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

We present and analyze a novel sparse polynomial technique for approximatinghigh-dimensional Hilbert-valued functions, with application to parameterizedpartial differential equations (PDEs) with deterministic and stochastic inputs.Our theoretical framework treats the function approximation problem as a jointsparse recovery problem, where the set of jointly sparse vectors is possiblyinfinite. To achieve the simultaneous reconstruction of Hilbert-valuedfunctions in both parametric domain and Hilbert space, we propose a novelmixed-norm based $\ell_1$ regularization method that exploits both energy andsparsity. Our approach requires extensions of concepts such as the restrictedisometry and null space properties, allowing us to prove recovery guaranteesfor sparse Hilbert-valued function reconstruction. We complement the enclosedtheory with an algorithm for Hilbert-valued recovery, based on standardforward-backward algorithm, meanwhile establishing its strong convergence inthe considered infinite-dimensional setting. Finally, we demonstrate theminimal sample complexity requirements of our approach, relative to otherpopular methods, with numerical experiments approximating the solutions ofhigh-dimensional parameterized elliptic PDEs.

Reconstructing high-dimensional Hilbert-valued functions via compressed sensing