GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
Aerodynamic design via control theory
Journal of Scientific Computing
Journal of Computational Physics
Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows
Journal of Computational Physics
Adjoint and defect error bounding and correction for functional estimates
Journal of Computational Physics
Journal of Computational Physics
Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations
SIAM Journal on Numerical Analysis
Error estimation and adaptation for functional outputs in time-dependent flow problems
Journal of Computational Physics
Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations
SIAM Journal on Scientific Computing
Output error estimation for summation-by-parts finite-difference schemes
Journal of Computational Physics
Summation-by-parts operators and high-order quadrature
Journal of Computational and Applied Mathematics
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The paper describes an algorithm for PDE-constrained optimization that controls numerical errors using error estimates and grid adaptation during the optimization process. A key aspect of the algorithm is the use of adjoint variables to estimate errors in the first-order optimality conditions. Multilevel optimization is used to drive the optimality conditions and their estimated errors below a specified tolerance. The error estimate requires two additional adjoint solutions, but only at the beginning and end of each optimization cycle. Moreover, the adjoint systems can be formed and solved with limited additional infrastructure beyond that found in typical PDE-constrained optimization algorithms. The approach is general and can accommodate both reduced-space and full-space formulations of the optimization problem. The algorithm is illustrated using the inverse design of a nozzle constrained by the quasi-one-dimensional Euler equations.