June  2020, 7(1): 1-33. doi: 10.3934/jcd.2020001

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

1. 

Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany

2. 

Institut für Mathematik, Karl-Franzens-Universität, Heinrichstr. 36/Ⅲ, A-8010 Graz, Austria

3. 

Institut für Mathematik, Brandenburgische Technische Universität Cottbus-Senftenberg, Konrad-Wachsmann-Allee 1, D-03046 Cottbus, Germany

4. 

Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, D-14195 Berlin, Germany

* Corresponding author

Received  May 2018 Revised  June 2019 Published  November 2019

Fund Project: This work was partly supported by the Einstein Center for Mathematics Berlin (ECMath)

We study and compare two different model reduction techniques for bilinear systems, specifically generalized balancing and $ \mathcal{H}_2 $-based model reduction, and apply it to semi-discretized controlled Fokker-Planck and Liouville–von Neumann equations. For this class of transport equations, the control enters the dynamics as an advection term that leads to the bilinear form. A specific feature of the systems is that they are stable, but not asymptotically stable, and we discuss aspects regarding structure and stability preservation in some depth as these aspects are particularly relevant for the equations of interest. Another focus of this article is on the numerical implementation and a thorough comparison of the aforementioned model reduction methods.

Citation: Peter Benner, Tobias Breiten, Carsten Hartmann, Burkhard Schmidt. Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations. Journal of Computational Dynamics, 2020, 7 (1) : 1-33. doi: 10.3934/jcd.2020001
References:
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S. Al-Baiyat and M. Bettayeb, A new model reduction scheme for k-power bilinear systems, Proc. 32nd IEEE Conf. Decis. Control, 32 (1993), 22-27.  doi: 10.1109/CDC.1993.325196.

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P. BennerT. DammM. Redmann and Y. R. Rodriguez Cruz, Positive operators and stable truncation, Linear Algebra Appl., 498 (2016), 74-87.  doi: 10.1016/j.laa.2014.12.005.

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P. BennerT. Damm and Y. R. Rodriguez Cruz, Dual pairs of generalized Lyapunov inequalities and balanced truncation of stochastic linear systems, IEEE Trans. Automat. Control, 62 (2017), 782-791.  doi: 10.1109/TAC.2016.2572881.

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T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numer. Linear Algebra Appl., 15 (2008), 853-871.  doi: 10.1002/nla.603.

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S. LallJ. Marsden and S. Glavaški, A subspace approach to balanced truncation for model reduction of nonlinear control systems, Internat. J. Robust Nonlinear Control, 12 (2002), 519-535.  doi: 10.1002/rnc.657.

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Y. LinL. Bao and Y. Wei, Order reduction of bilinear MIMO dynamical systems using new block Krylov subspaces, Comput. Math. Appl., 58 (2009), 1093-1102.  doi: 10.1016/j.camwa.2009.07.039.

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show all references

References:
[1]

M. I. AhmadU. Baur and P. Benner, Implicit Volterra series interpolation for model reduction of bilinear systems, J. Comput. Appl. Math., 316 (2017), 15-28.  doi: 10.1016/j.cam.2016.09.048.

[2]

S. Al-Baiyat and M. Bettayeb, A new model reduction scheme for k-power bilinear systems, Proc. 32nd IEEE Conf. Decis. Control, 32 (1993), 22-27.  doi: 10.1109/CDC.1993.325196.

[3]

I. Andrianov and P. Saalfrank, Theoretical study of vibration–phonon coupling of H adsorbed on a Si(100) surface, J. Chem. Phys., 124 (2006). doi: 10.1063/1.2161191.

[4]

M. Annunziato and A. Borzi, A Fokker–Planck control framework for multidimensional stochastic processes, J. Comput. Appl. Math., 237 (2013), 487-507.  doi: 10.1016/j.cam.2012.06.019.

[5]

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

[6]

Z. Bai and D. Skoogh, A projection method for model reduction of bilinear dynamical systems, Linear Algebra Appl., 415 (2006), 406-425.  doi: 10.1016/j.laa.2005.04.032.

[7]

U. BaurP. Benner and L. Feng, Model order reduction for linear and nonlinear systems: A system-theoretic perspective, Arch. Comput. Methods Eng., 21 (2014), 331-358.  doi: 10.1007/s11831-014-9111-2.

[8]

S. Becker and C. Hartmann, Infinite-dimensional bilinear and stochastic balanced truncation with explicit error bounds, Math. Control Signals Systems, 31 (2019), 1-37.  doi: 10.1007/s00498-019-0234-8.

[9]

P. Benner and T. Breiten, Interpolation-based $\mathcal{H}_2$-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33 (2012), 859-885.  doi: 10.1137/110836742.

[10]

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

[11]

P. BennerT. DammM. Redmann and Y. R. Rodriguez Cruz, Positive operators and stable truncation, Linear Algebra Appl., 498 (2016), 74-87.  doi: 10.1016/j.laa.2014.12.005.

[12]

P. BennerT. Damm and Y. R. Rodriguez Cruz, Dual pairs of generalized Lyapunov inequalities and balanced truncation of stochastic linear systems, IEEE Trans. Automat. Control, 62 (2017), 782-791.  doi: 10.1109/TAC.2016.2572881.

[13]

P. BennerS. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531.  doi: 10.1137/130932715.

[14]

P. BennerP. Kürschner and J. Saak, Efficient handling of complex shift parameters in the low-rank cholesky factor ADI method, Numer. Algorithms, 62 (2013), 225-251.  doi: 10.1007/s11075-012-9569-7.

[15]

N. Berglund, Kramers' law: Validity, derivations and generalisations, Markov Process. Related Fields, 19 (2013), 459-490. 

[16]

C. BoessA. LawlessN. Nichols and A. Bunse-Gerstner, State estimation using model order reduction for unstable systems, Computers & Fluids, 46 (2011), 155-160.  doi: 10.1016/j.compfluid.2010.11.033.

[17]

T. Breiten and T. Damm, Krylov subspace methods for model order reduction of bilinear control systems, Systems Control Lett., 59 (2010), 443-450.  doi: 10.1016/j.sysconle.2010.06.003.

[18]

T. BreitenK. Kunisch and L. Pfeiffer, Control strategies for the Fokker-Planck equation, ESAIM Control Optim. Calc. Var., 24 (2018), 741-763.  doi: 10.1051/cocv/2017046.

[19]

H.-P. BreuerW. Huber and F. Petruccione, Stochastic wave-function method versus density matrix: A numerical comparison, Comput. Phys. Comm., 104 (1997), 46-58.  doi: 10.1016/S0010-4655(97)00050-7.

[20] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, New York, 2002.  doi: 10.1093/acprof:oso/9780199213900.001.0001.
[21]

M. Condon and R. Ivanov, Empirical balanced truncation of nonlinear systems, J. Nonlinear Sci., 14 (2004), 405-414.  doi: 10.1007/s00332-004-0617-5.

[22]

M. Condon and R. Ivanov, Nonlinear systems – Algebraic Gramians and model reduction, COMPEL, 24 (2005), 202-219.  doi: 10.1108/03321640510571147.

[23]

T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Numer. Linear Algebra Appl., 15 (2008), 853-871.  doi: 10.1002/nla.603.

[24]

N. de Souza, Pulling on single molecules, Nature Meth., 9 (2012), 873-877.  doi: 10.1038/nmeth.2149.

[25]

G. Flagg, Interpolation Methods for the Model Reduction of Bilinear Systems, Ph.D thesis, Virginia Tech, 2012.

[26]

G. Flagg and S. Gugercin, Multipoint volterra series interpolation and $\mathcal{H}_2$ optimal model reduction of bilinear systems, SIAM J. Matrix Anal. Appl., 36 (2015), 549-579.  doi: 10.1137/130947830.

[27]

V. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J. Control Optim., 30 (1992), 1228-1249.  doi: 10.1137/0330065.

[28]

G. Grammel, Averaging of singularly perturbed systems, Nonlinear Anal., 28 (1997), 1851-1865.  doi: 10.1016/S0362-546X(95)00243-O.

[29]

W. Gray and J. P. Mesko, Observability functions for linear and nonlinear systems, Systems Control Lett., 38 (1999), 99-113.  doi: 10.1016/S0167-6911(99)00051-1.

[30]

S. GugercinA. Antoulas and S. Beattie, $\mathcal{H}_2$ model reduction for large-scale dynamical systems, SIAM J. Matrix Anal. Appl., 30 (2008), 609-638.  doi: 10.1137/060666123.

[31]

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

[32]

C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing, J. Stat. Mech. Theor. Exp., 2012 (2012). doi: 10.1088/1742-5468/2012/11/P11004.

[33]

C. HartmannV. Vulcanov and C. Schütte, Balanced truncation of linear second-order systems: A Hamiltonian approach, Multiscale Model. Simul., 8 (2010), 1348-1367.  doi: 10.1137/080732717.

[34]

G. Hummer and A. Szabo, Free energy profiles from single-molecule pulling experiments, Proc. Natl. Acad. Sci. USA, 107 (2010), 21441-21446.  doi: 10.1073/pnas.1015661107.

[35]

A. Isidori, Direct construction of minimal bilinear realizations from nonlinear input-output maps, IEEE Trans. Automatic Control, 18 (1973), 626-631.  doi: 10.1109/tac.1973.1100424.

[36]

S. LallJ. Marsden and S. Glavaški, A subspace approach to balanced truncation for model reduction of nonlinear control systems, Internat. J. Robust Nonlinear Control, 12 (2002), 519-535.  doi: 10.1002/rnc.657.

[37]

J. C. LatorreP. MetznerC. Hartmann and C. Schütte, A structure-preserving numerical discretization of reversible diffusions, Commun. Math. Sci., 9 (2011), 1051-1072.  doi: 10.4310/CMS.2011.v9.n4.a6.

[38]

A. S. LawlessN. K. NicholsC. Boess and A. Bunse-Gerstner, Using model reduction methods within incremental four-dimensional variational data assimilation, Monthly Weather Review, 136 (2008), 1511-1522.  doi: 10.1175/2007MWR2103.1.

[39]

C. Le Bris, Y. Maday and G. Turinici, Towards efficient numerical approaches for quantum control, in Quantum Control: Mathematical and Cumerical Cchallenges, CRM Proc. Lecture Notes, 33, Amer. Math. Soc., Providence, RI, 2003.

[40]

T. Lelivre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numer., 25 (2016), 681-880.  doi: 10.1017/S0962492916000039.

[41]

J.-R. Li and J. White, Low-rank solution of Lyapunov equations, SIAM Rev., 46 (2004), 693-713.  doi: 10.1137/S0036144504443389.

[42]

Y. LinL. Bao and Y. Wei, Order reduction of bilinear MIMO dynamical systems using new block Krylov subspaces, Comput. Math. Appl., 58 (2009), 1093-1102.  doi: 10.1016/j.camwa.2009.07.039.

[43]

G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119-130.  doi: 10.1007/BF01608499.

[44]

L. Meier and D. Luenberger, Approximation of linear constant systems, IEEE Transactions on Automatic Control, 12 (1967), 585-588.  doi: 10.1109/TAC.1967.1098680.

[45]

W. E. Moerner, Single-molecule spectroscopy, imaging, and photocontrol: Foundations for super-resolution microscopy (Nobel Lecture), Rev. Mod. Phys., 87 (2015), 1183-1212.  doi: 10.1002/anie.201501949.

[46]

M. Mohammadi, Analysis of discretization schemes for Fokker-Planck equations and related optimality systems, Ph.D thesis, Universität Würzburg, 2015.

[47]

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

[48]

R. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences, 89, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0977-5.

[49]

M. PetreczkyR. Wisniewski and J. Leth, Moment matching for bilinear systems with nice selections, IFAC-PapersOnLine, 49 (2016), 838-843.  doi: 10.1016/j.ifacol.2016.10.270.

[50]

J. Phillips, Projection-based approaches for model reduction of weakly nonlinear, time-varying systems, IEEE T. Comput. Aided. D., 22 (2003), 171-187.  doi: 10.1109/TCAD.2002.806605.

[51]

M. Redmann and P. Benner, Singular perturbation approximation for linear systems with Lévy noise, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500338.

[52]

L. Rey-Bellet, Open classical systems, in Open Quantum Systems II: The Markovian Approach, Lecture Notes in Math., 1881, Springer, Berlin, 2006, 41–78. doi: 10.1007/3-540-33966-3_2.

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Figure 1.  Periodically perturbed quadruple-well potential (36) used in our FPE example
Figure 2.  Eigenvectors of the discretization matrix $ A $ for our FPE example, associated with the first four right eigenvalues $ \lambda $ for $ \beta = 4 $. Note that the eigenvector for $ \lambda = 0 $ (upper left panel) corresponds to the canonical density (35)
Figure 3.  $ {\mathcal H}_2 $ error versus reduced dimension for the FPE example for $ \beta = 4 $. Comparison of BT method, SP method, and H2 method. Values that are not shown are those for which the computed error has dropped below machine precision (see Sec. 6.4)
Figure 4.  Time evolution of observables for the FPE example for $ \beta = 4 $ and for the control field given by Eq. (41) with $ t_0 = 150 $, $ \tau = 100 $, and $ a = 0.5 $: populations of the four quadrants of the $ x_1 $-$ x_2 $ plane for full ($ n = 2401 $) versus reduced dimensionality. From left to right: BT method, SP method, and H2 method
Figure 5.  Spectrum of the $ A $ matrix for the LvNE example for full versus reduced dimensionality. For relaxation rate $ \Gamma = 0.1 $ and temperature $ \Theta = 0.1 $. From left to right: BT method, SP method, and H2 method
Figure 6.  $ {\mathcal H}_2 $ error versus reduced dimensionality $ d $ for the LvNE example. Simulation results for which the error has dropped below machine precision are considered numerical artifact and thus are not shown. Left: For various values of the relaxation rate $ \Gamma $ (for constant temperature, $ \Theta = 0.1 $). Right: For various values of the temperature $ \Theta $ (for constant relaxation, $ \Gamma = 0.1 $)
Figure 7.  Time evolution of observables for the LvNE example for relaxation rate $ \Gamma = 0.1 $ and temperature $ \Theta = 0.1 $. The control field is given by Eq. (41) with $ a = 3 $, $ t_0 = 15 $, and $ \tau = 10 $. Populations of states localized in the left well, in the right well, and delocalized states over the barrier, for full ($ n = 441 $) versus reduced dimensionality. From left to right: BT method, SP method, and H2 method
Table 1.  Lowest twelve eigenvalues (in magnitude) of the discretization matrix $ A $ of the FPE example for inverse temperature $ \beta = 4 $ showing three clusters of four members each. Comparison of full versus reduced dynamics using the BT and H2 method. Results for the SP method (not shown) are very close to those for the BT method. For all practical purposes, the reduced systems for $ d>100 $ are virtually indistinguishable from the full-rank system
BT method H2 method
full $ d=100 $ $ d=50 $ $ d=25 $ $ d=100 $ $ d=50 $ $ d=25 $
-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000
-0.0037 -0.0037 -0.0037 -0.0037 -0.0037 -0.0037 -0.0037
-0.0073 -0.0073 -0.0073 -0.0074 -0.0073 -0.0073 -0.0074
-0.0118 -0.0118 -0.0118 -0.0118 -0.0118 -0.0118 -0.0118
-0.3266 -0.3265 -0.3260 -0.3264 -0.3265 -0.3255 -0.3263
-0.3303 -0.3297 -0.3294 -0.3504 -0.3298 -0.3298 -0.3629
-0.3358 -0.3353 -0.3423 -0.5455 -0.3349 -0.3432 -0.5450
-0.3447 -0.3447 -0.3432 -0.5582 -0.3445 -0.3432 -0.6058
-0.5421 -0.5422 -0.5434 -0.6083 -0.5412 -0.5435 -0.6336
-0.5453 -0.5455 -0.5606 -0.6487 -0.5450 -0.5622 -0.6676
-0.5666 -0.5657 -0.5867 -0.7683 -0.5663 -0.5888 -0.7791
-0.5948 -0.5951 -0.6107 -0.8003 -0.5951 -0.6192 -0.8052
BT method H2 method
full $ d=100 $ $ d=50 $ $ d=25 $ $ d=100 $ $ d=50 $ $ d=25 $
-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000
-0.0037 -0.0037 -0.0037 -0.0037 -0.0037 -0.0037 -0.0037
-0.0073 -0.0073 -0.0073 -0.0074 -0.0073 -0.0073 -0.0074
-0.0118 -0.0118 -0.0118 -0.0118 -0.0118 -0.0118 -0.0118
-0.3266 -0.3265 -0.3260 -0.3264 -0.3265 -0.3255 -0.3263
-0.3303 -0.3297 -0.3294 -0.3504 -0.3298 -0.3298 -0.3629
-0.3358 -0.3353 -0.3423 -0.5455 -0.3349 -0.3432 -0.5450
-0.3447 -0.3447 -0.3432 -0.5582 -0.3445 -0.3432 -0.6058
-0.5421 -0.5422 -0.5434 -0.6083 -0.5412 -0.5435 -0.6336
-0.5453 -0.5455 -0.5606 -0.6487 -0.5450 -0.5622 -0.6676
-0.5666 -0.5657 -0.5867 -0.7683 -0.5663 -0.5888 -0.7791
-0.5948 -0.5951 -0.6107 -0.8003 -0.5951 -0.6192 -0.8052
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