, Balancing of partially-observed stochastic differential equations., in Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, IEEE, 2008, 4867-4872.

[14]

E. Hausenblas, Approximation for semilinear stochastic evolution equations, Potential Anal., 18 (2003), 141-186.  doi: 10.1023/A:1020552804087.

[15]

M. HeinkenschlossT. Reis and A. C. Antoulas, Balanced truncation model reduction for systems with inhomogeneous initial conditions, Automatica J. IFAC, 47 (2011), 559-564.  doi: 10.1016/j.automatica.2010.12.002.

[16]

D. J. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.

[17]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.  doi: 10.1016/j.cam.2006.03.039.

[18]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes. 2nd ed., Grundlehren der Mathematischen Wissenschaften. 288. Berlin: Springer, 2003. doi: 10.1007/978-3-662-05265-5.

[19]

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), 649-667.  doi: 10.1098/rspa.2008.0325.

[20]

H.-H. Kuo, Introduction to Stochastic Integration, Universitext. New York, NJ: Springer, 2006.

[21]

Y. Liu and B. D. Anderson, Singular perturbation approximation of balanced systems, Int. J. Control, 50 (1989), 1379-1405.  doi: 10.1080/00207178908953437.

[22]

M. Metivier, Semimartingales: A Course on Stochastic Processes, De Gruyter Studies in Mathematics, 2. Berlin - New York: de Gruyter, 1982.

[23]

B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction., IEEE Trans. Autom. Control, 26 (1981), 17-32.  doi: 10.1109/TAC.1981.1102568.

[24]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications 113. Cambridge: Cambridge University Press, 2007. doi: 10.1017/CBO9780511721373.

[25]

A. J. Pritchard and J. Zabczyk, Stability and Stabilizability of Infinite-Dimensional Systems, SIAM Rev., 23 (1981), 25-52.  doi: 10.1137/1023003.

[26]

M. Redmann, Balancing Related Model Order Reduction Applied to Linear Controlled Evolution Equations with Lévy Noise, Ph.D. thesis, Otto-von-Guericke-Universität Magdeburg, 2016.

[27]

M. Redmann and P. Benner, Approximation and model order reduction for second order systems with Lévy-noise, AIMS Proceedings, 2015, 945-953. doi: 10.3934/proc.2015.0945.

[28]

M. Redmann and P. Benner, Singular perturbation approximation for linear systems with Lévy noise, Stochastics and Dynamics, 18 (2018), 1850033, 23pp. doi: 10.1142/S0219493718500338.

[29]

K.-I. Sato and M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, Probability Theory and Mathematical Statistics, 1021 (2006), 541-551.  doi: 10.1007/BFb0072949.

[30]

W. H. Schilders, H. A. Van der Vorst and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications, vol. 13, Springer, 2008. doi: 10.1007/978-3-540-78841-6.

[31]

J. Zabczyk, Controllability of stochastic linear systems, Systems Control Lett., 1 (1981), 25-31.  doi: 10.1016/S0167-6911(81)80008-4.

Figure 1.  Galerkin solution to the stochastic damped wave equation in (6).
Figure 2.  Components of the output (7) (position and velocity in the middle of the string) of the stochastic damped wave equation in (6).
Figure 3.  Output of stochastic damped wave equation in (6)-(7) in the phase plane.
Figure 4.  Logarithmic errors of BT for position $y^1$ and velocity $y^2$ with $r = 6$.
Figure 5.  Logarithmic errors of SPA for position $y^1$ and velocity $y^2$ with $r = 24$.
Figure 6.  Logarithmic errors of SPA for position $y^1$ and velocity $y^2$ with $r = 6$.
Figure 7.  Logarithmic errors of BT for position $y^1$ and velocity $y^2$ with $r = 24$.
Table 1.  Error and error bounds for both BT and SPA and several dimensions of the reduced order model (ROM).
Dim. ROM Error BT Error bound BT Error SPA Error bound SPA
2 7.6387e-02 9.3245e-02 1.0852e-01 1.2293e-01
4 8.5160e-03 1.2180e-02 8.6050e-03 1.2185e-02
8 5.1560e-03 9.6638e-03 5.6720e-03 9.7072e-03
16 1.8570e-03 6.6764e-03 2.4970e-03 6.7382e-03
32 6.7050e-04 4.3849e-03 1.4410e-03 4.9106e-03
64 9.9130e-05 2.3491e-03 3.1440e-04 2.6354e-03
Dim. ROM Error BT Error bound BT Error SPA Error bound SPA
2 7.6387e-02 9.3245e-02 1.0852e-01 1.2293e-01
4 8.5160e-03 1.2180e-02 8.6050e-03 1.2185e-02
8 5.1560e-03 9.6638e-03 5.6720e-03 9.7072e-03
16 1.8570e-03 6.6764e-03 2.4970e-03 6.7382e-03
32 6.7050e-04 4.3849e-03 1.4410e-03 4.9106e-03
64 9.9130e-05 2.3491e-03 3.1440e-04 2.6354e-03
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