Iterative Solution Methods for Large-Scale Constrained Saddle-Point Problems

Laitinen, Erkki K.; Лапин, А. В.

doi:10.1007/978-94-007-5288-7_2

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…The statements of Theorem 1 on the convergence (y ε , u ε , λ ε ) to (y, u, λ 0 ) as ε → 0 are valid for the solutions to problems (11) and (13), and error estimate (7) is also true:…”

Section: Remarkmentioning

confidence: 87%

“…Let (y, u, λ 0 ) be a solution to problem (11) and (y ε , u ε , λ ε ) be a solution to problem (13), and let the assumptions (12) be satisfied. Then the following estimates take place with the different choices of the matrix D:…”

Section: Remarkmentioning

confidence: 99%

“…The second approach to the solution of optimal control problems, including the state-constrained ones, is first construct a mesh approximation of the original differential problem and further solve the derived optimization problem. This approach is used, in particular, in [8][9][10][11][12][13] for constructing preconditioned Uzawa-type iterative solution methods for mesh optimal control problems with control and stet constraints.…”

Section: Introductionmentioning

confidence: 99%

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Method of penalization for the state equation for an elliptical optimal control problem

Лапин

Zalyalov

2015

Russ Math.

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We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss-Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.

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Section: Remarkmentioning

confidence: 87%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Method of penalization for the state equation for an elliptical optimal control problem

Лапин

Zalyalov

2015

Russ Math.

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“…with constants independent on τ and h. As this preconditioner we take = T D LL . Then method (20) for problem (13) with this preconditioner reads as follows:…”

Section: Saddle Point Problem and Iterative Solution Methodsmentioning

confidence: 99%

Iterative Solution of Mesh Constrained Optimal Control Problems with Two-Level Mesh Approximations of Parabolic State Equation

Лапин¹,

Laitinen²

2018

JAMP

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We consider a linear-quadratical optimal control problem of a system governed by parabolic equation with distributed in right-hand side control and control and state constraints. We construct a mesh approximation of this problem using different two-level approximations of the state equation, ADI and fractional steps approximations in time among others. Iterative solution methods are investigated for all constructed approximations of the optimal control problem. Their implementation can be carried out in parallel manner.There are few results in the area of numerical solution methods for the constrained parabolic optimal control problems. One of the approaches to overcome the difficulty connected with low regularity of the solutions is using Lavrentiev regularization. This approach has been used in [6] [7]. In [8] [9] [10] [11] a priori error estimates for space-time discretizations of linear-quadratic parabolic optimal control problems have been obtained for problems.

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“…Another approach to constructing iterative methods for finite-dimensional constrained saddle point problems have been proposed in [5] and developed in [6,7]. This approach is based on the transformation of the original saddle problem to an equivalent one with block-triangle and positive definite matrix acting on the direct variables.…”

Section: Introductionmentioning

confidence: 99%

Efficiently implementable iterative methods for linear elliptic variational inequalities with constraints on the gradient of solution

Kashtanov

Лапин

2015

Russ Math.

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We construct and investigate a new iterative solution method for a finite-dimensional constrained saddle point problem. The results are applied to prove the convergence of different iterative methods for mesh approximations of variational inequalities with constraints on the gradient of solution. In particular, we prove the convergence of two-stage iterative methods. The main advantage of the proposed methods is the simplicity of their implementation. The numerical testing demonstrates high convergence rate of the methods.

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Iterative Solution Methods for Large-Scale Constrained Saddle-Point Problems

Cited by 6 publications

References 15 publications

Method of penalization for the state equation for an elliptical optimal control problem

Method of penalization for the state equation for an elliptical optimal control problem

Iterative Solution of Mesh Constrained Optimal Control Problems with Two-Level Mesh Approximations of Parabolic State Equation

Efficiently implementable iterative methods for linear elliptic variational inequalities with constraints on the gradient of solution

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