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

• * Corresponding author: T. Breiten
• 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: • • 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
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