Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

Xenia Kerkhoff; Sandra May

doi:10.3934/mcrf.2021054

Article Contents

2023, Volume 13, Issue 1: 193-214. Doi: 10.3934/mcrf.2021054

This issue Previous Article Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems Next Article Eigenvalues of stochastic Hamiltonian systems with boundary conditions and its application

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

Xenia Kerkhoff^, and
Sandra May

Department of Mathematics, TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, Germany

^* Corresponding author: Xenia Kerkhoff
^* Corresponding author: Xenia Kerkhoff

Received: March 2021

Revised: July 2021

Early access: October 2021

Published: March 2023

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Abstract

We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.

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Mathematics Subject Classification: Primary: 49M41; Secondary: 65M60.

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Figure 1. Test 1: Results for gradient test. The $ x- $axis denotes $ \frac{1}{\rho} $ with $ \rho $ being the step length in the difference quotient, the $ y- $axis denotes the error of the gradient as given by (24)

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Figure 2. Test 2: Computed solution $ (q^h,u^h) $ for $ p = 2 $ and $ N = 160 $. Left column: control $ q^h $, right column: state $ u^h $. The second row shows the solutions $ q^h(\cdot,t) $ and $ u^h(\cdot,t) $ at time instances $ t = 0.25 $, $ t = 0.5 $, and $ t = 0.75 $ as a function of the space coordinate $ x $

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Table 1. Test 1: Errors and orders of convergence for $ \varepsilon = 10^{-3} $

$ p $	$ N $	error $ u $	order	error $ q $	order	error $ z $	order	iter.	gradient
OD-wo
1	40	2.35e-04	–	1.93e-03	–	2.31e-04	–	25	4.66e-05
1	80	5.43e-05	2.11	5.09e-04	1.92	7.36e-05	1.65	28	1.97e-05
1	160	1.28e-05	2.08	1.44e-04	1.82	2.48e-05	1.57	31	7.11e-06
1	320	3.19e-06	2.01	4.05e-05	1.83	8.22e-06	1.59	36	2.81e-06
2	20	2.31e-05	–	1.67e-04	–	1.67e-05	–	49	2.18e-07
2	40	2.95e-06	2.97	2.05e-05	3.03	2.06e-06	3.02	52	3.00e-08
2	80	3.41e-07	3.11	2.61e-06	2.98	2.61e-07	2.98	54	9.51e-09
2	160	4.65e-08	2.87	3.93e-07	2.73	3.47e-08	2.91	54	9.51e-09
3	10	4.15e-06	–	4.97e-05	–	4.96e-06	–	54	9.51e-09
3	20	1.81e-07	4.52	3.02e-06	4.04	3.00e-07	4.05	54	9.51e-09
3	40	2.97e-08	2.61	2.86e-07	3.40	1.70e-08	4.14	54	9.51e-09
3	80	2.62e-08	0.18	2.07e-07	0.47	3.33e-09	2.35	54	9.51e-09
OD-w
1	40	2.31e-04	–	1.87e-03	–	1.87e-04	–	54	9.51e-09
1	80	5.05e-05	2.19	4.63e-04	2.02	4.63e-05	2.02	54	9.51e-09
1	160	9.91e-06	2.35	1.15e-04	2.01	1.15e-05	2.01	54	9.51e-09
1	320	1.72e-06	2.52	2.87e-05	2.01	2.87e-06	2.01	54	9.51e-09
2	20	2.26e-05	–	1.67e-04	3.07	1.67e-05	–	54	9.51e-09
2	40	2.88e-06	2.98	2.01e-05	3.06	2.01e-06	3.06	54	9.51e-09
2	80	3.23e-07	3.15	2.45e-06	3.04	2.44e-07	3.04	54	9.51e-09
2	160	4.35e-08	2.89	3.62e-07	2.76	3.01e-08	3.02	54	9.51e-09
3	10	4.16e-06	–	4.96e-05	–	4.96e-06	–	54	9.51e-09
3	20	1.73e-07	4.59	2.99e-06	4.05	2.98e-07	4.05	54	9.51e-09
3	40	2.66e-08	2.70	2.65e-07	3.49	1.72e-08	4.11	54	9.51e-09
3	80	2.59e-08	0.04	2.05e-07	0.37	3.52e-09	2.29	54	9.51e-09

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Table 2. Test 1: Errors and orders of convergence for $ \varepsilon = 10^{-5} $

$ p $	$ N $	error $ u $	order	error $ q $	order	error $ z $	order	iter.	gradient
OD-wo
1	40	2.79e-04	–	1.97e-03	–	2.47e-04	–	24	5.33e-05
1	80	7.51e-05	1.89	5.51e-04	1.84	9.20e-05	1.42	27	2.81e-05
1	160	2.28e-05	1.72	1.93e-04	1.51	4.08e-05	1.18	29	1.26e-05
1	320	8.15e-06	1.49	7.90e-05	1.29	1.99e-05	1.04	33	7.11e-06
2	20	2.52e-05	–	1.72e-04	–	1.72e-05	–	44	2.25e-07
2	40	3.98e-06	2.66	2.18e-05	2.98	2.18e-06	2.97	51	4.63e-08
2	80	6.28e-07	2.67	3.09e-06	2.82	3.10e-07	2.82	54	9.51e-09
2	160	1.01e-07	2.64	5.60e-07	2.46	5.42e-08	2.51	54	9.51e-09
3	10	4.63e-06	–	4.87e-05	–	4.87e-06	–	54	9.51e-09
3	20	2.76e-07	4.07	3.12e-06	3.96	3.08e-07	3.98	54	9.51e-09
3	40	3.58e-08	2.94	3.19e-07	3.29	1.87e-08	4.04	54	9.51e-09
3	80	2.68e-08	0.42	2.12e-07	0.59	2.94e-09	2.67	54	9.51e-09
OD-w
1	40	2.74e-04	–	1.89e-03	–	1.89e-04	–	54	9.51e-09
1	80	6.99e-05	1.97	4.68e-04	2.01	4.68e-05	2.01	54	9.51e-09
1	160	1.76e-05	1.99	1.17e-04	2.01	1.17e-05	2.01	54	9.51e-09
1	320	4.39e-06	2.00	2.91e-05	2.00	2.91e-06	2.00	54	9.51e-09
2	20	2.46e-05	–	1.72e-04	–	1.72e-05	–	54	9.51e-09
2	40	3.84e-06	2.68	2.12e-05	3.02	2.12e-06	3.02	54	9.51e-09
2	80	5.86e-07	2.71	2.65e-06	3.00	2.65e-07	3.00	54	9.51e-09
2	160	8.82e-08	2.73	3.89e-07	2.77	3.33e-08	2.99	54	9.51e-09
3	10	4.65e-06	–	4.85e-05	–	4.85e-06	–	54	9.51e-09
3	20	2.65e-07	4.13	3.05e-06	3.99	3.04e-07	4.00	54	9.51e-09
3	40	2.96e-08	3.16	2.75e-07	3.47	1.86e-08	4.03	54	9.51e-09
3	80	2.59e-08	0.19	2.05e-07	0.42	3.53e-09	2.40	54	9.51e-09

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Table 3. Test 2: Values and orders of convergence (computed using values from 3 subsequent meshes) for functional $ J $ for $ p = 1 $

	OD-w				OD-wo
N	value $ J $	order	iter.	gradient	value $ J $	order	iter.	gradient
20	0.10640082692	-	44	8.58e-08	0.10641958284	-	11	2.10e-03
40	0.10818515778	-	44	7.94e-08	0.10819225166	-	14	1.02e-03
80	0.10855578246	2.27	43	8.03e-08	0.10855750188	2.28	18	3.44e-04
160	0.10862035158	2.52	43	8.00e-08	0.10862054051	2.53	22	1.09e-04

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Table 4. Test 2: Errors and orders of convergence for the control $ q $ for $ p = 1 $

	OD-w				OD-wo
N	error $ q $	order	iter.	gradient	error $ q $	order	iter.	gradient
20	1.67e-02	-	44	8.58e-08	1.77e-02	-	11	2.10e-03
40	4.56e-03	1.87	44	7.94e-08	6.10e-03	1.54	14	1.02e-03
80	1.47e-03	1.64	43	8.03e-08	2.83e-03	1.11	18	3.44e-04
160	4.20e-04	1.80	43	8.00e-08	1.00e-03	1.50	22	1.09e-04

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