1 research outputs found
A fast and memory-efficient spectral Galerkin scheme for distributed elliptic optimal control problems
Many scientific and engineering challenges can be formulated as optimization
problems which are constrained by partial differential equations (PDEs). These
include inverse problems, control problems, and design problems. As a major
challenge, the associated optimization procedures are inherently large-scale.
To ensure computational tractability, the design of efficient and robust
iterative methods becomes imperative. To meet this challenge, this paper
introduces a fast and memory-efficient preconditioned iterative scheme for a
class of distributed optimal control problems governed by
convection-diffusion-reaction (CDR) equations. As an alternative to low-order
discretizations and Schur-complement block preconditioners, the scheme combines
a high-order spectral Galerkin method with an efficient preconditioner tailored
specifically for the CDR application. The preconditioner is matrix-free and can
be applied within linear complexity where the proportionality constant is
small. Numerical results demonstrate that the preconditioner is ideal in the
sense that appropriate Krylov subspace methods converge within a low number of
iterations, independently of the problem size and the Tikhonov regularization
parameter