All-at-once solution of time-dependent Stokes control

Stoll, Martin; Wathen, Andrew J.

doi:10.1016/j.jcp.2012.08.039

Cited by 74 publications

(112 citation statements)

References 39 publications

(64 reference statements)

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“…As our major concern in this paper is to study efficient solvers resulting from the discretization of our model problems, we proceed next to recall the two common approaches in the literature to solve these optimization problems [26]. The first method is the so-called optimize-then-discretize (OTD) approach.…”

Section: Representation Of Random Inputsmentioning

confidence: 99%

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. The goal of this paper is the efficient numerical simulation of optimization problems governed by either steady-state or unsteady partial differential equations involving random coefficients. This class of problems often leads to prohibitively high dimensional saddle-point systems with tensor product structure, especially when discretized with the stochastic Galerkin finite element method. Here, we derive and analyze robust Schur complement-based block-diagonal preconditioners for solving the resulting stochastic optimality systems with all-at-once low-rank iterative solvers. Moreover, we illustrate the effectiveness of our solvers with numerical experiments.Key words. stochastic Galerkin system, iterative methods, PDE-constrained optimization, saddle-point system, low-rank solution, preconditioning, Schur complement AMS subject classifications. 35R60, 60H15, 60H35, 65N22, 65F10, 65F50 DOI. 10.1137/15M10185021. Introduction. Optimization problems constrained by deterministic steadystate partial differential equations (PDEs) are computationally challenging. This is even more so if the constraints are deterministic unsteady PDEs since one would then need to solve a system of PDEs coupled globally in time and space, and time-stepping methods quickly reach their limitations due to the enormous demand for storage [25]. Yet, more challenging than the aforementioned are problems constrained by unsteady PDEs involving (countably many) parametric or uncertain inputs. This class of problems often leads to prohibitively high dimensional linear systems with Kronecker product structure, especially when discretized with the stochastic Galerkin finite element method (SGFEM). Moreover, a typical model for an optimal control problem with stochastic inputs (SOCP) will usually be used for the quantification of the statistics of the system response-a task that could in turn result in additional enormous computational expense.Stochastic finite element-based solvers for a large range of PDEs with random data have been studied extensively [1,3,12,21,24]. However, optimization problems constrained by PDEs with random inputs have, in our opinion, not yet received adequate attention. Hence, this study is aimed at pushing the research frontier with respect to the numerical simulation of the latter class of stochastic problems (that is, SOCPs) toward larger and more challenging problems. Some of the papers on SOCPs include [12,13,24]. While [12] studies the existence and the uniqueness of solutions to control problems constrained by elliptic PDEs with random inputs, the emphasis in

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Section: Representation Of Random Inputsmentioning

confidence: 99%

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Benner¹,

Onwunta²,

Stoll³

2016

SIAM J. Matrix Anal. & Appl.

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“…To this end, we have investigated preconditioners to be used in iterative methods based on the ideas of Wathen et al [19,42]. As major result we could adapt known Schur complement approximations to our problem setting.…”

Section: Nested Iterationmentioning

confidence: 99%

“…Avoiding this explicit projection leads to solving a series of largescale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners based on the approaches of Wathen et al [19,42]. In addition, the main results can be extended to the non-symmetric case of linearized Navier-Stokes equations.…”

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confidence: 99%

“…After showing why the projection idea of [22] can be used as numerical realization of the Leray projector, we end up with large-scale saddle point systems as the major ingredients to compute the Riccati feedback via this projection approach. In Section 3, we introduce the solution strategy for these saddle point systems based on the ideas of [19,42]. Afterwards, we show numerical examples in Section 4 and summarize the ideas and results in Section 5.…”

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confidence: 99%

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Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

Benner¹,

Saak²,

Stoll³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the Stokes equations that describe instationary, incompressible flows for low and moderate Reynolds numbers. After a mixed finite element discretization [23] we get a differential-algebraic system of differential index two [45]. To reduce this index, we follow the analytic ideas of Raymond [34,35,36] coupled with the projection idea of Heinkenschloss et al. [22]. Avoiding this explicit projection leads to solving a series of largescale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners based on the approaches of Wathen et al. [19,42]. In addition, the main results can be extended to the non-symmetric case of linearized Navier-Stokes equations. We conclude with numerical examples showcasing the performance of our preconditioned iterative saddle point solver.

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“…, M ). Our discretization here follows an approach presented in Hinze et al (2008); Stoll and Wathen (2013), which follows the paradigm that the discretize-then-optimize and the optimize-then-discretize approach coincide. More details on the derivation of allat-once methods can be found in Stoll and Wathen (2010).…”

Section: Optimization Of the Allen-cahn Systemmentioning

confidence: 99%

Optimal Control for Allen-Cahn Equations Enhanced by Model Predictive Control

Benner

Stoll

2013

IFAC Proceedings Volumes

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The Allen-Cahn equation is a simple model of a nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. It has applications in many areas including material sciences, biology, geology, as well as image processing. We will consider a simple scalar Allen-Cahn equation subject to distributed control. Here, the nonlinear reaction term is obtained from using the standard double-well potential, leading to a cubic nonlinearity. We will describe a nonlinear feedback control strategy based on the concept of Model Predictive Control (MPC). We also show how to obtain the open-loop trajectory and control using numerical techniques for PDE-constrained optimization. The feedback control scheme is then applied to the spatially semi-discretized nonlinear optimal control problem. For the prediction and control step within the MPC scheme, we apply a linear-quadratic regulator/Gaussian design problem. The arising computational challenge consisting in solving the associated large-scale algebraic Riccati equations has already been shown in the literature to be feasible using reasonably fine discretizations.

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All-at-once solution of time-dependent Stokes control

Cited by 74 publications

References 39 publications

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

Optimal Control for Allen-Cahn Equations Enhanced by Model Predictive Control

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