Abstract
We consider a class of nonlinear saddle point problems with various applications in PDEs and optimal control problems and propose an algorithmic framework based on some inexact Uzawa methods in the literature. Under mild conditions, the convergence of this algorithmic frame- work is uniformly proved and the linear convergence rate is estimated. We take an elliptic optimal control problem with control constraints as an example to illustrate how to choose application-tailored preconditioners to generate specific and efficient algorithms by the algorithmic framework. The resulting algorithm does not need to solve any optimization subproblems or systems of linear equa- tions in its iteration; each of its iterations only requires the projection onto a simple admissible set, four algebraic multigrid V-cycles, and a few matrix-vector multiplications. Its numerical efficiency is then demonstrated by some preliminary numerical results. © 2019 Society for Industrial and Applied Mathematics.
| Original language | English |
|---|---|
| Pages (from-to) | 2656-2684 |
| Journal | SIAM Journal on Numerical Analysis |
| Volume | 57 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 2019 |
| Externally published | Yes |
Bibliographical note
Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].Funding
The work of the authors was partially supported by the seed fund for basic research at the University of Hong Kong, grant 201807159005, and the General Research Fund from the Hong Kong Research Grants Council, grant 12302318.
Research Keywords
- Convergence analysis
- Elliptic optimal control problem
- Inexact Uzawa method
- Linear convergence rate
- Nonlinear saddle point problems
- Subregularity