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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider a sparse grid collocation method in conjunction with a timediscretization of the differential equations for computing expectations offunctionals of solutions to differential equations perturbed by time-dependentwhite noise. We first analyze the error of Smolyak's sparse grid collocationused to evaluate expectations of functionals of solutions to stochasticdifferential equations discretized by the Euler scheme. We show theoreticallyand numerically that this algorithm can have satisfactory accuracy for smallmagnitude of noise or small integration time, however it does not convergeneither with decrease of the Euler scheme's time step size nor with increase ofSmolyak's sparse grid level. Subsequently, we use this method as a buildingblock for proposing a new algorithm by combining sparse grid collocation with arecursive procedure. This approach allows us to numerically integrate linearstochastic partial differential equations over longer times, which isillustrated in numerical tests on a stochastic advection-diffusion equation.

A Recursive Sparse Grid Collocation Method for Differential Equations with White Noise