Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

Kärcher, Mark; Tokoutsi, Zoi; Grepl, Martin; Veroy, Karen

doi:10.1007/s10915-017-0539-z

Cited by 44 publications

(68 citation statements)

References 33 publications

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“…Reduced order models offer promises in many fields such as system identification [18][19][20] , control [21][22][23][24][25][26] , optimization [27][28][29] , and data assimilation [30][31][32] applications. In these model reduction approaches, we aim at obtaining simplified (but dense) models from high-fidelity numerical simulation data or data collected from the experiment 12,33 .…”

Section: Introductionmentioning

confidence: 99%

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

et al. 2019

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In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various deep neural network architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equations, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a non-intrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly non-intrusive, since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate, and can be used as a robust predictive tool for non-intrusive model order reduction of complex fluid flows.

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Section: Introductionmentioning

confidence: 99%

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

et al. 2019

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“…However, these bounds provide no individual information on the error between the model corrections u * µ and u * R,µ , or between the state estimates y * µ and y * R,µ . We therefore pursue the approach from [19]…”

Section: A Posteriori Error Estimationmentioning

confidence: 99%

3D-VAR for parameterized partial differential equations: a certified reduced basis approach

2019

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In this paper, we propose a reduced order approach for 3D variational data assimilation governed by parametrized partial differential equations. In contrast to the classical 3D-VAR formulation that penalizes the measurement error directly, we present a modified formulation that penalizes the experimentallyobservable misfit in the measurement space. Furthermore, we include a model correction term that allows to obtain an improved state estimate. We begin by discussing the influence of the measurement space on the amplification of noise and prove a necessary and sufficient condition for the identification of a "good" measurement space. We then propose a certified reduced basis (RB) method for the estimation of the model correction, the state prediction, the adjoint solution and the observable misfit with respect to the true state for real-time and many-query applications. A posteriori bounds are proposed for the error in each of these approximations. Finally, we introduce different approaches for the generation of the reduced basis spaces and the stability-based selection of measurement functionals. The 3D-VAR method and the associated certified reduced basis approximation are tested in a parameter and state estimation problem for a steady-state thermal conduction problem with unknown parameters and unknown Neumann boundary conditions.

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“…In the second one we construct nonnegative slack variables for the control and so can generate feasible low dimensional surrogates for the control. Finally, we extend the proof of the a posteriori error bounds from Kärcher et al (2014) to derive efficient a posteriori bounds for the control error in the constraint case.…”

Section: Introductionmentioning

confidence: 97%

“…A new approach for efficient computation of error bounds for unconstrained distributed control problems was proposed in Kärcher et al (2014). This approach, however, and all other existing approaches in the literature, see e.g.…”

Section: Introductionmentioning

confidence: 99%

A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Control Problems with Control Constraints

et al. 2015

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In this talk, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations involving constraints on the control. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and propose rigorous error bounds for the error in the optimal control. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline-online computational procedure, thus making our approach relevant in the many-query or real-time context. We present numerical results for a model problem to show the validity of our approach.Many problems in science and engineering can be modeled in terms of optimal control problems governed by parametrized partial differential equations (PDEs). While the PDE describes the underlying system or component behavior, the parameters often serve to identify a particular configuration of the component -such as boundary and initial conditions, material properties, and geometry. In such cases -in addition to solving the optimal control problem itself -one is often interested in exploring many different parameter configurations and thus in speeding up the solution of the optimal control problem. However, using classical discretization techniques such as finite elements or finite volumes even a single solution is often computationally expensive and time-consuming, a parameter-space exploration thus prohibitive. One way to decrease the computational burden is the surrogate model approach, where the original high-dimensional model is replaced by a reduced order approximation. These ideas have received a lot of attention in the past and various model order reduction techniques have been used in this context. However, the solution of the reduced order optimal conThis work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111. trol problem is generally sub-optimal and reliable error estimation is thus crucial. Besides serving as a certificate of fidelity for the sub-optimal solution, our a posteriori error bounds are also a crucial ingredient in generating the reduced basis with greedy algorithms. A new approach for efficient computation of error bounds for unconstrained distributed control problems was proposed in Kärcher et al. (2014). This approach, however, and all other existing approaches in the literature, see e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), are not directly applicable to the important case with additional constraints on the control. In this work we extend the methodology presented in Zhang et al. (2014) to consider PDE-constrained optimal control problems. The authors in Zhang et al. (2014) develop a certified Reduced Basis (RB) method that provides sharp and inexpens...

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Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

Cited by 44 publications

References 33 publications

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

3D-VAR for parameterized partial differential equations: a certified reduced basis approach

A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Control Problems with Control Constraints

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