Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e1479"><mml:mi>φ</mml:mi></mml:math>-functions of matrices via quadrature

Croci, Matteo; Muñoz‐Matute, Judit

doi:10.1016/j.jocs.2023.101966

Cited by 7 publications

(2 citation statements)

References 31 publications

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“…(2018). When extending to three dimensions, it may also be useful to exploit the vertical horizontal tensor product structure in the exponential, as discussed in Croci and Muñoz‐Matute (2022).…”

Section: Description Of the Methodsmentioning

confidence: 99%

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Time‐parallel integration and phase averaging for the nonlinear shallow‐water equations on the sphere

Yamazaki

Cotter

2023

Quart J Royal Meteoro Soc

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We describe a proof‐of‐concept development and application of a phase‐averaging technique to the nonlinear rotating shallow‐water equations on the sphere, discretised using compatible finite‐element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: (1) we have developed a stable matrix exponential specific to finite elements and (2) we have developed a parallel finite averaging procedure. Following recent studies, we consider finite‐width phase‐averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm that there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi‐implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work.

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Section: Description Of the Methodsmentioning

confidence: 99%

“…Some more details of the implementation and examination of parallel performance are provided in Schreiber et al (2018). When extending to three dimensions, it may also be useful to exploit the vertical horizontal tensor product structure in the exponential, as discussed in Croci and Muñoz-Matute (2022).…”

Section: Chebyshev Exponentiationmentioning

confidence: 99%

Time‐parallel integration and phase averaging for the nonlinear shallow‐water equations on the sphere

Yamazaki

Cotter

2023

Quart J Royal Meteoro Soc

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

Caliari,

Cassini,

Zivcovich

2024

Calcolo

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We present a method for computing actions of the exponential-like $$\varphi $$ φ -functions for a Kronecker sum K of d arbitrary matrices $$A_\mu $$ A μ . It is based on the approximation of the integral representation of the $$\varphi $$ φ -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 $$\mu $$ μ -mode products involving exponentials of the small sized matrices $$A_\mu $$ 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 $$\varphi $$ φ -functions applied on the same vector or a linear combination of actions of $$\varphi $$ φ -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 $$\varphi $$ φ -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 $$\mu $$ μ -mode approach.

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A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

Caliari,

Cassini

2024

Journal of Computational Physics

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Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ-functions of matrices via quadrature

Cited by 7 publications

References 31 publications

Time‐parallel integration and phase averaging for the nonlinear shallow‐water equations on the sphere

Time‐parallel integration and phase averaging for the nonlinear shallow‐water equations on the sphere

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

A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

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