English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article, we derive a new, fast, and robust preconditioned iterativesolution strategy for the all-at-once solution of optimal control problems withtime-dependent PDEs as constraints, including the heat equation and thenon-steady convection--diffusion equation. After applying anoptimize-then-discretize approach, one is faced with continuous first-orderoptimality conditions consisting of a coupled system of PDEs. As opposed tomost work in preconditioning the resulting discretized systems, where a(first-order accurate) backward Euler method is used for the discretization ofthe time derivative, we employ a (second-order accurate) Crank--Nicolson methodin time. We apply a carefully tailored invertible transformation forsymmetrizing the matrix, and then derive an optimal preconditioner for thesaddle-point system obtained. The key components of this preconditioner are anaccurate mass matrix approximation, a good approximation of the Schurcomplement, and an appropriate multigrid process to apply this latterapproximation---these are constructed using our work in transforming the matrixsystem. We prove the optimality of the approximation of the Schur complementthrough bounds on the eigenvalues, and test our solver against a widely-usedpreconditioner for the linear system arising from a backward Eulerdiscretization. These demonstrate the effectiveness and robustness of oursolver with respect to mesh-sizes, regularization parameter, and diffusioncoefficient.

Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank-Nicolson Discretization in Time