Solving differential Riccati equations: A nonlinear space-time method using tensor trains

Tobias Breiten; Sergey Dolgov; Martin Stoll

doi:10.3934/naco.2020034

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September 2021, 11(3): 407-429. doi: 10.3934/naco.2020034

Solving differential Riccati equations: A nonlinear space-time method using tensor trains

Tobias Breiten ^1,, , Sergey Dolgov ^2, and Martin Stoll ^3,

1.	Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany
2.	University of Bath, Department of Mathematical Sciences, BA2 7AY Bath, United Kingdom
3.	Technische Universität Chemnitz, Department of Mathematics, Scientific Computing Group, 09107 Chemnitz, Germany

^* Corresponding author: T. Breiten

Received December 2019 Revised July 2020 Published September 2021 Early access August 2020

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

Keywords: Optimal Control, Low-rank methods, Riccati equations, Non-linear problems.

Mathematics Subject Classification: Primary: 15A24, 65H10, 15A69, 93A15.

Citation: Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034

References:

[1]	H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations, Systems Control: Foundations Applications, Birkhäuser Verlag, Basel, 2003, URL https://doi.org/10.1007/978-3-0348-8081-7, In Control and Systems Theory. doi: 10.1007/978-3-0348-8081-7. Google Scholar
[2]	N. Ahmed, G. Matthies, L. Tobiska and H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 1747-1756. doi: 10.1016/j.cma.2011.02.003. Google Scholar
[3]	J. Ballani and L. Grasedyck, A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27-43. doi: 10.1002/nla.1818. Google Scholar
[4]	R. Bellman, Dynamic programming, Courier Corporation, 2013. Google Scholar
[5]	P. Benner, Z. Bujanović, P. Kürschner and J. Saak, RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations, Numerische Mathematik, 138 (2018), 301-330. doi: 10.1007/s00211-017-0907-5. Google Scholar
[6]	P. Benner, Z. Bujanović, P. Kürschner and J. Saak, A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems, SIAM Journal on Scientific Computing, 42 (2020), A957–A996. doi: 10.1137/18M1220960. Google Scholar
[7]	P. Benner, S. Dolgov, A. Onwunta and M. Stoll, Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304 (2016), 26-54. doi: 10.1016/j.cma.2016.02.004. Google Scholar
[8]	P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, in in Proceedings of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar
[9]	P. Benner and H. Mena, Rosenbrock methods for solving Riccati differential equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2956. doi: 10.1109/TAC.2013.2258495. Google Scholar
[10]	P. Benner, A. Onwunta and M. Stoll, Low rank solution of unsteady diffusion equations with stochastic coefficients, SIAM Journal on Uncertainty Quantification, 3 (2015), 622-649. doi: 10.1137/130937251. Google Scholar
[11]	P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52. doi: 10.1002/gamm.201310003. Google Scholar
[12]	A. Brooks and T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8. Google Scholar
[13]	S. Dolgov and B. Khoromskij, Two-level QTT-Tucker format for optimized tensor calculus, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 593-623. doi: 10.1137/120882597. Google Scholar
[14]	S. Dolgov and M. Stoll, Low-rank solutions to an optimization problem constrained by the Navier-Stokes equations, SIAM Journal on Scientific Computing, 39 (2017), A255–A280. doi: 10.1137/15M1040414. Google Scholar
[15]	S. Dolgov, TT-GMRES: solution to a linear system in the structured tensor format, Russian Journal of Numerical Analysis and Mathematical Modelling, 28 (2013), 149-172. doi: 10.1515/rnam-2013-0009. Google Scholar
[16]	S. Dolgov and D. Savostyanov, Alternating minimal energy methods for linear systems in higher dimensions, SIAM Journal on Scientific Computing, 36 (2014), A2248–A2271. doi: 10.1137/140953289. Google Scholar
[17]	D. Donoho et al., High-dimensional data analysis: The curses and blessings of dimensionality, AMS Math Challenges Lecture, 1 (2000), 32. Google Scholar
[18]	F. Feitzinger, T. Hylla and E. Sachs, Inexact Kleinman–Newton method for Riccati equations, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 272-288. doi: 10.1137/070700978. Google Scholar
[19]	C. Fu and J. Pfaendtner, Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics, Journal of Chemical Theory and Computation, 14 (2018), 2516-2525. Google Scholar
[20]	L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations, SIAM Journal on Matrix Analysis and Applications, 29 (2007), 870-894. doi: 10.1137/040618102. Google Scholar
[21]	L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), 53-78. doi: 10.1002/gamm.201310004. Google Scholar
[22]	W. Hackbusch, Tensor Spaces And Numerical Tensor Calculus, Springer–Verlag, Berlin, 2012. doi: 10.1007/978-3-642-28027-6. Google Scholar
[23]	W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov, Approximate iterations for structured matrices, Numerische Mathematik, 109 (2008), 365-383. doi: 10.1007/s00211-008-0143-0. Google Scholar
[24]	E. Hansen and T. Stillfjord, Convergence analysis for splitting of the abstract differential Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139. doi: 10.1137/130935501. Google Scholar
[25]	R. Herzog and E. Sachs, Preconditioned conjugate gradient method for optimal control problems with control and state constraints, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2291–2317, URL https://doi.org/10.1137/090779127. doi: 10.1137/090779127. Google Scholar
[26]	M. Hinze, Optimal and Instantaneous Control of the Instationary Navier-Stokes Equations, Habilitation, Technisches Universität Dresden, 2002. Google Scholar
[27]	S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683–A713. doi: 10.1137/100818893. Google Scholar
[28]	B. Khoromskij and V. Khoromskaia, Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM Journal on Scientific Computing, 31 (2009), 3002-3026. doi: 10.1137/080730408. Google Scholar
[29]	G. Kirsten and V. Simoncini, Order reduction methods for solving large-scale differential matrix Riccati equations, accepted SIAM Journal on Scientific Computing. doi: 10.1137/19M1264217. Google Scholar
[30]	T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. Google Scholar
[31]	A. Koskela and H. Mena, Analysis of Krylov subspace approximation to large scale differential Riccati equations, arXiv preprint arXiv: 1705.07507. Google Scholar
[32]	D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688-1714. doi: 10.1137/090756843. Google Scholar
[33]	N. Lang, Numerical Methods for Large-Scale Linear Time-Varying Control Systems and Related Differential Matrix Equations, PhD thesis, Technische Universität Chemnitz, 2018. Google Scholar
[34]	N. Lang, H. Mena and J. Saak, On the benefits of the $LDL^T$ factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71. doi: 10.1016/j.laa.2015.04.006. Google Scholar
[35]	Y. Lin and V. Simoncini, A new subspace iteration method for the algebraic Riccati equation, Numerical Linear Algebra with Applications, 22 (2015), 26-47. doi: 10.1002/nla.1936. Google Scholar
[36]	H. Mena, A. Ostermann, L.-M. Pfurtscheller and C. Piazzola, Numerical low-rank approximation of matrix differential equations, Journal of Computational and Applied Mathematics, 340 (2018), 602-614. doi: 10.1016/j.cam.2018.01.035. Google Scholar
[37]	M. R. Opmeer, Decay of singular values of the Gramians of infinite-dimensional systems, in 2015 European Control Conference (ECC), 2015, 1183–1188. Google Scholar
[38]	I. Oseledets, DMRG approach to fast linear algebra in the TT-format, Computational Methods in Applied Mathematics, 11 (2011), 382-393. doi: 10.2478/cmam-2011-0021. Google Scholar
[39]	I. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317. doi: 10.1137/090752286. Google Scholar
[40]	I. Oseledets, S. Dolgov, V. Kazeev, D. Savostyanov, O. Lebedeva, P. Zhlobich, T. Mach and L. Song, TT-Toolbox, URL https://github.com/oseledets/TT-Toolbox, https://github.com/oseledets/TT-Toolbox. Google Scholar
[41]	I. Oseledets and E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM Journal on Scientific Computing, 31 (2009), 3744-3759. doi: 10.1137/090748330. Google Scholar
[42]	J. Pearson, M. Stoll and A. Wathen, Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function, Numerical Linear Algebra with Applications, 21 (2014), 81–97, URL https://doi.org/10.1002/nla.1863. doi: 10.1002/nla.1863. Google Scholar
[43]	J. Saak, M. Köhler and P. Benner, M-M.E.S.S.-2.0 – The matrix equations sparse solvers library, See also: www.mpi-magdeburg.mpg.de/projects/mess. doi: 10.5281/zenodo.3368844. Google Scholar
[44]	C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011. doi: 10.1088/0266-5611/29/6/065011. Google Scholar
[45]	C. Silvestre and A. Pascoal, Depth control of the INFANTE AUV using gain-scheduled reduced order output feedback, Control Engineering Practice, 15 (2007), 883-895. Google Scholar
[46]	V. Simoncini, Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 1655-1674. doi: 10.1137/16M1059382. Google Scholar
[47]	T. Stillfjord, Singular value decay of operator-valued differential Lyapunov and Riccati equations, SIAM Journal on Control and Optimization, 56 (2018), 3598-3618. doi: 10.1137/18M1178815. Google Scholar
[48]	M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM Journal on Scientific Computing, 37 (2015), 1-19. doi: 10.1137/130926365. Google Scholar
[49]	C. Tobler, Low-rank Tensor Methods for Linear Systems and Eigenvalue Problems, PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20320, 2012. Google Scholar
[50]	F. Tröltzsch, Optimale Steuerung Partieller Differentialgleichungen, Vieweg+Teubner Verlag, 2005. Google Scholar
[51]	B. Vandereycken and S. Vandewalle, Local Fourier analysis for tensor-product multigrid, in AIP Conference Proceedings, AIP, 1168 (2009), 354–356. Google Scholar
[52]	C. d. Villemagne and R. E. Skelton, Model reductions using a projection formulation, International Journal of Control, 46 (1987), 2141-2169. doi: 10.1080/00207178708934040. Google Scholar
[53]	Z. Zhang, X. Yang, I. Oseledets, G. Karniadakis and L. Daniel, Enabling high-dimensional hierarchical uncertainty quantification by ANOVA and tensor-train decomposition, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 34 (2015), 63-76. Google Scholar

show all references

References:

[1]	H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations, Systems Control: Foundations Applications, Birkhäuser Verlag, Basel, 2003, URL https://doi.org/10.1007/978-3-0348-8081-7, In Control and Systems Theory. doi: 10.1007/978-3-0348-8081-7. Google Scholar
[2]	N. Ahmed, G. Matthies, L. Tobiska and H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 1747-1756. doi: 10.1016/j.cma.2011.02.003. Google Scholar
[3]	J. Ballani and L. Grasedyck, A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27-43. doi: 10.1002/nla.1818. Google Scholar
[4]	R. Bellman, Dynamic programming, Courier Corporation, 2013. Google Scholar
[5]	P. Benner, Z. Bujanović, P. Kürschner and J. Saak, RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations, Numerische Mathematik, 138 (2018), 301-330. doi: 10.1007/s00211-017-0907-5. Google Scholar
[6]	P. Benner, Z. Bujanović, P. Kürschner and J. Saak, A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems, SIAM Journal on Scientific Computing, 42 (2020), A957–A996. doi: 10.1137/18M1220960. Google Scholar
[7]	P. Benner, S. Dolgov, A. Onwunta and M. Stoll, Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304 (2016), 26-54. doi: 10.1016/j.cma.2016.02.004. Google Scholar
[8]	P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, in in Proceedings of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar
[9]	P. Benner and H. Mena, Rosenbrock methods for solving Riccati differential equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2956. doi: 10.1109/TAC.2013.2258495. Google Scholar
[10]	P. Benner, A. Onwunta and M. Stoll, Low rank solution of unsteady diffusion equations with stochastic coefficients, SIAM Journal on Uncertainty Quantification, 3 (2015), 622-649. doi: 10.1137/130937251. Google Scholar
[11]	P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52. doi: 10.1002/gamm.201310003. Google Scholar
[12]	A. Brooks and T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8. Google Scholar
[13]	S. Dolgov and B. Khoromskij, Two-level QTT-Tucker format for optimized tensor calculus, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 593-623. doi: 10.1137/120882597. Google Scholar
[14]	S. Dolgov and M. Stoll, Low-rank solutions to an optimization problem constrained by the Navier-Stokes equations, SIAM Journal on Scientific Computing, 39 (2017), A255–A280. doi: 10.1137/15M1040414. Google Scholar
[15]	S. Dolgov, TT-GMRES: solution to a linear system in the structured tensor format, Russian Journal of Numerical Analysis and Mathematical Modelling, 28 (2013), 149-172. doi: 10.1515/rnam-2013-0009. Google Scholar
[16]	S. Dolgov and D. Savostyanov, Alternating minimal energy methods for linear systems in higher dimensions, SIAM Journal on Scientific Computing, 36 (2014), A2248–A2271. doi: 10.1137/140953289. Google Scholar
[17]	D. Donoho et al., High-dimensional data analysis: The curses and blessings of dimensionality, AMS Math Challenges Lecture, 1 (2000), 32. Google Scholar
[18]	F. Feitzinger, T. Hylla and E. Sachs, Inexact Kleinman–Newton method for Riccati equations, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 272-288. doi: 10.1137/070700978. Google Scholar
[19]	C. Fu and J. Pfaendtner, Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics, Journal of Chemical Theory and Computation, 14 (2018), 2516-2525. Google Scholar
[20]	L. Grasedyck and W. Hackbusch, A multigrid method to solve large scale Sylvester equations, SIAM Journal on Matrix Analysis and Applications, 29 (2007), 870-894. doi: 10.1137/040618102. Google Scholar
[21]	L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), 53-78. doi: 10.1002/gamm.201310004. Google Scholar
[22]	W. Hackbusch, Tensor Spaces And Numerical Tensor Calculus, Springer–Verlag, Berlin, 2012. doi: 10.1007/978-3-642-28027-6. Google Scholar
[23]	W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov, Approximate iterations for structured matrices, Numerische Mathematik, 109 (2008), 365-383. doi: 10.1007/s00211-008-0143-0. Google Scholar
[24]	E. Hansen and T. Stillfjord, Convergence analysis for splitting of the abstract differential Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139. doi: 10.1137/130935501. Google Scholar
[25]	R. Herzog and E. Sachs, Preconditioned conjugate gradient method for optimal control problems with control and state constraints, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2291–2317, URL https://doi.org/10.1137/090779127. doi: 10.1137/090779127. Google Scholar
[26]	M. Hinze, Optimal and Instantaneous Control of the Instationary Navier-Stokes Equations, Habilitation, Technisches Universität Dresden, 2002. Google Scholar
[27]	S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683–A713. doi: 10.1137/100818893. Google Scholar
[28]	B. Khoromskij and V. Khoromskaia, Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM Journal on Scientific Computing, 31 (2009), 3002-3026. doi: 10.1137/080730408. Google Scholar
[29]	G. Kirsten and V. Simoncini, Order reduction methods for solving large-scale differential matrix Riccati equations, accepted SIAM Journal on Scientific Computing. doi: 10.1137/19M1264217. Google Scholar
[30]	T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. Google Scholar
[31]	A. Koskela and H. Mena, Analysis of Krylov subspace approximation to large scale differential Riccati equations, arXiv preprint arXiv: 1705.07507. Google Scholar
[32]	D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688-1714. doi: 10.1137/090756843. Google Scholar
[33]	N. Lang, Numerical Methods for Large-Scale Linear Time-Varying Control Systems and Related Differential Matrix Equations, PhD thesis, Technische Universität Chemnitz, 2018. Google Scholar
[34]	N. Lang, H. Mena and J. Saak, On the benefits of the $LDL^T$ factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71. doi: 10.1016/j.laa.2015.04.006. Google Scholar
[35]	Y. Lin and V. Simoncini, A new subspace iteration method for the algebraic Riccati equation, Numerical Linear Algebra with Applications, 22 (2015), 26-47. doi: 10.1002/nla.1936. Google Scholar
[36]	H. Mena, A. Ostermann, L.-M. Pfurtscheller and C. Piazzola, Numerical low-rank approximation of matrix differential equations, Journal of Computational and Applied Mathematics, 340 (2018), 602-614. doi: 10.1016/j.cam.2018.01.035. Google Scholar
[37]	M. R. Opmeer, Decay of singular values of the Gramians of infinite-dimensional systems, in 2015 European Control Conference (ECC), 2015, 1183–1188. Google Scholar
[38]	I. Oseledets, DMRG approach to fast linear algebra in the TT-format, Computational Methods in Applied Mathematics, 11 (2011), 382-393. doi: 10.2478/cmam-2011-0021. Google Scholar
[39]	I. Oseledets, Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317. doi: 10.1137/090752286. Google Scholar
[40]	I. Oseledets, S. Dolgov, V. Kazeev, D. Savostyanov, O. Lebedeva, P. Zhlobich, T. Mach and L. Song, TT-Toolbox, URL https://github.com/oseledets/TT-Toolbox, https://github.com/oseledets/TT-Toolbox. Google Scholar
[41]	I. Oseledets and E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM Journal on Scientific Computing, 31 (2009), 3744-3759. doi: 10.1137/090748330. Google Scholar
[42]	J. Pearson, M. Stoll and A. Wathen, Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function, Numerical Linear Algebra with Applications, 21 (2014), 81–97, URL https://doi.org/10.1002/nla.1863. doi: 10.1002/nla.1863. Google Scholar
[43]	J. Saak, M. Köhler and P. Benner, M-M.E.S.S.-2.0 – The matrix equations sparse solvers library, See also: www.mpi-magdeburg.mpg.de/projects/mess. doi: 10.5281/zenodo.3368844. Google Scholar
[44]	C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011. doi: 10.1088/0266-5611/29/6/065011. Google Scholar
[45]	C. Silvestre and A. Pascoal, Depth control of the INFANTE AUV using gain-scheduled reduced order output feedback, Control Engineering Practice, 15 (2007), 883-895. Google Scholar
[46]	V. Simoncini, Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 1655-1674. doi: 10.1137/16M1059382. Google Scholar
[47]	T. Stillfjord, Singular value decay of operator-valued differential Lyapunov and Riccati equations, SIAM Journal on Control and Optimization, 56 (2018), 3598-3618. doi: 10.1137/18M1178815. Google Scholar
[48]	M. Stoll and T. Breiten, A low-rank in time approach to PDE-constrained optimization, SIAM Journal on Scientific Computing, 37 (2015), 1-19. doi: 10.1137/130926365. Google Scholar
[49]	C. Tobler, Low-rank Tensor Methods for Linear Systems and Eigenvalue Problems, PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20320, 2012. Google Scholar
[50]	F. Tröltzsch, Optimale Steuerung Partieller Differentialgleichungen, Vieweg+Teubner Verlag, 2005. Google Scholar
[51]	B. Vandereycken and S. Vandewalle, Local Fourier analysis for tensor-product multigrid, in AIP Conference Proceedings, AIP, 1168 (2009), 354–356. Google Scholar
[52]	C. d. Villemagne and R. E. Skelton, Model reductions using a projection formulation, International Journal of Control, 46 (1987), 2141-2169. doi: 10.1080/00207178708934040. Google Scholar
[53]	Z. Zhang, X. Yang, I. Oseledets, G. Karniadakis and L. Daniel, Enabling high-dimensional hierarchical uncertainty quantification by ANOVA and tensor-train decomposition, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 34 (2015), 63-76. Google Scholar

Figure 1. 1D heat equation with $ n = 1000 $ and $ n_t = 2000 $. Left. Controlled and desired states. Right. Singular values of solutions

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Figure 2. Left. Control and observation domains. Right. Singular values of matrix unfoldings

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Figure 3. Storage reduction by tensor truncation from $ P(\cdot) $ to $ \tilde{P}(\cdot) $

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Figure 4. Left. Control domains. Right. Observation domains

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Figure 6. Left. Desired state $ x_d(\xi_1,\xi_2,t_f) $. Right. Controlled state $ x(\xi_1,\xi_2,t_f) $

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Figure 5. Left. Change in Newton iterates. Right. TT ranks of $ \mathbf{p}_i $ during Newton iteration, $ r_1 $ (solid lines) and $ r_2 $ (dashed lines)

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Figure 7. Left. Change in Newton iterates. Right. TT ranks of $ \mathbf{p}_i $ during Newton iteration, $ r_1 $ (solid lines) and $ r_2 $ (dashed lines)

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Figure 8. Left. Desired state $ x_d(\xi_1,\xi_2,t_f) $. Right. Controlled state $ x(\xi_1,\xi_2,t_f) $

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Figure 9. Left. Change in Newton iterates. Right. TT ranks of $ \mathbf{p}_i $ during Newton iteration

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Figure 10. Left. Desired state $ x_d(\xi_1,\xi_2,t_f) $. Right. Controlled state $ x(\xi_1,\xi_2,t_f) $

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Figure 11. Relative error w.r.t. to the "true" solution from $\mathtt{ode23}$

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Table 1. Computing times and storage for $ n_t = 100 $ and $ n_t = 1000 $

	Time ($ s $)	Memory (MB)	Time ($ s $)	Memory (MB)
$\mathtt{ode23}$	731	2048	894	20480
$\mathtt{tt}$	6142	14.61	11940	18.6
$\mathtt{MMESS (split)}$	6.5	74.32	86	916
$\mathtt{MMESS (BDF)}$	48	115.3	172	1045
$\mathtt{RKSM-DRE}$	12	29.44	153	273

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	Time ($ s $)	Memory (MB)	Time ($ s $)	Memory (MB)
$\mathtt{ode23}$	731	2048	894	20480
$\mathtt{tt}$	6142	14.61	11940	18.6
$\mathtt{MMESS (split)}$	6.5	74.32	86	916
$\mathtt{MMESS (BDF)}$	48	115.3	172	1045
$\mathtt{RKSM-DRE}$	12	29.44	153	273

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Solving differential Riccati equations: A nonlinear space-time method using tensor trains

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