American Institute of Mathematical Sciences

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

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

Peter Benner 1, , Tobias Breiten 2, , Carsten Hartmann 3,, and  Burkhard Schmidt 4,

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.

Keywords: Bilinear systems, $ \mathcal{H}_{2} $ model reduction, balanced truncation, generalized Lyapunov equations, generalized Sylvester equations, Fokker-Planck equation, Liouville–von Neumann equations.
Mathematics Subject Classification: Primary: 93A15, 93B40; Secondary: 93C10, 93C70.
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:
[1]

M. I. Ahmad, U. 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]

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.  Google Scholar

[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.  Google Scholar

[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

[5]

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

[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.  Google Scholar

[7]

U. Baur, P. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Benner, T. Damm, M. 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.  Google Scholar

[12]

P. Benner, T. 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.  Google Scholar

[13]

P. Benner, S. 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. Benner, P. 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

[15]

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

[16]

C. Boess, A. Lawless, N. 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.  Google Scholar

[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.  Google Scholar

[18]

T. Breiten, K. 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

[19]

H.-P. Breuer, W. 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

[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.  Google Scholar
[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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. Gugercin, A. 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. Hartmann, B. 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. Hartmann, V. 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. Lall, J. 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. Latorre, P. Metzner, C. 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. Lawless, N. K. Nichols, C. 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. Lin, L. 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]

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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

[49]

M. Petreczky, R. 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.  Google Scholar

[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.  Google Scholar

[51]

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

[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.  Google Scholar

[53] W. Rugh, Nonlinear System Theory, Johns Hopkins University Press, Baltimore, MD, 1981.   Google Scholar
[54]

B. Schäfer-Bung, C. Hartmann, B. Schmidt and C. Schütte, Dimension reduction by balanced truncation: Application to light-induced control of open quantum systems, J Chem. Phys., 135 (2011). doi: 10.1063/1.3605243.  Google Scholar

[55]

B. Schmidt and C. Hartmann, Wavepacket: A MATLAB package for numerical quantum dynamics. Ⅱ: Open quantum systems and optimal control, Comput. Phys. Commun., 228 (2018), 229-244.  doi: 10.1016/j.cpc.2018.02.022.  Google Scholar

[56]

B. Schmidt and U. Lorenz, Wavepacket: A MATLAB package for numerical quantum dynamics. Ⅰ: Closed quantum systems and discrete variable representations, Comput. Phys. Commun., 213 (2017), 223-234.  doi: 10.1016/j.cpc.2016.12.007.  Google Scholar

[57]

T. Siu and M. Schetzen, Convergence of Volterra series representation and BIBO stability of bilinear systems, Internat. J. Systems Sci., 22 (1991), 2679-2684.  doi: 10.1080/00207729108910824.  Google Scholar

[58]

A. K. Tiwari, K. B. Møller and N. E. Henriksen, Selective bond breakage within the HOD molecule using optimized femtosecond ultraviolet laser pulses, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.065402.  Google Scholar

[59]

A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions, SIAM J. Control Optim., 35 (1997), 1-28.  doi: 10.1137/S0363012994264207.  Google Scholar

[60]

E. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), 87-90.  doi: 10.1016/0893-9659(88)90183-8.  Google Scholar

[61]

F. Watbled, On singular perturbations for differential inclusions on the infinite interval, J. Math. Anal. Appl., 310 (2005), 362–378. doi: 10.1016/j.jmaa.2005.01.067.  Google Scholar

[62]

U. Weiss, Quantum Dissipative Systems, Series in Modern Condensed Matter Physics, 10, World Scientific, Singapore, 1999. doi: 10.1142/9789812817877.  Google Scholar

[63]

D. Wilson, Optimum solution of model-reduction problem, Proceedings of the Institution of Electrical Engineers, 117 (1970), 1161-1165.  doi: 10.1049/piee.1970.0227.  Google Scholar

[64]

A. Zewail, Femtochemistry: Chemical reaction dynamics and their control, in Advances in Chemical Physics: Chemical Reactions and their Control on the Femtosecond Time Scale, 101, John Wiley & Sons, Hoboken, NJ, 1997,103–108. doi: 10.1002/9780470141601.ch1.  Google Scholar

[65]

L. Zhang and J. Lam, On $H_2$ model reduction of bilinear systems, Automatica J. IFAC, 38 (2002), 205-216.  doi: 10.1016/S0005-1098(01)00204-7.  Google Scholar

[66]

W. Zhang, H. Wang, C. Hartmann, M. Weber and C. Schütte, Applications of the cross-entropy method to importance sampling and optimal control of diffusions, SIAM J. Sci. Comput., 36 (2014), A2654–A2672. doi: 10.1137/14096493X.  Google Scholar

show all references

References:
[1]

M. I. Ahmad, U. 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]

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.  Google Scholar

[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.  Google Scholar

[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

[5]

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

[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.  Google Scholar

[7]

U. Baur, P. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Benner, T. Damm, M. 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.  Google Scholar

[12]

P. Benner, T. 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.  Google Scholar

[13]

P. Benner, S. 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. Benner, P. 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

[15]

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

[16]

C. Boess, A. Lawless, N. 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.  Google Scholar

[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.  Google Scholar

[18]

T. Breiten, K. 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

[19]

H.-P. Breuer, W. 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

[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.  Google Scholar
[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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. Gugercin, A. 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. Hartmann, B. 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. Hartmann, V. 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. Lall, J. 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. Latorre, P. Metzner, C. 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. Lawless, N. K. Nichols, C. 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. Lin, L. 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]

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.  Google Scholar

[46]

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

[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.  Google Scholar

[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.  Google Scholar

[49]

M. Petreczky, R. 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.  Google Scholar

[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.  Google Scholar

[51]

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

[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.  Google Scholar

[53] W. Rugh, Nonlinear System Theory, Johns Hopkins University Press, Baltimore, MD, 1981.   Google Scholar
[54]

B. Schäfer-Bung, C. Hartmann, B. Schmidt and C. Schütte, Dimension reduction by balanced truncation: Application to light-induced control of open quantum systems, J Chem. Phys., 135 (2011). doi: 10.1063/1.3605243.  Google Scholar

[55]

B. Schmidt and C. Hartmann, Wavepacket: A MATLAB package for numerical quantum dynamics. Ⅱ: Open quantum systems and optimal control, Comput. Phys. Commun., 228 (2018), 229-244.  doi: 10.1016/j.cpc.2018.02.022.  Google Scholar

[56]

B. Schmidt and U. Lorenz, Wavepacket: A MATLAB package for numerical quantum dynamics. Ⅰ: Closed quantum systems and discrete variable representations, Comput. Phys. Commun., 213 (2017), 223-234.  doi: 10.1016/j.cpc.2016.12.007.  Google Scholar

[57]

T. Siu and M. Schetzen, Convergence of Volterra series representation and BIBO stability of bilinear systems, Internat. J. Systems Sci., 22 (1991), 2679-2684.  doi: 10.1080/00207729108910824.  Google Scholar

[58]

A. K. Tiwari, K. B. Møller and N. E. Henriksen, Selective bond breakage within the HOD molecule using optimized femtosecond ultraviolet laser pulses, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.065402.  Google Scholar

[59]

A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fast motions, SIAM J. Control Optim., 35 (1997), 1-28.  doi: 10.1137/S0363012994264207.  Google Scholar

[60]

E. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), 87-90.  doi: 10.1016/0893-9659(88)90183-8.  Google Scholar

[61]

F. Watbled, On singular perturbations for differential inclusions on the infinite interval, J. Math. Anal. Appl., 310 (2005), 362–378. doi: 10.1016/j.jmaa.2005.01.067.  Google Scholar

[62]

U. Weiss, Quantum Dissipative Systems, Series in Modern Condensed Matter Physics, 10, World Scientific, Singapore, 1999. doi: 10.1142/9789812817877.  Google Scholar

[63]

D. Wilson, Optimum solution of model-reduction problem, Proceedings of the Institution of Electrical Engineers, 117 (1970), 1161-1165.  doi: 10.1049/piee.1970.0227.  Google Scholar

[64]

A. Zewail, Femtochemistry: Chemical reaction dynamics and their control, in Advances in Chemical Physics: Chemical Reactions and their Control on the Femtosecond Time Scale, 101, John Wiley & Sons, Hoboken, NJ, 1997,103–108. doi: 10.1002/9780470141601.ch1.  Google Scholar

[65]

L. Zhang and J. Lam, On $H_2$ model reduction of bilinear systems, Automatica J. IFAC, 38 (2002), 205-216.  doi: 10.1016/S0005-1098(01)00204-7.  Google Scholar

[66]

W. Zhang, H. Wang, C. Hartmann, M. Weber and C. Schütte, Applications of the cross-entropy method to importance sampling and optimal control of diffusions, SIAM J. Sci. Comput., 36 (2014), A2654–A2672. doi: 10.1137/14096493X.  Google Scholar

Figure 1.  Periodically perturbed quadruple-well potential (36) used in our FPE example
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056
[2]

Yang Liu, Chunyou Sun. Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4383-4408. doi: 10.3934/dcdss.2021124
[3]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
[4]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[5]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[6]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[7]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063
[8]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035
[9]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094
[10]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368
[11]

James Montaldi, Amna Shaddad. Generalized point vortex dynamics on $ \mathbb{CP} ^2 $. Journal of Geometric Mechanics, 2019, 11 (4) : 601-619. doi: 10.3934/jgm.2019030
[12]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[13]

Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013
[14]

Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131
[15]

Qiang Tu. A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5397-5407. doi: 10.3934/dcds.2021081
[16]

Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007
[17]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445
[18]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[19]

Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015
[20]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093

 Impact Factor: 

Tools

Metrics

  • PDF downloads (302)
  • HTML views (498)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy