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 work, we propose a novel preconditioned Krylov subspace method forsolving an optimal control problem of wave equations, after explicitlyidentifying the asymptotic spectral distribution of the involved sequence oflinear coefficient matrices from the optimal control problem. Namely, we firstshow that the all-at-once system stemming from the wave control problem isassociated to a structured coefficient matrix-sequence possessing an eigenvaluedistribution. Then, based on such a spectral distribution of which the symbolis explicitly identified, we develop an ideal preconditioner and twoparallel-in-time preconditioners for the saddle point system composed of twoblock Toeplitz matrices. For the ideal preconditioner, we show that theeigenvalues of the preconditioned matrix-sequence all belong to the set$\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup\left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading tomesh-independent convergence when the minimal residual method is employed. Theproposed {parallel-in-time} preconditioners can be implemented efficientlyusing fast Fourier transforms or discrete sine transforms, and theireffectiveness is theoretically shown in the sense that the eigenvalues of thepreconditioned matrix-sequences are clustered around $\pm 1$, which leads torapid convergence. When these parallel-in-time preconditioners are not fastdiagonalizable, we further propose modified versions which can be efficientlyinverted. Several numerical examples are reported to verify our derivedlocalization and spectral distribution result and to support the effectivenessof our proposed preconditioners and the related advantages with respect to therelevant literature.

A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory