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POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation

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Abstract

In this work we deal with parametrized time dependent optimal control problems governed by partial differential equations. We aim at extending the standard saddle point framework of steady constraints to time dependent cases. We provide an analysis of the well-posedness of this formulation both for parametrized scalar parabolic constraint and Stokes governing equations and we propose reduced order methods as an effective strategy to solve them. Indeed, on one hand, parametrized time dependent optimal control is a very powerful mathematical model which is able to describe several physical phenomena, on the other, it requires a huge computational effort. Reduced order methods are a suitable approach to have rapid and accurate simulations. We rely on POD–Galerkin reduction over the physical and geometrical parameters of the optimality system in a space-time formulation. Our theoretical results and our methodology are tested on two examples: a boundary time dependent optimal control for a Graetz flow and a distributed optimal control governed by time dependent Stokes equations. With these two test cases the convenience of the reduced order modelling is further extended to the field of time dependent optimal control.

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Notes

  1. The error for state, control and adjoint variables are presented in the following norms: \(\Vert y^{\mathcal N}- y_N \Vert ^2_{H^1}\), \(\Vert u^{\mathcal N}- u_N \Vert ^2_{L^2}\) and \(\Vert p^{\mathcal N}- p_N \Vert ^2_{H^1}\), respectively.

  2. The space \(L^2_0(\varOmega (\mu _2))\) is made by functions \(p \in L^2(\varOmega (\mu _2))\) which satisfy \(\displaystyle \int _{\varOmega } p \; d \varOmega = 0\). In the reduced model the aggregated basis associated to the state and adjoint pressure was built in order to satisfy this constraint, i.e. the reduced adjoint variable \(\xi _{N}\) verifies \(\displaystyle \int _{\varOmega } \xi _{N} \; d \varOmega = 0\).

  3. The error for state velocity and pressure, control and adjoint velocity and pressure variables are: \(\Vert y^{\mathcal N}- y_N \Vert ^2_{H^1}\), \(\Vert p^{\mathcal N}- p_N \Vert ^2_{L^2}\), \(\Vert u^{\mathcal N}- u_N \Vert ^2_{L^2}\), \(\Vert \lambda ^{\mathcal N}- \lambda _N \Vert ^2_{H^1}\) and \(\Vert \xi ^{\mathcal N}- \xi _N \Vert ^2_{L^2}\), respectively. We underline that in order to make the FE and ROM adjoint pressures comparable we define \(\xi ^{\mathcal N}\) := \({{\bar{\xi }}} ^{\mathcal N}- \displaystyle \int _{\varOmega } {{\bar{\xi }}} ^{\mathcal N}dx\), where \({{\bar{\xi }}} ^{\mathcal N}\) is the actual truth solution.

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Acknowledgements

We acknowledge the support by European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD Project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”. We also acknowledge the INDAM-GNCS project “Advanced intrusive and non-intrusive model order reduction techniques and applications”. The computations in this work have been performed with RBniCS [43] library, developed at SISSA mathLab, which is an implementation in FEniCS [34] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.

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  1. SISSA mathLab, Mathematics Area, International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy

    Maria Strazzullo, Francesco Ballarin & Gianluigi Rozza

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  1. Maria Strazzullo
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Strazzullo, M., Ballarin, F. & Rozza, G. POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation. J Sci Comput 83, 55 (2020). https://doi.org/10.1007/s10915-020-01232-x

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