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

  • * Corresponding author: T. Breiten

    * Corresponding author: T. Breiten 
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  • 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.

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

    Citation:

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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

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

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

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

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

    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)

    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)

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

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

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

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

    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
     | Show Table
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