Invited Session Thu.2.MA 415

Thursday, 13:15 - 14:45 h, Room: MA 415

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

Theory and methods for PDE-constrained optimization problems with inequalities


Chair: Michael Ulbrich



Thursday, 13:15 - 13:40 h, Room: MA 415, Talk 1

Francisco José Silva Alvarez
Characterization of quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

Coauthors: Terence Bayen, Frédéric Bonnans


In this work, we are concerned with the following optimal control problem:

minu J(u) := ∫Ω l(x,yu(x),u(x)) d x,

under bounds constraints on the control u, and where yu is the unique solution of

 -Δ y(x) + \varphi(x,y(x),u(x)) = 0,  \mathrm{for}  x ∈ Ω,\
\hspace{3.75cm} y(x) = 0, \hspace{0.12cm} \mathrm{for}  x ∈ \partial Ω.

We extend to strong solutions classical second order analysis results, which are usually established for weak solutions. We mean by strong solution a control \bar{u} that satisfies:

There exists ε>0 such that J(\bar{u) ≤ J(u) for all u with || yu-y\bar{u}|| ≤ ε.}

The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of the optimization of elliptic equations. Our main result is a characterization of local quadratic growth for the cost function J around a strong minimum.



Thursday, 13:45 - 14:10 h, Room: MA 415, Talk 2

Martin Weiser
Goal-oriented estimation for nonlinear optimal control problems


In optimal control problems with elliptic PDE constraints,

min J(y,u)  
s.t.  c(y,u) = 0,

the value of the cost functional is a natural quantity of interest for goal-oriented
error estimation and mesh refinement. The talk will discuss the difference between
the all-at-once error quantity J(yh,uh)-J\mathrm{opt} introduced by Becker/\allowbreak Kapp/\allowbreak Rannacher
and the black-box error quantity J(y(uh),uh)-J\mathrm{opt}. Both qualitative and
quantitative differences will be addressed for linear-quadratic problems.
In the second part, the black-box approach will be extended to smooth nonlinear problems
and will result in a novel accuracy matching for inexact Newton methods. Quantitative aspects
are illustrated on numerical examples including interior point regularizations of inequality constrained problems.



Thursday, 14:15 - 14:40 h, Room: MA 415, Talk 3

Florian Kruse
An infeasible interior point method for optimal control problems with state constraints

Coauthor: Michael Ulbrich


We present an infeasible interior point method for pointwise state
constrained optimal control problems with elliptic PDEs. A smoothed
constraint violation functional is used to develop a self-concordant
barrier approach in an infinite-dimensional setting.
For the resulting algorithm we provide a detailed convergence analysis
in function space. This includes a rate of convergence and a rigorous
measure for the proximity of the actual iterate to both the path of
minimizers and the solution of the problem.
Moreover, we report on numerical experiments to illustrate the
efficiency and the mesh independence of this algorithm.


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