Parameter-robust preconditioning for the optimal control of the wave equation

Li, Jun; Pearson, John W.

doi:10.1007/s11075-019-00720-y

Cited by 10 publications

(13 citation statements)

References 46 publications

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“…In subsection 5.1, we provide a full characterization of the spectrum of the preconditioned system which highlights both the dependence on β and on the random field extremal values. In subsection 5.2, we propose instead a β-robust preconditioner based on a more involved factorization of the Schur complement of (4.6), inspired by works on deterministic OCPs in [41,47,46,27,31,42] 5.1. A first Schur complement approximation.…”

Section: Algebraic Preconditionersmentioning

confidence: 99%

“…Nevertheless, theoretical results are not available for several problems, see e.g. [31,40], despite the improved β robustness has been confirmed by numerical examples. In this subsection, we apply this technique to our model problem and we partially characterize the spectrum of the preconditioned Schur complement.…”

Section: Matching Schur Complement Techniquementioning

confidence: 99%

“…The first one, used to derive the preconditioner P LR is algebraic and has its roots in the seminal work of [38], where the authors proposed an optimal, but expensive, preconditioner for saddle point matrices which relies on the exact Schur complement. For deterministic OCP, several preconditioners based on approximations of the exact Schur complements have been study in the last decade [47,46,41,27,31,42]. Concerning OCPUU, the same strategy has been applied in [3] in combination with SGMs, but we remark once more that their problem is not an instance of robust OCP.…”

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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: 99%

Section: Matching Schur Complement Techniquementioning

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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“…This theorem shows that after K iterations, we can find an approximate solution of the variational inequality (18) with an accuracy of O(1/K). This approximate solution is given in (41), and it is the average of all the points w k which can be computed by all the known iterates generated by Algorithm 1. Hence, this is an O(1/K) worst-case convergence rate in the ergodic sense for Algorithm 1.…”

Section: Ergodic Convergence Ratementioning

confidence: 99%

Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond

Song¹,

Yuan²,

Yue³

2020

Preprint

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Optimal control problems subject to both parabolic partial differential equation (PDE) constraints and additional constraints on the control variables are generally challenging, from either theoretical analysis or algorithmic design perspectives. Conceptually, the well-known alternating direction method of multipliers (ADMM) can be directly applied to such a problem. An attractive advantage of this direct ADMM application is that the additional constraint on the control variable can be untied from the parabolic PDE constraint; these two inherently different constraints thus can be treated individually in iterations. At each iteration of the ADMM, the main computation is for solving an optimal control problem with a parabolic PDE constraint while it is not interacted with the constraint on the control variable. Because of its inevitably high dimensionality after the space-time discretization, the parabolic optimal control problem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative scheme is required. It then becomes important to find an easily implementable and efficient inexactness criterion to execute the internal iterations, and to prove the overall convergence rigorously for the resulting two-layer nested iterative scheme. To implement the ADMM efficiently, we propose an inexactness criterion that is independent of the mesh size of the involved discretization, and it can be executed automatically with no need to set empirically perceived constant accuracy a prior. The inexactness criterion turns out to allow us to solve the resulting optimal control problems with the only parabolic PDE constraints to medium or even low accuracy and thus saves computation significantly, yet convergence of the overall two-layer nested iterative scheme can be still guaranteed rigorously. Efficiency of this ADMM implementation is promisingly validated by preliminary numerical results. Our methodology can also be extended to a range of optimal control problems constrained by other linear PDEs such as elliptic equations, hyperbolic equations, convection-diffusion equations and fractional parabolic equations.

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“…For optimal control problems with an elliptic state equation α-robust preconditioners are provided by [15,14,12,16]. Some time-depending problems are addressed in [13,11], however, the rigorous analysis of α-robust preconditioners always required full observation (observation throughout the whole domain). A special case with a hyperbolic state equation was studied in [2].…”

Section: Introductionmentioning

confidence: 99%

Robust Preconditioners for Multiple Saddle Point Problems and Applications to Optimal Control Problems

Beigl¹,

Sogn²,

Zulehner³

2019

Preprint

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In this paper we consider multiple saddle point problems with block tridiagonal Hessian in a Hilbert space setting. Well-posedness and the related issue of preconditioning are discussed. We give a characterization of all block structured norms which ensure well-posedness of multiple saddle point problems as a helpful tool for constructing block diagonal preconditioners. We subsequently apply our findings to a general class of PDE-constrained optimal control problems containing a regularization parameter α and derive α-robust preconditioners for the corresponding optimality systems. Finally, we demonstrate the generality of our approach with two optimal control problems related to the heat and the wave equation, respectively. Preliminary numerical experiments support the feasibility of our method.

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Parameter-robust preconditioning for the optimal control of the wave equation

Cited by 10 publications

References 46 publications

Preconditioners for robust optimal control problems under uncertainty

Preconditioners for robust optimal control problems under uncertainty

Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond

Robust Preconditioners for Multiple Saddle Point Problems and Applications to Optimal Control Problems

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