A preconditioned iterative method for saddlepoint problems
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
Domain Decomposition Operator Splittings for the Solution of Parabolic Equations
SIAM Journal on Scientific Computing
Parallel methods for integrating ordinary differential equations
Communications of the ACM
Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific Computing
Analysis of the Parareal Time-Parallel Time-Integration Method
SIAM Journal on 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
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In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an “all at once” discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.