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

@article{Strazzullo2020PODGalerkinMO,
  title={POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation},
  author={Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza},
  journal={Journal of Scientific Computing},
  year={2020},
  volume={83},
  pages={1-35}
}
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… 
POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations
TLDR
This work proposes reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by Shallow Waters Equations in a solution tracking setting and shows how reduced order modelling could help in studying different configurations and phenomena in a fast way.
Space-time POD-Galerkin approach for parametric flow control
TLDR
This contribution proposes reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations and shows how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not losing accuracy.
Reduced order methods for parametric flow control problems and applications
TLDR
This contribution proposes reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations and shows how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation.
A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition
TLDR
This work recasts time-dependent optimal control problems governed by partial differential equations in a Dynamic Mode Decomposition with control framework and reconstruct and perform future predictions for all the variables of interest at a lower computational cost with respect to the standard space-time discretized models.
Parallelized POD-based suboptimal economic model predictive control of a state-constrained Boussinesq approximation
Motivated by an energy efficient building application, we want to optimize a quadratic cost functional subject to the Boussinesq approximation of the Navier-Stokes equations and to bilateral state
Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid-structure interaction problems
In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its
A Certified Reduced Basis Method for Linear Parametrized Parabolic Optimal Control Problems in Space-Time Formulation
TLDR
This work exploits an error estimator procedure, based on easy-to-compute quantities which guarantee a rigorous and efficient bound for the error of the involved variables, to efficiently solve time dependent parametrized optimal control problems governed by parabolic partial differential equations through the certified reduced basis method.
Data Assimilation Predictive GAN (DA-PredGAN): applied to determine the spread of COVID-19
TLDR
To predict the spread of COVID-19 in an idealised town, the proposed methods can accurately predict the evolution of the high-fidelity numerical simulation, and can efficiently assimilate observed data and determine the corresponding model parameters.
Fast active thermal cloaking through PDE-constrained optimization and reduced-order modeling
In this paper we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this
Hull shape design optimization with parameter space and model reductions, and self-learning mesh morphing
TLDR
A data-driven framework involving multiple reduction techniques is proposed to reduce computational burden of parametric partial differential equations shape optimization, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps.
...
1
2
...

References

SHOWING 1-10 OF 84 REFERENCES
Reduced Basis Method for Parametrized Elliptic Optimal Control Problems
TLDR
The methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable is developed and recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems.
Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering
TLDR
This work proposes reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering and proposes a POD-Galerkin reduction of the optimality system.
Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls
TLDR
This paper introduces reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and proposes two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional.
Reduced basis methods for optimal control of advection-diffusion problems ∗
The reduced basis (RB) method is proposed for the approximation of multi-parametrized equations governing an optimal control problem. The idea behind the RB method is to project the so- lution onto a
Optimal flow control for Navier–Stokes equations: drag minimization
Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag
A certified reduced basis method for parametrized elliptic optimal control problems
In this paper, we employ the reduced basis method as a surrogate model for the solu- tion of linear-quadratic optimal control problems governed by parametrized elliptic partial dierential equations.
Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations
TLDR
The reduced basis method is extended to the case of noncoercive (elliptic) equations, such as the Stokes equations, and applied to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.
On the stability of the reduced basis method for Stokes equations in parametrized domains
We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global
A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations
  • Masayuki Yano
  • Computer Science, Mathematics
    SIAM J. Sci. Comput.
  • 2014
We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but
Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints
TLDR
This paper employs the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations and proposes two different reduced basis approximations and associated error estimation procedures.
...
1
2
3
4
5
...