Magnus integrators on multicore CPUs and GPUs

Auer, N.; Einkemmer, Lukas; Kandolf, Peter; Ostermann, Alexander

doi:10.1016/j.cpc.2018.02.019

Cited by 22 publications

(18 citation statements)

References 19 publications

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“…In this work, in addition to this EMR, we have focused on the following higher order formulas: Order 4:One fourth‐order method that requires of only two exponentials was reported in References 36 and 37. It is given by:

Γ_{2}^{[4]} (A) = \exp {- i Δ t (a_{11} A [t_{1}] + a_{12} A [t_{2}])} \exp {- i Δ t (a_{12} A [t_{1}] + a_{11} A [t_{2}])} .

The constants a ij and t i are:

\{\begin{matrix} a_{11} = \frac{3 prefix- 2 \sqrt{3}}{12}, a_{12} = \frac{3 + 2 \sqrt{3}}{12}, \\ c_{1} = \frac{1}{2} prefix- \frac{\sqrt{3}}{6}, c_{2} = \frac{1}{2} + \frac{\sqrt{3}}{6}, \\ t_{1} = t + c_{1} normalΔ t, t_{2} = t + c_{2} normalΔ t . \end{matrix}

Another fourth‐order method, which is computationally more expensive yet allegedly more precise, was suggested by Bader et al 38 It makes use of four exponentials (and therefore we will denote it

…”

Section: Commutator‐free Magnus Expansion Methods For Ehrenfest Dynamcismentioning

confidence: 99%

Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

Pueyo

Blanes

Castro

2020

Comp and Math Methods

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Hybrid quantum‐classical systems combine both classical and quantum degrees of freedom. Typically, in Chemistry, Molecular Physics, or Materials Science, the classical degrees of freedom describe atomic nuclei (or cations with frozen core electrons), whereas the quantum particles are the electrons. Although many possible hybrid dynamical models exist, the basic one is the so‐called Ehrenfest dynamics that results from the straightforward partial classical limit applied to the full quantum Schrödinger equation. Few numerical methods have been developed specifically for the integration of this type of systems. Here we present a preliminary study of the performance of a family of recently developed propagators: the (quasi) commutator‐free Magnus expansions. These methods, however, were initially designed for nonautonomous linear equations. We employ them for the nonlinear Ehrenfest system, by approximating the state value at each time step in the propagation, using an extrapolation from previous time steps.

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Γ_{2}^{[4]} (A) = \exp {- i Δ t (a_{11} A [t_{1}] + a_{12} A [t_{2}])} \exp {- i Δ t (a_{12} A [t_{1}] + a_{11} A [t_{2}])} .

The constants a ij and t i are:

\{\begin{matrix} a_{11} = \frac{3 prefix- 2 \sqrt{3}}{12}, a_{12} = \frac{3 + 2 \sqrt{3}}{12}, \\ c_{1} = \frac{1}{2} prefix- \frac{\sqrt{3}}{6}, c_{2} = \frac{1}{2} + \frac{\sqrt{3}}{6}, \\ t_{1} = t + c_{1} normalΔ t, t_{2} = t + c_{2} normalΔ t . \end{matrix}

Another fourth‐order method, which is computationally more expensive yet allegedly more precise, was suggested by Bader et al 38 It makes use of four exponentials (and therefore we will denote it

…”

Section: Commutator‐free Magnus Expansion Methods For Ehrenfest Dynamcismentioning

confidence: 99%

Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

Pueyo

Blanes

Castro

2020

Comp and Math Methods

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“…32, and labelled as the "Method 4" in page 6 of Ref. 61. This method, that we will refer to in the following by "CFM4", is given by:…”

Section: Commutator-freementioning

confidence: 99%

Propagators for the Time-Dependent Kohn–Sham Equations: Multistep, Runge–Kutta, Exponential Runge–Kutta, and Commutator Free Magnus Methods

Pueyo

Marques

Rubio

et al. 2018

J. Chem. Theory Comput.

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We examine various integration schemes for the time-dependent Kohn-Sham equations. Contrary to the time-dependent Schrödinger's equation, this set of equations is nonlinear, due to the dependence of the Hamiltonian on the electronic density. We discuss some of their exact properties, and in particular their symplectic structure. Four different families of propagators are considered, specifically the linear multistep, Runge-Kutta, exponential Runge-Kutta, and the commutator-free Magnus schemes. These have been chosen because they have been largely ignored in the past for time-dependent electronic structure calculations. The performance is analyzed in terms of cost-versus-accuracy. The clear winner, in terms of robustness, simplicity, and efficiency is a simplified version of a fourth-order commutator-free Magnus integrator. However, in some specific cases, other propagators, such as some implicit versions of the multistep methods, may be useful.

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“…On the other hand, as mentioned in Subsection 2.1 it is possible to derive an explicit form of the solution of the DLE given by (4). Following [45], we use a high-order quadrature rule to compute an approximation to the integral term.…”

Section: Implementationsmentioning

confidence: 99%

“…In a splitting scheme, the basic building block is instead the computation of the action of a matrix exponential on a skinny matrix. Speed-ups have previously been observed for applications where matrix exponentials are multiplied by vectors [4,22], see also [25]. In these works, a speedup is generally not observed for "small" matrices (n 1000), and the speed-up is of limited size when the matrices are sparse rather than dense.…”

Section: Introductionmentioning

confidence: 96%

GPU acceleration of splitting schemes applied to differential matrix equations

2019

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We consider differential Lyapunov and Riccati equations, and generalized versions thereof. Such equations arise in many different areas and are especially important within the field of optimal control. In order to approximate their solution, one may use several different kinds of numerical methods. Of these, splitting schemes are often a very competitive choice. In this article, we investigate the use of graphical processing units (GPUs) to parallelize such schemes and thereby further increase their effectiveness. According to our numerical experiments, large speed-ups are often observed for sufficiently large matrices. We also provide a comparison between different splitting strategies, demonstrating that splitting the equations into a moderate number of subproblems is generally optimal.

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Magnus integrators on multicore CPUs and GPUs

Cited by 22 publications

References 19 publications

Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

Performance of fourth and sixth‐order commutator‐free Magnus expansion integrators for Ehrenfest dynamics

Propagators for the Time-Dependent Kohn–Sham Equations: Multistep, Runge–Kutta, Exponential Runge–Kutta, and Commutator Free Magnus Methods

GPU acceleration of splitting schemes applied to differential matrix equations

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