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 查看全文>>