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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Reduced-order modeling (ROM) commonly refers to the construction, based on afew solutions (referred to as snapshots) of an expensive discretized partialdifferential equation (PDE), and the subsequent application of low-dimensionaldiscretizations of partial differential equations (PDEs) that can be used tomore efficiently treat problems in control and optimization, uncertaintyquantification, and other settings that require multiple approximate PDEsolutions. In this work, a ROM is developed and tested for the treatment ofnonlinear PDEs whose solutions bifurcate as input parameter values change. Insuch cases, the parameter domain can be subdivided into subregions, each ofwhich corresponds to a different branch of solutions. Popular ROM approachessuch as proper orthogonal decomposition (POD), results in a globallow-dimensional basis that does no respect not take advantage of the oftenlarge differences in the PDE solutions corresponding to different subregions.Instead, in the new method, the k-means algorithm is used to cluster snapshotsso that within cluster snapshots are similar to each other and are dissimilarto those in other clusters. This is followed by the construction of local PODbases, one for each cluster. The method also can detect which cluster a newparameter point belongs to, after which the local basis corresponding to thatcluster is used to determine a ROM approximation. Numerical experiments showthe effectiveness of the method both for problems for which bifurcation causecontinuous and discontinuous changes in the solution of the PDE.

A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions