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

Existing implementations of gradient-based optimisation methods typicallyassume that the problem is posed in Euclidean space. When solving optimalityproblems on function spaces, the functional derivative is then inaccuratelyrepresented with respect to $\ell^2$ instead of the inner product induced bythe function space. This error manifests as a mesh dependence in the number ofiterations required to solve the optimisation problem. In this paper, ananalytic estimate is derived for this iteration count in the case of a simpleand generic discretised optimisation problem. The system analysed is thesteepest descent method applied to a finite element problem. The estimate isbased on Kantorovich's inequality and on an upper bound for the conditionnumber of Galerkin mass matrices. Computer simulations validate the iterationnumber estimate. Similar numerical results are found for a more complexoptimisation problem constrained by a partial differential equation.Representing the functional derivative with respect to the inner productinduced by the continuous control space leads to mesh independent convergence.

An iteration count estimate for a mesh-dependent steepest descent method based on finite elements and Riesz inner product representation