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

Beigl, Alexander; Sogn, Jarle; Zulehner, Walter

doi:10.1137/19m1308426

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…We observe that the block operator A has a block tridiagonal form. Such tridiagonal operators are studied in [25,4]. We use the Schur complement preconditioner proposed in [25]:…”

Section: Analysis Of the Continuous Problemmentioning

confidence: 99%

“…There are two main problems with (1.1)-(1.2), namely: 1) potential sharp gradients leading to non-physical oscillations in the numerical solution and 2) ill-posedness due to limited observations, this is, when O is a subset of Ω. Motivated by the fact that higher regularity has been exploited in the cases with limited observations [17,25,4], we derive order optimal preconditioners via stability analysis in non-standard Sobolev spaces.…”

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confidence: 99%

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Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

Mardal¹,

Sogn²,

Takacs³

2020

Preprint

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In this paper we analyze an optimization problem with limited observation governed by a convection-diffusion-reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.

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“…We observe that the block operator A has a block tridiagonal form. Such tridiagonal operators are studied in [25,4]. We use the Schur complement preconditioner proposed in [25]:…”

Section: Analysis Of the Continuous Problemmentioning

confidence: 99%

mentioning

confidence: 99%

Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

Mardal¹,

Sogn²,

Takacs³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…There has been a recent surge of interest in the iterative solution of multiple saddle-point problems, and our work adds to the increasing body of literature that considers these problems. Recent papers that provide interesting analysis are, for example, [2,3,6,24,36].…”

mentioning

confidence: 99%

Eigenvalue Bounds for Double Saddle-Point Systems

Bradley¹,

Greif²

2021

Preprint

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We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.

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Preconditioning of discrete state- and control-constrained optimal control convection-diffusion problems

Dravins,

Neytcheva

2023

Calcolo

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We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.

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

Cited by 8 publications

References 14 publications

Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

Eigenvalue Bounds for Double Saddle-Point Systems

Preconditioning of discrete state- and control-constrained optimal control convection-diffusion problems

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