Preconditioning for boundary control problems in incompressible fluid dynamics

Heidel, Gennadij; Wathen, Andrew J.

doi:10.1002/nla.2218

Cited by 12 publications

(11 citation statements)

References 54 publications

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“…Its effect is to couple all the equations, increasing the difficulties to construct β robust preconditioners. We remark that a similar low-rank perturbation appears in deterministic OCP with a control acting on a subset of the boundary, see [20]. The first approximation S, and corresponding preconditioner P , we consider is obtained dropping the β-dependent low-rank term, (5.2)…”

Section: Algebraic Preconditionersmentioning

confidence: 94%

“…For robust OCPUU one has Im I Y , see Lemma 6.1, and similarly for a deterministic OCP with local control one cannot generate (H 1 (D)) using only elements of (H 1 ( D)) [14]. From the algebraic point of view, this leads to low-rank perturbed Schur complements, where the rank of the perturbation is equal to the size of the finite element discretization of the control space (see (5.1) and [20]).…”

Section: Lrmentioning

confidence: 99%

“…A mean approximation, combined possibly with a Chebyshev semi-iteration, is shown to be sufficient to efficiently approximate this inverse for quite a wide range of parameters, leading to practical P LRM , P LRC , P OPM , P OPC preconditioners, where the subscript M stands for "mean" and C for Chebyshev. We remark that the development of robust preconditioners for small values of the regularization parameter is not obvious and poses some interesting mathematical and computational challenges which, surprisingly, are similar to those encountered in deterministic OCP when the control acts locally, either on a portion of the domain [14], or on a portion of the boundary [20]. We further stress that our analysis does not assume that the random bilinear form is uniformly bounded and coercive with respect to the randomness, which is a frequent simplifying hypothesis in the literature, see, e.g., [44,55,3].…”

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confidence: 99%

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Preconditioners for robust optimal control problems under uncertainty

Nobile¹,

Vanzan²

2021

Preprint

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The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient κ(x, ω) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables 1/ min x∈D κ(x, ω) and max x∈D κ(x, ω), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

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Section: Algebraic Preconditionersmentioning

confidence: 94%

Section: Lrmentioning

confidence: 99%

mentioning

confidence: 99%

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Preconditioners for robust optimal control problems under uncertainty

Nobile¹,

Vanzan²

2021

Preprint

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“…An important building block for the optimal control of PDEs is the preconditioning techniques to accelerate the simulation of PDEs. Many efforts have been dedicated to the development of efficient and robust preconditioning techniques for the distributed control problems where the control is distributed throughout the domain, 12‐21 or the boundary control problems where only boundary conditions can be regulated 22‐24 . For the in‐domain control problems where the control only acts on a few parts of the domain, preconditioning techniques developed for the distributed control problems do not give satisfactory performance.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Qiu

Gijzen

Wingerden

et al. 2020

Numerical Linear Algebra App

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In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.

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“…Such interesting applications of these problems are in fluid flow, biological and chemical processes, finance and many more. The need for efficient simulations of the optimal control problems stimulates the development and evolvement of the numerical solution in various divisions such as Stokes control [19,26,28,39], Navier-Stokes control [16,27], wave control [37], parabolic control [19,20,38] and elliptic control [3,4,13,14,35]. The efficient solution of PDE-constrained optimization problems is a highly challenging task computationally.…”

Section: Introductionmentioning

confidence: 99%

Robust and efficient finite element multigrid and preconditioned minimum residual solvers for the distributed elliptic optimal control problems

Muzhinji¹

2020

J. Math. Computer Sci.

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In this study, the optimal control problem is considered. This is an important class of partial differential equations constrained optimization problems. The constraint here is an elliptic partial differential equation with Neumann boundary conditions. The discretization of the optimality system produces a block coupled algebraic system of equations of saddle point form. The solution of such systems is a major computational task since they require specialized methods. Constructing robust, fast and efficient solvers for their numerical solution has preoccupied the computational science community for decades and various approaches have been developed. The approaches involve solving simultaneously for all the unknowns using a coupled block system, the segregated approach where a reduced system is solved and the approach of reducing to a fixed point form. Here the minimum residual solver with ideal preconditioning is applied to the unreduced 3 by 3 and reduced 2 by 2 coupled systems and compared to the multigrid method applied to the compact fixed-point form. The two methods are compared numerically in terms of iterative counts and computational times. The numerical results indicate that the two methods produce similar outcomes and the multigrid solver becoming very competitive in terms of the iterative counts though slower than preconditioned minimum residual solver in terms of computational times. For all the approaches, the two methods exhibited mesh and parameter independent convergence. The optimal performance of the two methods is verified computationally and theoretically.

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Preconditioning for boundary control problems in incompressible fluid dynamics

Cited by 12 publications

References 54 publications

Preconditioners for robust optimal control problems under uncertainty

Preconditioners for robust optimal control problems under uncertainty

Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations

Robust and efficient finite element multigrid and preconditioned minimum residual solvers for the distributed elliptic optimal control problems

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