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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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.  Google Scholar

[2]

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[3]

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[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.  Google Scholar

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[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.  Google Scholar

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[9]

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[10]

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[11]

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[12]

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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[24]

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

[25]

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

[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.  Google Scholar

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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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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