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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

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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