## Multigrid solution of a distributed optimal control problem constrained by the Stokes equations

## Abstract

We construct multigrid preconditioners to accelerate the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schur-complement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.

- Authors:

- Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States). Dept. of Mathematics and Statistics

- Publication Date:

- Research Org.:
- Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF)

- OSTI Identifier:
- 1493158

- Grant/Contract Number:
- SC0005455; DMS-1016177

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Applied Mathematics and Computation

- Additional Journal Information:
- Journal Volume: 219; Journal Issue: 10; Journal ID: ISSN 0096-3003

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; multigrid methods; PDE-constrained optimization; Stokes equations; finite elements; optimal order preconditioners

### Citation Formats

```
Drăgănescu, Andrei, and Soane, Ana Maria. Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. United States: N. p., 2013.
Web. doi:10.1016/j.amc.2012.11.070.
```

```
Drăgănescu, Andrei, & Soane, Ana Maria. Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. United States. doi:10.1016/j.amc.2012.11.070.
```

```
Drăgănescu, Andrei, and Soane, Ana Maria. Thu .
"Multigrid solution of a distributed optimal control problem constrained by the Stokes equations". United States. doi:10.1016/j.amc.2012.11.070. https://www.osti.gov/servlets/purl/1493158.
```

```
@article{osti_1493158,
```

title = {Multigrid solution of a distributed optimal control problem constrained by the Stokes equations},

author = {Drăgănescu, Andrei and Soane, Ana Maria},

abstractNote = {We construct multigrid preconditioners to accelerate the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system (the KKT system) connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schur-complement of the block associated with the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.},

doi = {10.1016/j.amc.2012.11.070},

journal = {Applied Mathematics and Computation},

number = 10,

volume = 219,

place = {United States},

year = {2013},

month = {1}

}