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

Marco Caliari, Fabio Cassini, Franco Zivcovich

TL;DR

The paper introduces PHIKS, a $\mu$-mode framework to approximate actions of the exponential-like $\varphi$-functions for Kronecker-sum matrices without forming the large operator. By combining Gaussian quadrature on scaled Kronecker sums with a scaling-and-squaring strategy and efficient Tucker operations, PHIKS can compute multiple $\varphi$-functions and linear combinations on the same vector, suitable for high-order exponential integrators. The method is validated on stiff PDE-inspired ODEs (ADR, Allen–Cahn, Brusselator) and multiple exponential Runge–Kutta schemes, showing substantial speedups over state-of-the-art approaches while maintaining accuracy. The results suggest PHIKS offers significant practical impact for large-scale, tensor-structured problems and paves the way for extensions to fractional diffusion and higher-order time discretizations.

Abstract

We present a method for computing actions of the exponential-like $\varphi$-functions for a Kronecker sum $K$ of $d$ arbitrary matrices $A_μ$. 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 $μ$-mode products involving exponentials of the small sized matrices $A_μ$, 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 on 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 $μ$-mode approach.

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

TL;DR

The paper introduces PHIKS, a -mode framework to approximate actions of the exponential-like -functions for Kronecker-sum matrices without forming the large operator. By combining Gaussian quadrature on scaled Kronecker sums with a scaling-and-squaring strategy and efficient Tucker operations, PHIKS can compute multiple -functions and linear combinations on the same vector, suitable for high-order exponential integrators. The method is validated on stiff PDE-inspired ODEs (ADR, Allen–Cahn, Brusselator) and multiple exponential Runge–Kutta schemes, showing substantial speedups over state-of-the-art approaches while maintaining accuracy. The results suggest PHIKS offers significant practical impact for large-scale, tensor-structured problems and paves the way for extensions to fractional diffusion and higher-order time discretizations.

Abstract

We present a method for computing actions of the exponential-like -functions for a Kronecker sum of arbitrary matrices . It is based on the approximation of the integral representation of the -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 -mode products involving exponentials of the small sized matrices , without forming the large sized matrix itself. PHIKS, which profits from the highly efficient level 3 BLAS, is designed to compute different -functions applied on the same vector or a linear combination of actions of -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 on popular exponential Runge--Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of -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 -mode approach.
Paper Structure (16 sections, 1 theorem, 72 equations, 6 figures, 2 tables)

This paper contains 16 sections, 1 theorem, 72 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A mu-mode approach for evolutionary equations in Kronecker form
  3. Approximation of $\varphi$-functions of a Kronecker sum
  4. Actions of $\varphi$-functions on the same vector
  5. Linear combination of actions of $\varphi$-functions
  6. Choice of $s$, $q$, and quadrature formula
  7. Numerical experiments
  8. Code validation
  9. Evolutionary advection--diffusion--reaction equation
  10. Exponential Euler
  11. Exponential Runge--Kutta scheme of order two
  12. Allen--Cahn equation
  13. Brusselator model
  14. An exponential Runge--Kutta scheme of order four with five stages
  15. An exponential Runge--Kutta scheme of order four with six stages
  16. ...and 1 more sections

Key Result

Lemma 1

Let $L_\mu \in \mathbb{C}^{n_\mu\times n_\mu}$ be matrices, with $\mu=1,\ldots,d$, and let $\boldsymbol v\in \mathbb{C}^N$, with $N=n_1\cdots n_d$. Let $\boldsymbol V \in \mathbb{C}^{n_1 \times \cdots \times n_d}$ be an order-$d$ tensor such that $\mathrm{vec}(\boldsymbol V) = \boldsymbol v$, where

Figures (6)

  • Figure 1: Relative difference between kiops and phiks, measured in infinity norm, for the actions $\varphi_\ell(K/2^j)\boldsymbol v$, $\ell=1,\ldots,p$, and for the linear combinations $\Phi_j(K)\boldsymbol v_{1,2,\ldots,p}$ in the code validation. The plots refer to $j=0$ and $d=3$ (top left), $j=1$ and $d=3$ (top right), $j=0$ and $d=6$ (bottom left), $j=1$ and $d=6$ (bottom right).
  • Figure 2: Rate of convergence of the exponential Euler scheme \ref{['eq:expEullc']} for the semidiscretization of ADR equation \ref{['eq:ADR']} with $n_1=n_2=n_3=n=20$ discretization points (left) and wall-clock time in seconds for increasing number $n$ of discretization points and 250 time steps up to final time $T=0.1$ (right).
  • Figure 3: Rate of convergence of the ETD2RK scheme \ref{['eq:ETD2RKlc']} for the semidiscretization of ADR equation \ref{['eq:ADR']} with $n_1=n_2=n_3=n=20$ discretization points (left) and wall-clock time in seconds for increasing number $n$ of discretization points and 100 time steps up to final time $T=0.1$ (right).
  • Figure 4: Rate of convergence of the exponential Runge--Kutta scheme \ref{['eq:expRK3s3']} for the semidiscretization of Allen--Cahn equation \ref{['eq:AC']} with $n_1=n_2=n=21$ discretization points (left) and wall-clock time in seconds for increasing number $n$ of discretization points and 20 time steps up to final time $T=0.025$ (right).
  • Figure 5: Rate of convergence of the exponential Runge--Kutta scheme \ref{['eq:expRK4s5']} for the semidiscretization of Brusselator model \ref{['eq:bruss']} with $n_1=n_2=n_3=n=11$ discretization points (left) and wall-clock time in seconds for increasing number $n$ of discretization points and 20 time steps up to final time $T=1$ (right).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Remark 1