Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

Autor: Monika Wolfmayr
Rok vydání: 2020
Předmět:
Optimization problem
time-periodic condition
multiharmonic finite element method
Discretization
two-sided bounds
Systems and Control (eess.SY)
010103 numerical & computational mathematics
System of linear equations
Electrical Engineering and Systems Science - Systems and Control
01 natural sciences
Upper and lower bounds
Saddle point
FOS: Mathematics
FOS: Electrical engineering
electronic engineering
information engineering
Applied mathematics
Mathematics - Numerical Analysis
Boundary value problem
0101 mathematics
Mathematics - Optimization and Control
Mathematics
osittaisdifferentiaaliyhtälöt
35Kxx
65M60
65M70
65M15
65K10
parabolic optimal control problems
Numerical Analysis (math.NA)
matemaattinen optimointi
Optimal control
Finite element method
010101 applied mathematics
Computational Mathematics
Computational Theory and Mathematics
Optimization and Control (math.OC)
Modeling and Simulation
a posteriori error analysis
numeerinen analyysi
guaranteed lower bounds
Zdroj: Computers & Mathematics with Applications. 80:1050-1072
ISSN: 0898-1221
DOI: 10.1016/j.camwa.2020.04.021
Popis: In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.
Comment: 27 pages, 10 tables. This work extends the analysis of arXiv:1511.05699
Databáze: OpenAIRE
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