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Lin Lin - One of the best experts on this subject based on the ideXlab platform.
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Parallel Transport dynamics for mixed quantum states with applications to time dependent density functional theory
arXiv: Numerical Analysis, 2021Co-Authors: Di Fang, Lin LinAbstract:Direct simulation of the von Neumann dynamics for a general (pure or mixed) quantum state can often be expensive. One prominent example is the real-time time-dependent density functional theory (rt-TDDFT), a widely used framework for the first principle description of many-electron dynamics in chemical and materials systems. Practical rt-TDDFT calculations often avoid the direct simulation of the von Neumann equation, and solve instead a set of Schrodinger equations, of which the dynamics is equivalent to that of the von Neumann equation. However, the time step size employed by the Schrodinger dynamics is often much smaller. In order to improve the time step size and the overall efficiency of the simulation, we generalize a recent work of the Parallel Transport (PT) dynamics for simulating pure states [An, Lin, Multiscale Model. Simul. 18, 612, 2020] to general quantum states. The PT dynamics provides the optimal gauge choice, and can employ a time step size comparable to that of the von Neumann dynamics. Going beyond the linear and near adiabatic regime in previous studies, we find that the error of the PT dynamics can be bounded by certain commutators between Hamiltonians, density matrices, and their derived quantities. Such a commutator structure is not present in the Schrodinger dynamics. We demonstrate that the Parallel Transport-implicit midpoint (PT-IM) method is a suitable method for simulating the PT dynamics, especially when the spectral radius of the Hamiltonian is large. The commutator structure of the error bound, and numerical results for model rt-TDDFT calculations in both linear and nonlinear regimes, confirm the advantage of the PT dynamics.
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quantum dynamics with the Parallel Transport gauge
Multiscale Modeling & Simulation, 2020Co-Authors: Lin LinAbstract:The dynamics of a closed quantum system is often studied with the direct evolution of the Schrodinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant ...
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Quantum Dynamics with the Parallel Transport Gauge
2020Co-Authors: An Dong, Lin LinAbstract:The dynamics of a closed quantum system is often studied with the direct evolution of the Schrodinger equation. In this paper, we propose that the gauge choice (i.e. degrees of freedom irrelevant to physical observables) of the Schrodinger equation can be generally non-optimal for numerical simulation. This can limit, and in some cases severely limit the time step size. We find that the optimal gauge choice is given by a Parallel Transport formulation. This Parallel Transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The Parallel Transport dynamics can be derived from a Hamiltonian structure, thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the Parallel Transport dynamics in the context of the singularly perturbed linear Schrodinger equation, and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrodinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse.Comment: SIAM Multiscale Model. Simul. accepte
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Parallel Transport time dependent density functional theory calculations with hybrid functional on summit
IEEE International Conference on High Performance Computing Data and Analytics, 2019Co-Authors: Weile Jia, Linwang Wang, Lin LinAbstract:Real-time time-dependent density functional theory (rt-TDDFT) with hybrid exchange-correlation functional has wide-ranging applications in chemistry and material science simulations. However, it can be thousands of times more expensive than a conventional ground state DFT simulation, and hence is limited to small systems. In this paper, we accelerate hybrid functional rt-TDDFT calculations using the Parallel Transport gauge formalism, and the GPU implementation on Summit. Our implementation can efficiently scale to 786 GPUs for a large system with 1536 silicon atoms, and the wall clock time is only 1.5 hours per femtosecond. This unprecedented speed enables the simulation of large systems with more than 1000 atoms using rt-TDDFT and hybrid functional.
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fast real time time dependent hybrid functional calculations with the Parallel Transport gauge and the adaptively compressed exchange formulation
Computer Physics Communications, 2019Co-Authors: Weile Jia, Lin LinAbstract:Abstract We present a new method to accelerate real-time time-dependent density functional theory (rt-TDDFT) calculations with hybrid exchange–correlation functionals. In the context of a large basis set such as planewaves and real space grids, the main computational bottleneck for large scale calculations is the application of the Fock exchange operator to the time-dependent orbitals. Our main goal is to reduce the frequency of applying the Fock exchange operator, without loss of accuracy. We achieve this by combining the recently developed Parallel Transport (PT) gauge formalism (Jia et al. J. Chem. Theory Comput. 2018) and the adaptively compressed exchange operator (ACE) formalism (Lin, J. Chem. Theory Comput. 2016). The PT gauge yields the slowest possible dynamics among all choices of gauge. When coupled with implicit time integrators such as the Crank–Nicolson (CN) scheme, the resulting PT–CN scheme can significantly increase the time step from sub-attoseconds to 10 − 100 attoseconds. At each time step t n , PT–CN requires the self-consistent solution of the orbitals at time t n + 1 . We use ACE to delay the update of the Fock exchange operator in this nonlinear system, while maintaining the same self-consistent solution. We verify the performance of the resulting PT–CN–ACE method by computing the absorption spectrum of a benzene molecule and the response of bulk silicon systems to an ultrafast laser pulse, using the planewave basis set and the HSE exchange–correlation functional. We report the strong and weak scaling of the PT–CN–ACE method for silicon systems ranging from 32 to 1024 atoms, on a Parallel computer with up to 2048 computational cores. Compared to standard explicit time integrators such as the 4 t h order Runge–Kutta method (RK4), we find that the PT–CN–ACE can reduce the frequency of the Fock exchange operator application by nearly 70 times, and the thus reduce the overall wall clock time by 46 times for the system with 1024 atoms. Hence our work enables hybrid functional rt-TDDFT calculations to be routinely performed with a large basis set for the first time.
Xavier Pennec - One of the best experts on this subject based on the ideXlab platform.
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Parallel Transport on Kendall Shape Spaces
2021Co-Authors: Nicolas Guigui, Elodie Maignant, Alain Trouvé, Xavier PennecAbstract:Kendall shape spaces are a widely used framework for the statistical analysis of shape data arising from many domains, often requiring the Parallel Transport as a tool to normalise time series data or Transport gradient in optimisation procedures. We present an implementation of the pole ladder, an algorithm to compute Parallel Transport based on geodesic Parallelograms and compare it to methods by integration of the Parallel Transport ordinary differential equation.
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A reduced Parallel Transport equation on Lie Groups with a left-invariant metric
2021Co-Authors: Nicolas Guigui, Xavier PennecAbstract:This paper presents a derivation of the Parallel Transport equation expressed in the Lie algebra of a Lie group endowed with a left-invariant metric. The use of this equation is exemplified on the group of rigid body motions SE(3), using basic numerical integration schemes, and compared to the pole ladder algorithm. This results in a stable and efficient implementation of Parallel Transport. The implementation leverages the python package geomstats and is available online.
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cardiac motion modeling with Parallel Transport and shape splines
arXiv: Computer Vision and Pattern Recognition, 2021Co-Authors: Nicolas Guigui, Pamela Moceri, Maxime Sermesant, Xavier PennecAbstract:In cases of pressure or volume overload, probing cardiac function may be difficult because of the interactions between shape and this http URL this work, we use the LDDMM framework and Parallel Transport to estimate and reorient deformations of the right ventricle. We then propose a normalization procedure for the amplitude of the deformation, and a second-order spline model to represent the full cardiac contraction. The method is applied to 3D meshes of the right ventricle extracted from echocardiographic sequences of 314 patients divided into three disease categories and a control group. We find significant differences between pathologies in the model parameters, revealing insights into the dynamics of each disease.
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Parallel Transport with pole ladder a third order scheme in affine connection spaces which is exact in affine symmetric spaces
arXiv: Differential Geometry, 2018Co-Authors: Xavier PennecAbstract:Parallel Transport is an important step in many discrete algorithms for statistical computing on manifolds. Numerical methods based on Jacobi fields or geodesics Parallelograms are currently used in geometric data processing. In this last class, pole ladder is a simplification of Schild's ladder for the Parallel Transport along geodesics that was shown to be particularly simple and numerically stable in Lie groups. So far, these methods were shown to be first order approximations of the Riemannian Parallel Transport, but higher order error terms are difficult to establish. In this paper, we build on a BCH-type formula on affine connection spaces to establish the behavior of one pole ladder step up to order 5. It is remarkable that the scheme is of order three in general affine connection spaces with a symmetric connection, much higher than expected. Moreover, the fourth-order term involves the covariant derivative of the curvature only, which is vanishing in locally symmetric space. We show that pole ladder is actually locally exact in these spaces, and even almost surely globally exact in Riemannian symmetric manifolds. These properties make pole ladder a very attractive alternative to other methods in general affine manifolds with a symmetric connection.
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Noname manuscript No. (will be inserted by the editor) Efficient Parallel Transport of Deformations in Time Series of Images: from Schild’s to Pole Ladder
2015Co-Authors: Marco Lorenzi, Xavier PennecAbstract:the date of receipt and acceptance should be inserted later Abstract Group-wise analysis of time series of images requires to compare lon-gitudinal evolutions of images observed on different subjects. In medical imaging, longitudinal anatomical changes can be modeled thanks to non-rigid registration of follow-up images. The comparison of longitudinal trajectories requires the trans-port (or ”normalization”) of longitudinal deformations in a common reference frame. We previously proposed an effective computational scheme based on the Schild’s ladder for the Parallel Transport of diffeomorphic deformations parame-terized by tangent velocity fields, based on the construction of a geodesic Parallel-ogram on a manifold. Schild’s ladder may be however inefficient for Transporting longitudinal deformations from image time series of multiple time points, in which the computation of the geodesic diagonals is required several times. We propose here a new algorithm, the pole ladder, in which one diagonal of the Parallelogram is the baseline-to-reference frame geodesic. This drastically reduces the number of geodesics to compute. Moreover, differently from the Schild’s ladder, the pole ladder is symmetric with respect to the baseline-to-reference frame geodesic. From the theoretical point of view, we show that the pole ladder is rigorously equivalent to the Schild’s ladder when Transporting along geodesics. From the practical point of view, we establish the computational advantages and demonstrate the effective-ness of this very simple method by comparing with standard methods of Transport on simulated images with progressing brain atrophy. Finally, we illustrate its ap-plication to a clinical problem: the measurement of the longitudinal progressio
Weile Jia - One of the best experts on this subject based on the ideXlab platform.
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Parallel Transport time dependent density functional theory calculations with hybrid functional on summit
IEEE International Conference on High Performance Computing Data and Analytics, 2019Co-Authors: Weile Jia, Linwang Wang, Lin LinAbstract:Real-time time-dependent density functional theory (rt-TDDFT) with hybrid exchange-correlation functional has wide-ranging applications in chemistry and material science simulations. However, it can be thousands of times more expensive than a conventional ground state DFT simulation, and hence is limited to small systems. In this paper, we accelerate hybrid functional rt-TDDFT calculations using the Parallel Transport gauge formalism, and the GPU implementation on Summit. Our implementation can efficiently scale to 786 GPUs for a large system with 1536 silicon atoms, and the wall clock time is only 1.5 hours per femtosecond. This unprecedented speed enables the simulation of large systems with more than 1000 atoms using rt-TDDFT and hybrid functional.
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fast real time time dependent hybrid functional calculations with the Parallel Transport gauge and the adaptively compressed exchange formulation
Computer Physics Communications, 2019Co-Authors: Weile Jia, Lin LinAbstract:Abstract We present a new method to accelerate real-time time-dependent density functional theory (rt-TDDFT) calculations with hybrid exchange–correlation functionals. In the context of a large basis set such as planewaves and real space grids, the main computational bottleneck for large scale calculations is the application of the Fock exchange operator to the time-dependent orbitals. Our main goal is to reduce the frequency of applying the Fock exchange operator, without loss of accuracy. We achieve this by combining the recently developed Parallel Transport (PT) gauge formalism (Jia et al. J. Chem. Theory Comput. 2018) and the adaptively compressed exchange operator (ACE) formalism (Lin, J. Chem. Theory Comput. 2016). The PT gauge yields the slowest possible dynamics among all choices of gauge. When coupled with implicit time integrators such as the Crank–Nicolson (CN) scheme, the resulting PT–CN scheme can significantly increase the time step from sub-attoseconds to 10 − 100 attoseconds. At each time step t n , PT–CN requires the self-consistent solution of the orbitals at time t n + 1 . We use ACE to delay the update of the Fock exchange operator in this nonlinear system, while maintaining the same self-consistent solution. We verify the performance of the resulting PT–CN–ACE method by computing the absorption spectrum of a benzene molecule and the response of bulk silicon systems to an ultrafast laser pulse, using the planewave basis set and the HSE exchange–correlation functional. We report the strong and weak scaling of the PT–CN–ACE method for silicon systems ranging from 32 to 1024 atoms, on a Parallel computer with up to 2048 computational cores. Compared to standard explicit time integrators such as the 4 t h order Runge–Kutta method (RK4), we find that the PT–CN–ACE can reduce the frequency of the Fock exchange operator application by nearly 70 times, and the thus reduce the overall wall clock time by 46 times for the system with 1024 atoms. Hence our work enables hybrid functional rt-TDDFT calculations to be routinely performed with a large basis set for the first time.
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
Journal of Chemical Theory and Computation, 2018Co-Authors: Weile Jia, Lin Lin, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation, because of the fast oscillation of electron wave functions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schrodinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of 10-100 attoseconds using a planewave basis set. Thanks to the significant increase of the size of the time step, we also demonstrate that the new method is more than 10 times faster, in terms of the wall clock time, when compared to the standard explicit fourth-order Runge-Kutta time integrator for silicon systems ranging from 32 to 1024 atoms.
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fast real time time dependent hybrid functional calculations with the Parallel Transport gauge and the adaptively compressed exchange formulation
arXiv: Computational Physics, 2018Co-Authors: Weile Jia, Lin LinAbstract:We present a new method to accelerate real time-time dependent density functional theory (rt-TDDFT) calculations with hybrid exchange-correlation functionals. For large basis set, the computational bottleneck for large scale calculations is the application of the Fock exchange operator to the time-dependent orbitals. Our main goal is to reduce the frequency of applying the Fock exchange operator, without loss of accuracy. We achieve this by combining the recently developed Parallel Transport (PT) gauge formalism and the adaptively compressed exchange operator (ACE) formalism. The PT gauge yields the slowest possible dynamics among all choices of gauge. When coupled with implicit time integrators such as the Crank-Nicolson (CN) scheme, the resulting PT-CN scheme can significantly increase the time step from sub-attoseconds to 10-100 attoseconds. At each time step $t_{n}$, PT-CN requires the self-consistent solution of the orbitals at time $t_{n+1}$. We use ACE to delay the update of the Fock exchange operator in this nonlinear system, while maintaining the same self-consistent solution. We verify the performance of the resulting PT-CN-ACE method by computing the absorption spectrum of a benzene molecule and the response of bulk silicon systems to an ultrafast laser pulse, using the planewave basis set and the HSE functional. We report the strong and weak scaling of the PT-CN-ACE method for silicon systems ranging from 32 to 1024 atoms, with up to 2048 computational cores. Compared to standard explicit time integrators such as the 4th order Runge-Kutta method (RK4), the PT-CN-ACE can reduce the Fock exchange operator application by nearly 70 times, thus reduce the overall wall clock time time by 46 times for the system with 1024 atoms. Hence our work enables hybrid functional rt-TDDFT calculations to be routinely performed with a large basis set for the first time.
Linwang Wang - One of the best experts on this subject based on the ideXlab platform.
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Parallel Transport time dependent density functional theory calculations with hybrid functional on summit
IEEE International Conference on High Performance Computing Data and Analytics, 2019Co-Authors: Weile Jia, Linwang Wang, Lin LinAbstract:Real-time time-dependent density functional theory (rt-TDDFT) with hybrid exchange-correlation functional has wide-ranging applications in chemistry and material science simulations. However, it can be thousands of times more expensive than a conventional ground state DFT simulation, and hence is limited to small systems. In this paper, we accelerate hybrid functional rt-TDDFT calculations using the Parallel Transport gauge formalism, and the GPU implementation on Summit. Our implementation can efficiently scale to 786 GPUs for a large system with 1536 silicon atoms, and the wall clock time is only 1.5 hours per femtosecond. This unprecedented speed enables the simulation of large systems with more than 1000 atoms using rt-TDDFT and hybrid functional.
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
Journal of Chemical Theory and Computation, 2018Co-Authors: Dong An, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation, because of the fast oscillation of electron wave functions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schrodinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of 10–100 attoseconds using a planewave basis set. Thanks to the significant increase of the size of the tim...
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
Journal of Chemical Theory and Computation, 2018Co-Authors: Weile Jia, Lin Lin, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation, because of the fast oscillation of electron wave functions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schrodinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of 10-100 attoseconds using a planewave basis set. Thanks to the significant increase of the size of the time step, we also demonstrate that the new method is more than 10 times faster, in terms of the wall clock time, when compared to the standard explicit fourth-order Runge-Kutta time integrator for silicon systems ranging from 32 to 1024 atoms.
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
arXiv: Computational Physics, 2018Co-Authors: Dong An, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation due to the fast oscillation of electron wavefunctions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schr\"odinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of $10\sim 100$ attoseconds in a planewave basis set, and is no less than $5\sim 10$ times faster when compared to the standard explicit 4th order Runge-Kutta time integrator. Thanks to the significant increase of the size of the time step, we also demonstrate that the new method is more than 10 times faster in terms of the wall clock time when compared to the standard explicit 4th order Runge-Kutta time integrator for silicon systems ranging from 32 to 1024 atoms
Dong An - One of the best experts on this subject based on the ideXlab platform.
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
Journal of Chemical Theory and Computation, 2018Co-Authors: Dong An, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation, because of the fast oscillation of electron wave functions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schrodinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of 10–100 attoseconds using a planewave basis set. Thanks to the significant increase of the size of the tim...
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fast real time time dependent density functional theory calculations with the Parallel Transport gauge
arXiv: Computational Physics, 2018Co-Authors: Dong An, Linwang WangAbstract:Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation due to the fast oscillation of electron wavefunctions, which significantly limits its range of applicability for the study of ultrafast dynamics. In this paper, we demonstrate that such oscillation can be considerably reduced by optimizing the gauge choice using the Parallel Transport formalism. RT-TDDFT calculations can thus be significantly accelerated using a combination of the Parallel Transport gauge and implicit integrators, and the resulting scheme can be used to accelerate any electronic structure software that uses a Schr\"odinger representation. Using absorption spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for example, we show that the new method can utilize a time step that is on the order of $10\sim 100$ attoseconds in a planewave basis set, and is no less than $5\sim 10$ times faster when compared to the standard explicit 4th order Runge-Kutta time integrator. Thanks to the significant increase of the size of the time step, we also demonstrate that the new method is more than 10 times faster in terms of the wall clock time when compared to the standard explicit 4th order Runge-Kutta time integrator for silicon systems ranging from 32 to 1024 atoms
