TY - JOUR
TI - A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control
AU - Grossmann, Christian
AU - Winkler, Max
TI - A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control
AB - 
	The quadratic loss penalty is a well known technique for optimization and control problems to treat constraints. In the present paper they are applied to handle control bounds in a boundary control problems with semilinear elliptic state equations. Unlike in the case of finite dimensional optimization for infinite dimensional problems the order of convergence could only be roughly estimated, but numerical experiments revealed a clearly better convergence behavior with constants independent of the dimension of the used discretization. The main result in the present paper is the proof of sharp convergence bounds for both, the finite und infinite dimensional problem with a mesh-independence in case of the discretization. Further, to achieve an efficient realization of penalty methods the principle of control reduction is applied, i.e. the control variable is represented by the adjoint state variable by means of
	some nonlinear function. The resulting optimality system this way depends only on the state and adjoint state. This system is discretized by conforming linear finite elements. Numerical experiments show exactly the theoretically predicted behavior of the studied penalty technique.
VL - 2012
IS - Volume 21
PY - 2012
SN - 1732-3916
C1 - 2083-8476
SP - 9
EP - 26
DO - 10.4467/20838476SI.12.001.0811
UR - https://ejournals.eu/en/journal/schedae-informaticae/article/a-mesh-independence-principle-for-quadratic-penalties-applied-to-semilinear-elliptic-boundary-control
KW - Optimal boundary control
KW - mesh-independence principle
KW - weakly nonlinear elliptic equations
KW - penalty methods for control constraints.