Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints

Herzog, Roland; Sachs, E. W.

doi:10.1137/090779127

Cited by 87 publications

(87 citation statements)

References 19 publications

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“…We refer to [31] and [24]. For the Stokes problem (Example 1.2) one can find such estimates in [7,Section 6.2.4] and for an optimal control problem as in Example 1.3, we refer to [19].…”

Section: Linear Convergence Analysismentioning

confidence: 99%

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Herzog¹,

Sachs²

2015

SIAM J. Numer. Anal.

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“…We refer to [31] and [24]. For the Stokes problem (Example 1.2) one can find such estimates in [7,Section 6.2.4] and for an optimal control problem as in Example 1.3, we refer to [19].…”

Section: Linear Convergence Analysismentioning

confidence: 99%

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Herzog¹,

Sachs²

2015

SIAM J. Numer. Anal.

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“…However, the size of the discretized optimality system can make its solution computationally prohibitive for large-scale problems. Finding a good quality preconditioner matrix for the full (or reduced) optimality system is also far from trivial, and is the subject of current research (see, e.g., [45,46,47]). Another approach, and the one we employ in this paper, is reduced-space optimization.…”

Section: The Adaptive Inverse Problem Frameworkmentioning

confidence: 99%

Space–time adaptive solution of inverse problems with the discrete adjoint method

Alexe

Sandu

2014

Journal of Computational Physics

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Abstract. Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, h/p-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem. Space-time adaptive solution of inverse problems with the discrete adjoint method2

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“…Their efficient solution thus calls for preconditioned iterative solvers. Due to space restrictions, we refer to [20][21][22][23][24][25][26][27] for more on the issue of preconditioning.…”

Section: Wwwgamm-mitteilungenorgmentioning

confidence: 99%

Algorithms for PDE‐constrained optimization

2010

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MSC (2000)49-M05, 49-M37, 76-D55, 90-C06, Some first and second order algorithmic approaches for the solution of PDE-constrained optimization problems are reviewed. An optimal control problem for the stationary Navier-Stokes system with pointwise control constraints serves as an illustrative example. Some issues in treating inequality constraints for the state variable and alternative objective functions are also discussed.

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Preconditioned Conjugate Gradient Method for Optimal Control Problems with Control and State Constraints

Cited by 87 publications

References 19 publications

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Space–time adaptive solution of inverse problems with the discrete adjoint method

Algorithms for PDE‐constrained optimization

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