September  2016, 6(3): 363-389. doi: 10.3934/mcrf.2016007

A sparse Markov chain approximation of LQ-type stochastic control problems

1. 

School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, United Kingdom

2. 

Institut für Mathematik, Brandenburgische Technische Universitat Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany

Received  March 2015 Revised  October 2015 Published  August 2016

We propose a novel Galerkin discretization scheme for stochastic optimal control problems on an indefinite time horizon. The control problems are linear-quadratic in the controls, but possibly nonlinear in the state variables, and the discretization is based on the fact that problems of this kind admit a dual formulation in terms of linear boundary value problems. We show that the discretized linear problem is dual to a Markov decision problem, prove an $L^{2}$ error bound for the general scheme and discuss the sparse discretization using a basis of so-called committor functions as a special case; the latter is particularly suited when the dynamics are metastable, e.g., when controlling biomolecular systems. We illustrate the method with several numerical examples, one being the optimal control of Alanine dipeptide to its helical conformation.
Citation: Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
References:
[1]

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments,, Comput. Methods Appl. M., 139 (1995), 3. doi: 10.1016/S0045-7825(96)01078-X. Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar

[3]

O. Bokanowski, J. Garcke, M. Griebel and I. Klompmaker, An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi-Bellman equations,, J. Sci. Comput., 55 (2013), 575. doi: 10.1007/s10915-012-9648-x. Google Scholar

[4]

M. Boulbrachene and M. Haiour, The finite element approximation of Hamilton-Jacobi-Bellman equations,, Comput. Math. Appl., 41 (2001), 993. doi: 10.1016/S0898-1221(00)00334-5. Google Scholar

[5]

A. Bovier, Methods of Contemporary Statistical Mechanics,, Metastability, (2009). doi: 10.1007/978-3-540-92796-9. Google Scholar

[6]

D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics,, Cambridge University Press, (2007). doi: 10.1088/0957-0233/13/9/704. Google Scholar

[7]

G. Bussi, D. Donadio and M. Parrinello, Canonical sampling through velocity rescaling,, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2408420. Google Scholar

[8]

E. Carlini, M. Falcone and R. Ferretti, An efficient algorithm for Hamilton-Jacobi equations in high dimension,, Comput. Visual. Sci., 7 (2004), 15. doi: 10.1007/s00791-004-0124-5. Google Scholar

[9]

T. Cecil, J. Qian and S. Osher, Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions,, J. Comput. Phys., 196 (2004), 327. doi: 10.1016/j.jcp.2003.11.010. Google Scholar

[10]

P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games,, Math. Control Signals Systems, 9 (1996), 303. doi: 10.1007/BF01211853. Google Scholar

[11]

T. Darden, D. York and L. Pedersen, Particle mesh Ewald: An $N\cdot\log(N)$ method for Ewald sums in large systems},, J. Chem. Phys., 98 (1993), 10089. doi: 10.1063/1.464397. Google Scholar

[12]

N. Djurdjevac, M. Sarich and C. Schütte, Estimating the eigenvalue error of Markov state models,, Multiscale Model. Simul., 10 (2012), 61. doi: 10.1137/100798910. Google Scholar

[13]

P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions,, Multiscale Model. Simul., 10 (2012), 1. doi: 10.1137/110842545. Google Scholar

[14]

A. Faradjian and R. Elber, Computing time scales from reaction coordinates by milestoning,, J. Chem. Phys., 120 (2004), 10880. doi: 10.1063/1.1738640. Google Scholar

[15]

W. Fleming, Exit probabilities and optimal stochastic control,, Appl. Math. Optim., 4 (1977), 329. doi: 10.1007/BF01442148. Google Scholar

[16]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (2006). Google Scholar

[17]

I. Gikhman and A. Skorokhod, The Theory of Stochastic Processes II,, Springer-Verlag, (2004). Google Scholar

[18]

C. Hartmann, R. Banisch, M. Sarich, T. Badowski and C. Schütte, Characterization of rare events in molecular dynamics,, Entropy, 16 (2014), 350. doi: 10.3390/e16010350. Google Scholar

[19]

C. Hartmann, J. Latorre, G. Pavliotis and W. Zhang, Optimal control of multiscale systems using reduced-order models,, J. Comput. Dynamics, 1 (2014), 279. doi: 10.3934/jcd.2014.1.279. Google Scholar

[20]

C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing,, J. Stat. Mech. Theor. Exp., 11 (2012). Google Scholar

[21]

B. Hess, H. Bekker, H. Berendsen and J. Fraaije, LINCS: a linear constraint solver for molecular simulations,, J. Comp. Chem., 18 (1997), 1463. Google Scholar

[22]

R. H. Hoppe, Multi-grid methods for Hamilton-Jacobi-Bellman equations,, Numer. Math., 49 (1986), 239. doi: 10.1007/BF01389627. Google Scholar

[23]

V. Hornak, R. Abel, A. Okur, B. Strockbine, A. Roitberg and C. Simmerling, Comparison of multiple amber force fields and development of improved protein backbone parameters,, Proteins, 65 (2006), 712. doi: 10.1002/prot.21123. Google Scholar

[24]

C.-S. Huang, S. Wang, C. Chen and Z.-C. Li, A radial basis collocation method for Hamilton-Jacobi-Bellman equations,, Automatica, 42 (2006), 2201. doi: 10.1016/j.automatica.2006.07.013. Google Scholar

[25]

H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions,, Physica, 7 (1940), 284. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar

[26]

H. J. Kushner, A survey of some applications of probability and stochastic control theory to finite difference methods for degenerate elliptic and parabolic equations,, SIAM Review, 18 (1976), 545. doi: 10.1137/1018112. Google Scholar

[27]

H. J. Kushner, Numerical methods for stochastic control problems in finance,, in Mathematics of Derivative Securities, 15 (1997), 504. Google Scholar

[28]

H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer Verlag, (1992). doi: 10.1007/978-1-4684-0441-8. Google Scholar

[29]

J. Mattingly, A. Stuart and D. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise,, Stochastic Process. Appl., 101 (2002), 185. doi: 10.1016/S0304-4149(02)00150-3. Google Scholar

[30]

J. Menaldi, Some estimates for finite difference approximations,, SIAM J. Control Optim., 27 (1989), 579. doi: 10.1137/0327031. Google Scholar

[31]

R. H. Momeya and Z. B. Salah, The minimal entropy martingale measure (memm) for a markov-modulated exponential lévy model,, Asia-Pacific Financial Markets, 19 (2012), 63. doi: 10.1007/s10690-011-9142-8. Google Scholar

[32]

B. Øksendal, Stochastic Differential Equations: An Introduction With Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar

[33]

M. Peletier, G. Savaré and M. Veneroni, Chemical reactions as $\Gamma$-limit of diffusion,, SIAM Review, 54 (2012), 327. doi: 10.1137/110858781. Google Scholar

[34]

H. Rabitz, R. de Vivie-Riedle, M. Motzkus and K. Kompa, Whither the future of controlling quantum phenomena?,, Science, 288 (2000), 824. doi: 10.1126/science.288.5467.824. Google Scholar

[35]

M. Sarich, Projected Transfer Operators,, PhD thesis, (2011). Google Scholar

[36]

M. Sarich, F. Noé and C. Schütte, On the approximation quality of Markov state models,, Multiscale Model. Simul., 8 (2010), 1154. doi: 10.1137/090764049. Google Scholar

[37]

M. Sarich and C. Schütte, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, AMS, (2013). Google Scholar

[38]

C. Schütte, F. Noé, J. Lu, M. Sarich and E. Vanden-Eijnden, Markov state models based on milestoning,, J. Chem. Phys., 134 (2011). Google Scholar

[39]

C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using Markov state models,, Math. Program. (Series B), 134 (2012), 259. doi: 10.1007/s10107-012-0547-6. Google Scholar

[40]

S. Sheu, Stochastic control and exit probabilities of jump processes,, J. Control Optim., 23 (1985), 306. doi: 10.1137/0323022. Google Scholar

[41]

Q. Song, Convergence of Markov chain approximation on generalized HJB equation and its applications,, Automatica, 44 (2008), 761. doi: 10.1016/j.automatica.2007.07.014. Google Scholar

[42]

Q. Song, G. Yin and Z. Zhang, Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions,, Automatica, 42 (2006), 1147. doi: 10.1016/j.automatica.2006.03.016. Google Scholar

[43]

H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry,, Physica Scripta, 2004 (2004), 132. doi: 10.1238/Physica.Topical.110a00132. Google Scholar

[44]

A. Steinbrecher, Optimal control of robot guided laser material treatment,, in Progress in Industrial Mathematics at ECMI 2008 (eds. A. Fitt, 15 (2010), 505. doi: 10.1007/978-3-642-12110-4_79. Google Scholar

[45]

M. Sun, Domain decomposition algorithms for solving Hamilton-Jacobi-Bellman equations,, Numer. Func. Anal. Optim., 14 (1993), 145. doi: 10.1080/01630569308816513. Google Scholar

[46]

D. Van Der Spoel, E. Lindahl, B. Hess, G. Groenhof, A. Mark and H. C. Berendsen, Gromacs: Fast, flexible, and free,, J. Comp. Chem., 26 (2005), 1701. Google Scholar

[47]

E. Vanden-Eijnden, Transition path theory,, Lect. Notes Phys., 703 (2006), 439. Google Scholar

[48]

S. Wang, L. Jennings and K. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method,, J. Global Optim., 27 (2003), 177. doi: 10.1023/A:1024980623095. Google Scholar

[49]

H. Wendland, Meshless galerkin methods using radial basis functions,, Math. Comput., 68 (1999), 1521. doi: 10.1090/S0025-5718-99-01102-3. Google Scholar

[50]

M. Zhong and E. Todorov, Moving least-squares approximations for linearly-solvable stochastic optimal control problems,, J. Control Theory Appl., 9 (2011), 451. doi: 10.1007/s11768-011-0275-0. Google Scholar

show all references

References:
[1]

T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments,, Comput. Methods Appl. M., 139 (1995), 3. doi: 10.1016/S0045-7825(96)01078-X. Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979). Google Scholar

[3]

O. Bokanowski, J. Garcke, M. Griebel and I. Klompmaker, An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi-Bellman equations,, J. Sci. Comput., 55 (2013), 575. doi: 10.1007/s10915-012-9648-x. Google Scholar

[4]

M. Boulbrachene and M. Haiour, The finite element approximation of Hamilton-Jacobi-Bellman equations,, Comput. Math. Appl., 41 (2001), 993. doi: 10.1016/S0898-1221(00)00334-5. Google Scholar

[5]

A. Bovier, Methods of Contemporary Statistical Mechanics,, Metastability, (2009). doi: 10.1007/978-3-540-92796-9. Google Scholar

[6]

D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics,, Cambridge University Press, (2007). doi: 10.1088/0957-0233/13/9/704. Google Scholar

[7]

G. Bussi, D. Donadio and M. Parrinello, Canonical sampling through velocity rescaling,, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2408420. Google Scholar

[8]

E. Carlini, M. Falcone and R. Ferretti, An efficient algorithm for Hamilton-Jacobi equations in high dimension,, Comput. Visual. Sci., 7 (2004), 15. doi: 10.1007/s00791-004-0124-5. Google Scholar

[9]

T. Cecil, J. Qian and S. Osher, Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions,, J. Comput. Phys., 196 (2004), 327. doi: 10.1016/j.jcp.2003.11.010. Google Scholar

[10]

P. Dai Pra, L. Meneghini and W. Runggaldier, Connections between stochastic control and dynamic games,, Math. Control Signals Systems, 9 (1996), 303. doi: 10.1007/BF01211853. Google Scholar

[11]

T. Darden, D. York and L. Pedersen, Particle mesh Ewald: An $N\cdot\log(N)$ method for Ewald sums in large systems},, J. Chem. Phys., 98 (1993), 10089. doi: 10.1063/1.464397. Google Scholar

[12]

N. Djurdjevac, M. Sarich and C. Schütte, Estimating the eigenvalue error of Markov state models,, Multiscale Model. Simul., 10 (2012), 61. doi: 10.1137/100798910. Google Scholar

[13]

P. Dupuis, K. Spiliopoulos and H. Wang, Importance sampling for multiscale diffusions,, Multiscale Model. Simul., 10 (2012), 1. doi: 10.1137/110842545. Google Scholar

[14]

A. Faradjian and R. Elber, Computing time scales from reaction coordinates by milestoning,, J. Chem. Phys., 120 (2004), 10880. doi: 10.1063/1.1738640. Google Scholar

[15]

W. Fleming, Exit probabilities and optimal stochastic control,, Appl. Math. Optim., 4 (1977), 329. doi: 10.1007/BF01442148. Google Scholar

[16]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions,, Springer, (2006). Google Scholar

[17]

I. Gikhman and A. Skorokhod, The Theory of Stochastic Processes II,, Springer-Verlag, (2004). Google Scholar

[18]

C. Hartmann, R. Banisch, M. Sarich, T. Badowski and C. Schütte, Characterization of rare events in molecular dynamics,, Entropy, 16 (2014), 350. doi: 10.3390/e16010350. Google Scholar

[19]

C. Hartmann, J. Latorre, G. Pavliotis and W. Zhang, Optimal control of multiscale systems using reduced-order models,, J. Comput. Dynamics, 1 (2014), 279. doi: 10.3934/jcd.2014.1.279. Google Scholar

[20]

C. Hartmann and C. Schütte, Efficient rare event simulation by optimal nonequilibrium forcing,, J. Stat. Mech. Theor. Exp., 11 (2012). Google Scholar

[21]

B. Hess, H. Bekker, H. Berendsen and J. Fraaije, LINCS: a linear constraint solver for molecular simulations,, J. Comp. Chem., 18 (1997), 1463. Google Scholar

[22]

R. H. Hoppe, Multi-grid methods for Hamilton-Jacobi-Bellman equations,, Numer. Math., 49 (1986), 239. doi: 10.1007/BF01389627. Google Scholar

[23]

V. Hornak, R. Abel, A. Okur, B. Strockbine, A. Roitberg and C. Simmerling, Comparison of multiple amber force fields and development of improved protein backbone parameters,, Proteins, 65 (2006), 712. doi: 10.1002/prot.21123. Google Scholar

[24]

C.-S. Huang, S. Wang, C. Chen and Z.-C. Li, A radial basis collocation method for Hamilton-Jacobi-Bellman equations,, Automatica, 42 (2006), 2201. doi: 10.1016/j.automatica.2006.07.013. Google Scholar

[25]

H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions,, Physica, 7 (1940), 284. doi: 10.1016/S0031-8914(40)90098-2. Google Scholar

[26]

H. J. Kushner, A survey of some applications of probability and stochastic control theory to finite difference methods for degenerate elliptic and parabolic equations,, SIAM Review, 18 (1976), 545. doi: 10.1137/1018112. Google Scholar

[27]

H. J. Kushner, Numerical methods for stochastic control problems in finance,, in Mathematics of Derivative Securities, 15 (1997), 504. Google Scholar

[28]

H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time,, Springer Verlag, (1992). doi: 10.1007/978-1-4684-0441-8. Google Scholar

[29]

J. Mattingly, A. Stuart and D. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise,, Stochastic Process. Appl., 101 (2002), 185. doi: 10.1016/S0304-4149(02)00150-3. Google Scholar

[30]

J. Menaldi, Some estimates for finite difference approximations,, SIAM J. Control Optim., 27 (1989), 579. doi: 10.1137/0327031. Google Scholar

[31]

R. H. Momeya and Z. B. Salah, The minimal entropy martingale measure (memm) for a markov-modulated exponential lévy model,, Asia-Pacific Financial Markets, 19 (2012), 63. doi: 10.1007/s10690-011-9142-8. Google Scholar

[32]

B. Øksendal, Stochastic Differential Equations: An Introduction With Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6. Google Scholar

[33]

M. Peletier, G. Savaré and M. Veneroni, Chemical reactions as $\Gamma$-limit of diffusion,, SIAM Review, 54 (2012), 327. doi: 10.1137/110858781. Google Scholar

[34]

H. Rabitz, R. de Vivie-Riedle, M. Motzkus and K. Kompa, Whither the future of controlling quantum phenomena?,, Science, 288 (2000), 824. doi: 10.1126/science.288.5467.824. Google Scholar

[35]

M. Sarich, Projected Transfer Operators,, PhD thesis, (2011). Google Scholar

[36]

M. Sarich, F. Noé and C. Schütte, On the approximation quality of Markov state models,, Multiscale Model. Simul., 8 (2010), 1154. doi: 10.1137/090764049. Google Scholar

[37]

M. Sarich and C. Schütte, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches,, AMS, (2013). Google Scholar

[38]

C. Schütte, F. Noé, J. Lu, M. Sarich and E. Vanden-Eijnden, Markov state models based on milestoning,, J. Chem. Phys., 134 (2011). Google Scholar

[39]

C. Schütte, S. Winkelmann and C. Hartmann, Optimal control of molecular dynamics using Markov state models,, Math. Program. (Series B), 134 (2012), 259. doi: 10.1007/s10107-012-0547-6. Google Scholar

[40]

S. Sheu, Stochastic control and exit probabilities of jump processes,, J. Control Optim., 23 (1985), 306. doi: 10.1137/0323022. Google Scholar

[41]

Q. Song, Convergence of Markov chain approximation on generalized HJB equation and its applications,, Automatica, 44 (2008), 761. doi: 10.1016/j.automatica.2007.07.014. Google Scholar

[42]

Q. Song, G. Yin and Z. Zhang, Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions,, Automatica, 42 (2006), 1147. doi: 10.1016/j.automatica.2006.03.016. Google Scholar

[43]

H. Stapelfeldt, Laser aligned molecules: Applications in physics and chemistry,, Physica Scripta, 2004 (2004), 132. doi: 10.1238/Physica.Topical.110a00132. Google Scholar

[44]

A. Steinbrecher, Optimal control of robot guided laser material treatment,, in Progress in Industrial Mathematics at ECMI 2008 (eds. A. Fitt, 15 (2010), 505. doi: 10.1007/978-3-642-12110-4_79. Google Scholar

[45]

M. Sun, Domain decomposition algorithms for solving Hamilton-Jacobi-Bellman equations,, Numer. Func. Anal. Optim., 14 (1993), 145. doi: 10.1080/01630569308816513. Google Scholar

[46]

D. Van Der Spoel, E. Lindahl, B. Hess, G. Groenhof, A. Mark and H. C. Berendsen, Gromacs: Fast, flexible, and free,, J. Comp. Chem., 26 (2005), 1701. Google Scholar

[47]

E. Vanden-Eijnden, Transition path theory,, Lect. Notes Phys., 703 (2006), 439. Google Scholar

[48]

S. Wang, L. Jennings and K. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method,, J. Global Optim., 27 (2003), 177. doi: 10.1023/A:1024980623095. Google Scholar

[49]

H. Wendland, Meshless galerkin methods using radial basis functions,, Math. Comput., 68 (1999), 1521. doi: 10.1090/S0025-5718-99-01102-3. Google Scholar

[50]

M. Zhong and E. Todorov, Moving least-squares approximations for linearly-solvable stochastic optimal control problems,, J. Control Theory Appl., 9 (2011), 451. doi: 10.1007/s11768-011-0275-0. Google Scholar

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