A Reduced Basis Method for Parametrized Variational Inequalities

Haasdonk, Bernard; Salomon, Julien; Wohlmuth, Barbara

doi:10.1137/110835372

Cited by 30 publications

(91 citation statements)

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“…A combination between RB method and domain decomposition techniques is the so-called reduced basis element method; see [22] for the Stokes case. Recently, a RB formulation for variational inequalities by a saddle point scheme has been proposed in [15].…”

Section: Introductionmentioning

confidence: 99%

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

2013

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In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi's and Babuška's stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi's saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška's inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.

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Section: Introductionmentioning

confidence: 99%

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

2013

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“…Section 3 is devoted to the derivation of error/residual estimates for which we consider a well-known saddle-point formulation of (1.1). We generalize estimates from [6]. Next, we detail our general findings for the particular case of space-time variational formulations of parabolic variational inequalities in section 4.…”

Section: Introductionmentioning

confidence: 92%

“…One advantage of (3.1) for the numerical treatment is that one does not need to construct a conforming discretization of the convex set K, i.e., some finite-dimensional K N ⊂ K. Often, it is easier to construct M N ⊂ M or to solve (3.1) by a primal-dual active set strategy (i.e., no need for a conforming discretization of the space M ) [6]. Moreover, as we shall see below, the saddle-point form allows us to derive a posteriori error estimates.…”

Section: Saddle-point Formulationmentioning

confidence: 99%

“…1 In order that (1.1) makes sense, one has to require that X ⊆ Y 2 since the test function v − u needs to be in Y for u ∈ X and v ∈ K ⊂ Y . A lot of research has been done in the case that Y = X and a(·, ·) being coercive or at least satisfying a Gårding inequality; see, e.g., [6,10,11], just to mention a few. This includes well-posedness (existence, uniqueness, stability) as well as numerical methods.…”

Section: Introductionmentioning

confidence: 99%

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On Noncoercive Variational Inequalities

Glas¹,

Urban²

2014

SIAM J. Numer. Anal.

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We consider variational inequalities with different trial and test spaces and a possibly noncoercive bilinear form. Well-posedness is shown under general conditions that are, e.g., valid for the space-time variational formulation of parabolic variational inequalities. Moreover, we prove an estimate for the error of a Petrov-Galerkin approximation in terms of the residual. For parabolic variational inequalities the arising estimate is independent of the final time.

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“…The authors in Zhang et al (2014) develop a certified Reduced Basis (RB) method that provides sharp and inexpensive a posteriori error bounds for variational inequalities. In particular, the approach has advantages compared to prior work on variational inequalities with the RB method Haasdonk et al (2012). The methodology in Zhang et al (2014) not only (i ) provides sharper error bounds that mimic the convergence rate of the RB approximation, but also (ii) does so at an 720 Eduard Bader et al / IFAC-PapersOnLine 48-1 (2015) [719][720] online cost that is independent of the high dimension of the original problem.…”

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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A Reduced Basis Method for Parametrized Variational Inequalities

Cited by 30 publications

References 32 publications

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

On Noncoercive Variational Inequalities

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

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