Publications

Authors:

Caliari, Marco; Cassini, Fabio; Zivcovich, Franco

Title:

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

Year:

2024

Type of item:

Articolo in Rivista

Tipologia ANVUR:

Articolo su rivista

Language:

Inglese

Referee:

No

Name of journal:

CALCOLO

ISSN of journal:

0008-0624

N° Volume:

61

Page numbers:

1-28

Keyword:

Semilinear evolutionary problems; Kronecker sum; Exponential integrators; $\varphi$-functions; $\mu$-mode approach; Tucker operator

Short description of contents:

We present a method for computing actions of the exponential-like $\varphi$-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 and squaring technique. The resulting algorithm, which we call PHIKS, evaluates the required actions by means of $\mu$-mode products involving exponentials of the small sized matrices $A_\mu$, without forming the large sized matrix $K$ itself. PHIKS, which profits from the highly efficient level 3 BLAS, is designed to compute different $\varphi$-functions applied on the same vector or a linear combination of actions of $\varphi$-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of PHIKS in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge--Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of $\varphi$-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection--diffusion--reaction, Allen--Cahn, and Brusselator equations show the superiority of the proposed $\mu$-mode approach.

Web page:

https://link.springer.com/article/10.1007/s10092-024-00610-3

Product ID:

141475

Handle IRIS:

11562/1139948

Last Modified:

October 5, 2024

Bibliographic citation:

Caliari, Marco; Cassini, Fabio; Zivcovich, Franco, A $\mu$-mode approach for exponential integrators: actions of $\varphi$-functions of Kronecker sums «CALCOLO» , vol. 61 , 2024 , pp. 1-28

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