Adaptive multilevel inexact SQP methods for PDE-constrained optimization. (English) Zbl 1214.49027
Summary: We present a class of inexact adaptive multilevel trust-region Sequential Quadratic Programming (SQP) methods for the efficient solution of optimization problems governed by nonlinear PDEs. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process (1) implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and (2) implementable accuracy requirements for iterative solvers of the linearized PDE and adjoint PDE on the current grid. We prove global convergence to a stationary point of the infinite-dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equations. Numerical results are presented.
MSC:
49M25 | Discrete approximations in optimal control |
65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
90C30 | Nonlinear programming |
90C55 | Methods of successive quadratic programming type |