A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences

Carere, Giuseppe; Strazzullo, Maria; Ballarin, Francesco; Rozza, Gianluigi; Stevenson, Rob

doi:10.1016/j.camwa.2021.10.020

Cited by 22 publications

(12 citation statements)

References 21 publications

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“…For environmental sciences applications, another important development for nonlinear time dependent problems could be the use of random input parameters as an extension of [17], since, in marine ecosystems, it is not always possible to assign deterministic values for the parameters describing the physical model.…”

Section: Discussionmentioning

confidence: 99%

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

Strazzullo¹,

Ballarin²,

Rozza³

2020

Preprint

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In this work we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable e.g. in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

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Section: Discussionmentioning

confidence: 99%

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

Strazzullo¹,

Ballarin²,

Rozza³

2020

Preprint

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“…Summarizing, the barotropic vorticity problem is given by eqs. ( 2), (4) endowed with boundary conditions ( 5)- (6) and initial data (7).…”

Section: Problem Definition 21 Quasi-geostrophic Equationsmentioning

confidence: 99%

“…One of the simplest models for geophysical flow is given by the Quasi-Geostrophic equations (QGE): see, e.g., [33,8,23] for mathematical and physical fundamentals, [29,7,31] for some advanced applications and [26] for a recent review on this model. Despite the simplification in the QGE, when the Munk scale (a length that depends on two nondimensional quantities, the Rossby number and the Reynolds number) is small the numerical simulation of the QGE becomes computationally challenging since very fine meshes are required.…”

Section: Introductionmentioning

confidence: 99%

A novel Large Eddy Simulation model for the Quasi-Geostrophic Equations in a Finite Volume setting

Girfoglio¹,

Quaini²,

Rozza³

2022

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We present a Large Eddy Simulation (LES) approach based on a nonlinear differential low-pass filter for the simulation of two-dimensional barotropic flows with under-refined meshes. For the implementation of such model, we choose a segregated three-step algorithm combined with a computationally efficient Finite Volume method. We assess the performance of our approach on the classical double-gyre wind forcing benchmark. The numerical experiments we present demonstrate that our nonlinear filter is an improvement over a linear filter since it is able to recover the four-gyre pattern of the time-averaged stream function even with extremely coarse meshes. In addition, our LES approach provides an average kinetic energy that compares well with the one computed with a Direct Numerical Simulation.

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“…We apply this procedure in a partitioned approach, following good results shown in literature [30,34,53,63]. As this algorithm aims to minimize the expectation of the square error between the truth and the ROM solutions, we can identify different types of weighted ROMs (wROMs) [11,15,13,16,17,18,49,59,61,60] based on the chosen quadrature rules. In this work, we will exploit Monte-Carlo and Quasi Monte-Carlo procedures, tensor rules based on Gauss-Jacobi and Clenshaw-Curtis quadrature techniques, and Smolyak isotropic sparse grids.…”

Section: Introductionmentioning

confidence: 99%

“…Stabilized Advection-Dominated problems in a ROM framework without control are studied, for instance, in [37,59], both for steady and unsteady cases. Instead, in [11] wROMs for generic OCP(µ)s are applied to experiments concerning environmental sciences. Instead, SUPG Advection-Dominated distributed OCP(µ)s are analyzed in a deterministic context in [63], both for elliptic and parabolic experiments.…”

Section: Introductionmentioning

confidence: 99%

Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs

Zoccolan¹,

Strazzullo²,

Rozza³

2023

Preprint

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In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov-Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

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A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences

Cited by 22 publications

References 21 publications

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

A novel Large Eddy Simulation model for the Quasi-Geostrophic Equations in a Finite Volume setting

Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs

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