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 a novel method for computing actions of the so-called$\varphi$-functions for a Kronecker sum $K$ of $d$ arbitrary matrices $A_\mu$.It is based on the approximation of the integral representation of the$\varphi$-functions by Gaussian quadrature formulas combined with a scaling andsquaring technique. The resulting algorithm, which we call PHIKS, evaluates therequired actions by means of $\mu$-mode products involving exponentials of thesmall sized matrices $A_\mu$, without using the large sized matrix $K$ itself.PHIKS, which profits from the highly efficient level 3 BLAS, is designed tocompute different $\varphi$-functions applied on the same vector or a linearcombination of actions of $\varphi$-functions applied on different vectors. Inaddition, due to the underlying scaling and squaring technique, the desiredquantities are available simultaneously at suitable time scales. All thesefeatures allow the effective usage of PHIKS in the exponential integrationcontext. In particular, we tested our newly designed method on popularexponential Runge-Kutta integrators of stiff order from one to four, incomparison with state-of-the-art algorithms for computing actions of$\varphi$-functions. Our numerical experiments with discretized semilinearevolutionary 2D or 3D advection-diffusion-reaction, Allen-Cahn, and Brusselatorequations show the superiority of the $\mu$-mode approach of PHIKS.

A $\mu$-mode approach for exponential integrators: actions of $\varphi$-functions of Kronecker sums