# Nonlinear optimized Schwarz preconditioner for elliptic optimal control problems

@article{Ciaramella2021NonlinearOS, title={Nonlinear optimized Schwarz preconditioner for elliptic optimal control problems}, author={Gabriele Ciaramella and Felix Kwok and Georg M{\"u}ller}, journal={ArXiv}, year={2021}, volume={abs/2104.00274} }

where ‖·‖Lr denotes the usual norm for L (Ω) with 1 ≤ r ≤ ∞, the functions yd, f ∈ L (Ω) are given, and the scalar parameters b, c, β ≥ 0 and ν, β ≥ 0 are known. Our model includes problems such as the simplified GinzburgLandau superconductivity equation as well as inverse problems where Lregularization is used to enhance sparsity of the control function u. For simplicity, the domain Ω ⊂ R is assumed to be a rectangle (0, L̃)× (0, L̂). The function φ : R → R is assumed to be of class C, with…

## One Citation

Convergence analysis of the Schwarz alternating method for unconstrained elliptic optimal control problems

- Mathematics, Computer Science
- 2022

Numerical results corroborate the theoretical results and show that with α decreasing to zero, the method will converge faster, and gives some exposition of this phenomenon.

## References

SHOWING 1-7 OF 7 REFERENCES

Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L1 Cost Functional

- Mathematics, Computer ScienceSIAM J. Optim.
- 2012

Semilinear elliptic optimal control problems involving the $L^1$ norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori…

Elliptic optimal control problems with L1-control cost and applications for the placement of control devices

- Mathematics, Computer ScienceComput. Optim. Appl.
- 2009

For solving the non-differentiable optimal control problem, a semismooth Newton method is proposed that can be stated and analyzed in function space and converges locally with a superlinear rate.

Optimal Control of Partial Differential Equations: Theory, Methods and Applications

- Mathematics
- 2010

Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in…

Newton Methods for the Optimal Control of Closed Quantum Spin Systems

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2015

An efficient and robust computational framework for solving closed quantum spin optimal-control and exact-controllability problems with control constraints is presented and existence and regularity of optimal controls are proved.

Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-Sized Subdomains: Part I

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2017

It is proved numerically that the convergence of the Schwarz method in this case does not depend on the number of subdomains, even without coarse correction, by analyzing the Schwarz iteration matrices in Fourier space and evaluating corresponding norms in a simplified setting.

Optimized Schwarz Methods

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2006

This paper analyzes these new methods for symmetric positive definite problems and shows their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.

On the Scalability of Classical One-Level Domain-Decomposition Methods

- MathematicsVietnam Journal of Mathematics
- 2018

One-level domain-decomposition methods are in general not scalable, and coarse corrections are needed to obtain scalability. It has however recently been observed in applications in computational…