High Resolution Forward And Inverse Earthquake Modeling on Terascale Computers
Proceedings of the 2003 ACM/IEEE conference on Supercomputing
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Adjoint concepts for the optimal control of Burgers equation
Computational Optimization and Applications
Adaptive finite element methods for nonlinear inverse problems
Proceedings of the 2009 ACM symposium on Applied Computing
Large-Scale Scientific Computing
Block iterative algorithms for the solution of parabolic optimal control problems
VECPAR'06 Proceedings of the 7th international conference on High performance computing for computational science
Optimization under adaptive error control for finite element based simulations
Computational Mechanics
Approximate Nullspace Iterations for KKT Systems
SIAM Journal on Matrix Analysis and Applications
Multilevel Algorithms for Large-Scale Interior Point Methods
SIAM Journal on Scientific Computing
Optimal Solvers for PDE-Constrained Optimization
SIAM Journal on Scientific Computing
Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems
SIAM Journal on Scientific Computing
A multigrid method for constrained optimal control problems
Journal of Computational and Applied Mathematics
A preconditioning technique for a class of PDE-constrained optimization problems
Advances in Computational Mathematics
Fast Algorithms for Source Identification Problems with Elliptic PDE Constraints
SIAM Journal on Imaging Sciences
SIAM Journal on Matrix Analysis and Applications
An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations
SIAM Journal on Scientific Computing
Adaptive Multilevel Inexact SQP Methods for PDE-Constrained Optimization
SIAM Journal on Optimization
A Robust Multigrid Method for Elliptic Optimal Control Problems
SIAM Journal on Numerical Analysis
Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations
SIAM Journal on Scientific Computing
Inversion of airborne contaminants in a regional model
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part III
Domain decomposition methods for PDE constrained optimization problems
VECPAR'04 Proceedings of the 6th international conference on High Performance Computing for Computational Science
Journal of Computational Physics
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Large-scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. Reduced quasi-Newton sequential quadratic programming (SQP) methods are state-of-the-art approaches for such problems. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of using a full space Newton solver combined with an approximate reduced space quasi-Newton SQP preconditioner. The basic components of the method are Newton solution of the first-order optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the Karush--Kuhn--Tucker (KKT) linear systems arising at each Newton iteration using a symmetric quasi-minimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation initialized by a two-step stationary method. Accordingly, we term the new method {\it Lagrange--Newton--Krylov--Schur} (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In part I of this two-part article, we investigate the effectiveness of the KKT linear system solver. We test our method on two optimal control problems in which the state constraints are described by the steady-state Stokes equations. The objective is to minimize dissipation or the deviation from a given velocity field; the control variables are the boundary velocities. Numerical experiments on up to 256 Cray T3E processors and on an SGI Origin 2000 include scalability and performance assessment of the LNKS algorithm and comparisons with reduced SQP for up to $1,000,000$ state and 50,000 decision variables. In part II of the article, we address globalization and inexactness issues, and apply LNKS to the optimal control of the steady incompressible Navier--Stokes equations.