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

Tobias Breiten; Sergey Dolgov; Martin Stoll

doi:10.3934/naco.2020034

Article Contents

2021, Volume 11, Issue 3: 407-429. Doi: 10.3934/naco.2020034

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

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
^* Corresponding author: T. Breiten

Received: December 2019

Revised: July 2020

Early access: August 2020

Published: September 2021

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Abstract

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.

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

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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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