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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We present a combination technique based on mixed differences of both spatialapproximations and quadrature formulae for the stochastic variables to solveefficiently a class of Optimal Control Problems (OCPs) constrained by randompartial differential equations. The method requires to solve the OCP forseveral low-fidelity spatial grids and quadrature formulae for the objectivefunctional. All the computed solutions are then linearly combined to get afinal approximation which, under suitable regularity assumptions, preserves thesame accuracy of fine tensor product approximations, while drastically reducingthe computational cost. The combination technique involves only tensor productquadrature formulae, thus the discretized OCPs preserve the convexity of thecontinuous OCP. Hence, the combination technique avoids the inconveniences ofMultilevel Monte Carlo and/or sparse grids approaches, but remains suitable forhigh dimensional problems. The manuscript presents an a-priori procedure tochoose the most important mixed differences and an asymptotic complexityanalysis, which states that the asymptotic complexity is exclusively determinedby the spatial solver. Numerical experiments validate the results.

A combination technique for optimal control problems constrained by random PDEs