A μ\mu-mode integrator for solving evolution equations in Kronecker form

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed

Productive and efficient computational science through domain-specific abstractions

In an ideal world, scientific applications are computationally efficient, maintainable and composable and allow scientists to work very productively. We argue that these goals are achievable for a specific application field by choosing suitable domain-specific abstractions that encapsulate domain knowledge with a high degree of expressiveness. This thesis demonstrates the design and composition of domain-specific abstractions by abstracting the stages a scientist goes through in formulating a problem of numerically solving a partial differential equation. Domain knowledge is used to transform this problem into a different, lower level representation and decompose it into parts which can be solved using existing tools. A system for the portable solution of partial differential equations using the finite element method on unstructured meshes is formulated, in which contributions from different scientific communities are composed to solve sophisticated problems. The concrete implementations of these domain-specific abstractions are Firedrake and PyOP2. Firedrake allows scientists to describe variational forms and discretisations for linear and non-linear finite element problems symbolically, in a notation very close to their mathematical models. PyOP2 abstracts the performance-portable parallel execution of local computations over the mesh on a range of hardware architectures, targeting multi-core CPUs, GPUs and accelerators. Thereby, a separation of concerns is achieved, in which Firedrake encapsulates domain knowledge about the finite element method separately from its efficient parallel execution in PyOP2, which in turn is completely agnostic to the higher abstraction layer. As a consequence of the composability of those abstractions, optimised implementations for different hardware architectures can be automatically generated without any changes to a single high-level source. Performance matches or exceeds what is realistically attainable by hand-written code. Firedrake and PyOP2 are combined to form a tool chain that is demonstrated to be competitive with or faster than available alternatives on a wide range of different finite element problems.Open Acces

Computational and Theoretical Developements for (Time Dependent) Density Functional Theory

En esta tesis se presentan avances computacionales y teoricos en la teoria de funcionales de la densidad (DFT) y en la teoria de funcionales de la densidad dependientes del tiempo (TDDFT). Hemos explorado una posible nueva ruta para la mejora de los funcionales de intercambio y correlacion (XCF) en DFT, comprobado y desarrollado propagadores numericos para TDDFT, y aplicado una combinacion de la teoria de control optimo con TDDFT.En los ultimos anos, DFT se ha convertido en el metodo mas utilizado en el area de estructura electronica gracias a su inigualable relacion entre coste y precision. Podemos usar DFT para calcular multitud de propiedades fisicas y quimicas de atomos, moleculas, nanoestructuras, y materia macroscopica. El factor principal que determina la precision que podemos alcanzar usando DFT es el XCF, un objeto desconocido para el cual se han propuesto cientos de aproximaciones distintas. Algunas de estas aproximaciones funcionan correctamente en ciertas situaciones, pero a dia de hoy no existe un XCF que pueda aplicarse con certeza sobre su validez a un sistema arbitrario. Mas aun, no hay una forma sistematica de refinar estos funcionales. Proponemos y exploramos, para sistemas unidimensionales, una nueva manera de estudiarlos y optimizarlos basada en establecer una relacion con la interaccion entre electrones.TDDFT es la extension de DFT a problemas dependientes del tiempo y problemas conestados excitados, y es tambien uno de los metodos mas populares (a veces el unico metodo que se puede poner en practica) en la comunidad de estructura electronica para tratar conellos. De nuevo, la razon detras de su popularidad reside en su relacion precision/coste computacional, que nos permite tratar sistemas mayores y mas complejos. Puede usarse en combinacion con la dinamica de Ehrenfest, un tipo de dinamica molecular no adiabatica.Hemos ido mas alla y hemos combinado TDDFT y la dinamica de Ehrenfest con la teoria de control optimo, creando un instrumento que nos permite, por ejemplo, predecir la forma de los pulsos laser que inducen una explosion de Coulomb en clusters de sodio. A pesar del buen rendimiento computacional de TDDFT en comparacion con otros metodos, hallamos que el coste de estos calculos era bastante elevado.Motivados por este hecho, tambien dedicamos una parte del trabajo de la tesis a la investigacion computacional. En particular, hemos estudiado e implementado familias de propagadores numericos que no se habian examinado en el contexto de TDDFT. Mas concretamente, metodos con varios pasos previos, formulas Runge-Kutta exponenciales, y las expansiones de Magnus sin conmutadores. Finalmente, hemos implementado modificaciones de estas expansiones de Magnus sin conmutadores para la propagacion de las ecuaciones clasico-cuanticas que resultan de la combinacion de la dinamica de Ehrenfest con TDDFT.In this thesis we present computational and theoretical developments for density functional theory (DFT) and time dependent density functional theory (TDDFT). We have explored a new possible route to improve exchange and correlation functionals (XCF) in DFT, tested and developed numerical propagators for TDDFT, and applied a combination of optimal control theory with TDDFT. In recent years, DFT has become the most used method in the electronic structure field thanks to its unparalleled precision/computational cost relationship. We can use DFT to accurately calculate many physical and chemical properties of atoms, molecules, nanostructures, and bulk materials. The main factor that determines the precision that we can obtain using DFT is the XCF, an unknown object for which hundreds of different approximations have been proposed. Some of these approximations work well enough for certain situations, but to this day there is no XCF that can be reliably applied to any arbitrary system. Moreover, there is no clear way for a systematic refinement of these functionals. We propose and explore, for one-dimensional systems, a new way to optimize them, based on establishing a relationship with the electron-electron interaction. TDDFT is the extension of DFT to time-dependent and excited-states problems, and it is also one of the most popular methods (sometimes the only practical one) in the electronic structure community to deal with them. Once again, the reason behind its popularity is its accuracy/computational cost ratio, which allows us to tackle bigger, more complex systems. It can be used in combination with Ehrenfest dynamics, a non-adiabatic type of molecular dynamics. We have furthermore combined both TDDFT and Ehrenfest dynamics with optimal control theory, a scheme that has allowed us, for example, to predict the shapes of the laser pulses that induce a Coulomb explosion in different sodium clusters. Despite the good numerical performance of TDDFT compared to other methods, we found that these computations were still quite expensive. Motivated by this fact, we have also dedicated a part of the thesis work to computational research. In particular, we have studied and implemented families of numerical propagators that had not been tested in the context of TDDFT. More concretely, linear multistep schemes, exponential Runge-Kutta formulas, and commutator-free Magnus expansions. Moreover, we have implemented modifications of these commutator-free Magnus methods for the propagation of the classical-quantum equations that result of combining Ehrenfest dynamics with TDDFT.<br /

Proceedings of the Third International Workshop on Sustainable Ultrascale Computing Systems (NESUS 2016) Sofia, Bulgaria

Proceedings of: Third International Workshop on Sustainable Ultrascale Computing Systems (NESUS 2016). Sofia (Bulgaria), October, 6-7, 2016

Propagators for Quantum-Classical Models: Commutator-Free Magnus Methods

[EN] We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrodinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrodinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.A.C. acknowledges support from the MINECO FIS2017-82426-P grant. S.B. acknowledges the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program "Geometry, compatibility and structure preservation in computational differential equations (2019)", EPSRC grant number EP/R014604/1. S.B. also acknowledges funding by the Ministerio de Economia y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE) and the Ministerio de Ciencia Innovacion y Universidades, through Programa de Estancias de profesores e investigadores senior en centros extranjeros, incluido el Programa "Salvador de Madariaga" 2019 (PRX19/00295).Gómez Pueyo, A.; Blanes Zamora, S.; Castro, A. (2020). Propagators for Quantum-Classical Models: Commutator-Free Magnus Methods. Journal of Chemical Theory and Computation. 16(3):1420-1430. https://doi.org/10.1021/acs.jctc.9b01031S14201430163Molner, S. P. (1999). The Art of Molecular Dynamics Simulation (Rapaport, D. C.). Journal of Chemical Education, 76(2), 171. doi:10.1021/ed076p171Fermi, E.; Pasta, J.; Ulam, S. Studies of Nonlinear Problems I, Los Alamos Report LA 1940; Los Alamos Scientific Laboratory, 1955; reproduced in Nonlinear Wave Motion. Providence, RI, 1974.Alder, B. J., & Wainwright, T. E. (1959). Studies in Molecular Dynamics. I. General Method. The Journal of Chemical Physics, 31(2), 459-466. doi:10.1063/1.1730376Bornemann, F. A., Nettesheim, P., & Schütte, C. (1996). Quantum‐classical molecular dynamics as an approximation to full quantum dynamics. The Journal of Chemical Physics, 105(3), 1074-1083. doi:10.1063/1.471952DOLTSINIS, N. L., & MARX, D. (2002). FIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS. Journal of Theoretical and Computational Chemistry, 01(02), 319-349. doi:10.1142/s0219633602000257Runge, E., & Gross, E. K. U. (1984). Density-Functional Theory for Time-Dependent Systems. Physical Review Letters, 52(12), 997-1000. doi:10.1103/physrevlett.52.997Marques, M. A. L., Maitra, N. T., Nogueira, F. M. S., Gross, E. K. U., & Rubio, A. (Eds.). (2012). Fundamentals of Time-Dependent Density Functional Theory. Lecture Notes in Physics. doi:10.1007/978-3-642-23518-4Verlet, L. (1967). Computer «Experiments» on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review, 159(1), 98-103. doi:10.1103/physrev.159.98Hairer, E., & Wanner, G. (1996). Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics. doi:10.1007/978-3-642-05221-7Castro, A., Marques, M. A. L., & Rubio, A. (2004). Propagators for the time-dependent Kohn–Sham equations. The Journal of Chemical Physics, 121(8), 3425-3433. doi:10.1063/1.1774980Schleife, A., Draeger, E. W., Kanai, Y., & Correa, A. A. (2012). Plane-wave pseudopotential implementation of explicit integrators for time-dependent Kohn-Sham equations in large-scale simulations. The Journal of Chemical Physics, 137(22), 22A546. doi:10.1063/1.4758792Russakoff, A., Li, Y., He, S., & Varga, K. (2016). Accuracy and computational efficiency of real-time subspace propagation schemes for the time-dependent density functional theory. The Journal of Chemical Physics, 144(20), 204125. doi:10.1063/1.4952646Kidd, D., Covington, C., & Varga, K. (2017). Exponential integrators in time-dependent density-functional calculations. Physical Review E, 96(6). doi:10.1103/physreve.96.063307Dewhurst, J. K., Krieger, K., Sharma, S., & Gross, E. K. U. (2016). An efficient algorithm for time propagation as applied to linearized augmented plane wave method. Computer Physics Communications, 209, 92-95. doi:10.1016/j.cpc.2016.09.001Akama, T., Kobayashi, O., & Nanbu, S. (2015). Development of efficient time-evolution method based on three-term recurrence relation. The Journal of Chemical Physics, 142(20), 204104. doi:10.1063/1.4921465Kolesov, G., Grånäs, O., Hoyt, R., Vinichenko, D., & Kaxiras, E. (2015). Real-Time TD-DFT with Classical Ion Dynamics: Methodology and Applications. Journal of Chemical Theory and Computation, 12(2), 466-476. doi:10.1021/acs.jctc.5b00969Schaffhauser, P., & Kümmel, S. (2016). Using time-dependent density functional theory in real time for calculating electronic transport. Physical Review B, 93(3). doi:10.1103/physrevb.93.035115O’Rourke, C., & Bowler, D. R. (2015). Linear scaling density matrix real time TDDFT: Propagator unitarity and matrix truncation. The Journal of Chemical Physics, 143(10), 102801. doi:10.1063/1.4919128Oliveira, M. J. T., Mignolet, B., Kus, T., Papadopoulos, T. A., Remacle, F., & Verstraete, M. J. (2015). Computational Benchmarking for Ultrafast Electron Dynamics: Wave Function Methods vs Density Functional Theory. Journal of Chemical Theory and Computation, 11(5), 2221-2233. doi:10.1021/acs.jctc.5b00167Zhu, Y., & Herbert, J. M. (2018). Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation. The Journal of Chemical Physics, 148(4), 044117. doi:10.1063/1.5004675Rehn, D. A., Shen, Y., Buchholz, M. E., Dubey, M., Namburu, R., & Reed, E. J. (2019). ODE integration schemes for plane-wave real-time time-dependent density functional theory. The Journal of Chemical Physics, 150(1), 014101. doi:10.1063/1.5056258Gómez Pueyo, A., Marques, M. A. L., Rubio, A., & Castro, A. (2018). Propagators for the Time-Dependent Kohn–Sham Equations: Multistep, Runge–Kutta, Exponential Runge–Kutta, and Commutator Free Magnus Methods. Journal of Chemical Theory and Computation, 14(6), 3040-3052. doi:10.1021/acs.jctc.8b00197Stoer, J., & Bulirsch, R. (2002). Introduction to Numerical Analysis. doi:10.1007/978-0-387-21738-3Blanes, S., & Casas, F. (2017). A Concise Introduction to Geometric Numerical Integration. doi:10.1201/b21563Flocard, H., Koonin, S. E., & Weiss, M. S. (1978). Three-dimensional time-dependent Hartree-Fock calculations: Application toO16+O16collisions. Physical Review C, 17(5), 1682-1699. doi:10.1103/physrevc.17.1682Chen, R., & Guo, H. (1999). The Chebyshev propagator for quantum systems. Computer Physics Communications, 119(1), 19-31. doi:10.1016/s0010-4655(98)00179-9Hochbruck, M., & Lubich, C. (1997). On Krylov Subspace Approximations to the Matrix Exponential Operator. SIAM Journal on Numerical Analysis, 34(5), 1911-1925. doi:10.1137/s0036142995280572Frapiccini, A. L., Hamido, A., Schröter, S., Pyke, D., Mota-Furtado, F., O’Mahony, P. F., … Piraux, B. (2014). Explicit schemes for time propagating many-body wave functions. Physical Review A, 89(2). doi:10.1103/physreva.89.023418Caliari, M., Kandolf, P., Ostermann, A., & Rainer, S. (2016). The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential. SIAM Journal on Scientific Computing, 38(3), A1639-A1661. doi:10.1137/15m1027620Schaefer, I., Tal-Ezer, H., & Kosloff, R. (2017). Semi-global approach for propagation of the time-dependent Schrödinger equation for time-dependent and nonlinear problems. Journal of Computational Physics, 343, 368-413. doi:10.1016/j.jcp.2017.04.017Trotter, H. F. (1959). On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4), 545-545. doi:10.1090/s0002-9939-1959-0108732-6Strang, G. (1968). On the Construction and Comparison of Difference Schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517. doi:10.1137/0705041Feit, M. ., Fleck, J. ., & Steiger, A. (1982). Solution of the Schrödinger equation by a spectral method. Journal of Computational Physics, 47(3), 412-433. doi:10.1016/0021-9991(82)90091-2Suzuki, M. (1990). Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Physics Letters A, 146(6), 319-323. doi:10.1016/0375-9601(90)90962-nSuzuki, M. (1992). General theory of higher-order decomposition of exponential operators and symplectic integrators. Physics Letters A, 165(5-6), 387-395. doi:10.1016/0375-9601(92)90335-jYoshida, H. (1990). Construction of higher order symplectic integrators. Physics Letters A, 150(5-7), 262-268. doi:10.1016/0375-9601(90)90092-3Sugino, O., & Miyamoto, Y. (1999). Density-functional approach to electron dynamics: Stable simulation under a self-consistent field. Physical Review B, 59(4), 2579-2586. doi:10.1103/physrevb.59.2579McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Ascher, U. M., Ruuth, S. J., & Wetton, B. T. R. (1995). Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. SIAM Journal on Numerical Analysis, 32(3), 797-823. doi:10.1137/0732037Cooper, G. J., & Sayfy, A. (1983). Additive Runge-Kutta methods for stiff ordinary differential equations. Mathematics of Computation, 40(161), 207-218. doi:10.1090/s0025-5718-1983-0679441-1Hochbruck, M., Lubich, C., & Selhofer, H. (1998). Exponential Integrators for Large Systems of Differential Equations. SIAM Journal on Scientific Computing, 19(5), 1552-1574. doi:10.1137/s1064827595295337Hochbruck, M., & Ostermann, A. (2005). Exponential Runge–Kutta methods for parabolic problems. Applied Numerical Mathematics, 53(2-4), 323-339. doi:10.1016/j.apnum.2004.08.005Hochbruck, M., & Ostermann, A. (2005). Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems. SIAM Journal on Numerical Analysis, 43(3), 1069-1090. doi:10.1137/040611434Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Houston, W. V. (1940). Acceleration of Electrons in a Crystal Lattice. Physical Review, 57(3), 184-186. doi:10.1103/physrev.57.184Chen, Z., & Polizzi, E. (2010). Spectral-based propagation schemes for time-dependent quantum systems with application to carbon nanotubes. Physical Review B, 82(20). doi:10.1103/physrevb.82.205410Sato, S. A., & Yabana, K. (2014). Efficient basis expansion for describing linear and nonlinear electron dynamics in crystalline solids. Physical Review B, 89(22). doi:10.1103/physrevb.89.224305Wang, Z., Li, S.-S., & Wang, L.-W. (2015). Efficient Real-Time Time-Dependent Density Functional Theory Method and its Application to a Collision of an Ion with a 2D Material. Physical Review Letters, 114(6). doi:10.1103/physrevlett.114.063004Magnus, W. (1954). On the exponential solution of differential equations for a linear operator. Communications on Pure and Applied Mathematics, 7(4), 649-673. doi:10.1002/cpa.3160070404Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Nettesheim, P., Bornemann, F. A., Schmidt, B., & Schütte, C. (1996). An explicit and symplectic integrator for quantum-classical molecular dynamics. Chemical Physics Letters, 256(6), 581-588. doi:10.1016/0009-2614(96)00471-xHochbruck, M., & Lubich, C. (1999). A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 421-432. doi:10.1007/978-3-642-58360-5_24Hochbruck, M., & Lubich, C. (1999). Bit Numerical Mathematics, 39(4), 620-645. doi:10.1023/a:1022335122807Nettesheim, P., & Reich, S. (1999). Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 412-420. doi:10.1007/978-3-642-58360-5_23Nettesheim, P., & Schütte, C. (1999). Numerical Integrators for Quantum-Classical Molecular Dynamics. Lecture Notes in Computational Science and Engineering, 396-411. doi:10.1007/978-3-642-58360-5_22Reich, S. (1999). Multiple Time Scales in Classical and Quantum–Classical Molecular Dynamics. Journal of Computational Physics, 151(1), 49-73. doi:10.1006/jcph.1998.6142Jiang, H., & Zhao, X. S. (2000). New propagators for quantum-classical molecular dynamics simulations. The Journal of Chemical Physics, 113(3), 930-935. doi:10.1063/1.481873Grochowski, P., & Lesyng, B. (2003). Extended Hellmann–Feynman forces, canonical representations, and exponential propagators in the mixed quantum-classical molecular dynamics. The Journal of Chemical Physics, 119(22), 11541-11555. doi:10.1063/1.1624062Blanes, S., & Moan, P. C. (2006). Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Applied Numerical Mathematics, 56(12), 1519-1537. doi:10.1016/j.apnum.2005.11.004Bader, P., Blanes, S., & Kopylov, N. (2018). Exponential propagators for the Schrödinger equation with a time-dependent potential. The Journal of Chemical Physics, 148(24), 244109. doi:10.1063/1.5036838Marques, M. (2003). octopus: a first-principles tool for excited electron–ion dynamics. Computer Physics Communications, 151(1), 60-78. doi:10.1016/s0010-4655(02)00686-0Castro, A., Appel, H., Oliveira, M., Rozzi, C. A., Andrade, X., Lorenzen, F., … Rubio, A. (2006). octopus: a tool for the application of time-dependent density functional theory. physica status solidi (b), 243(11), 2465-2488. doi:10.1002/pssb.200642067Schwerdtfeger, P. (2011). The Pseudopotential Approximation in Electronic Structure Theory. ChemPhysChem, 12(17), 3143-3155. doi:10.1002/cphc.201100387Alonso, J. L., Castro, A., Clemente-Gallardo, J., Cuchí, J. C., Echenique, P., & Falceto, F. (2011). Statistics and Nosé formalism for Ehrenfest dynamics. Journal of Physics A: Mathematical and Theoretical, 44(39), 395004. doi:10.1088/1751-8113/44/39/395004Iserles, A., & Nørsett, S. P. (1999). On the solution of linear differential equations in Lie groups. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 983-1019. doi:10.1098/rsta.1999.0362Munthe–Kaas, H., & Owren, B. (1999). Computations in a free Lie algebra. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 957-981. doi:10.1098/rsta.1999.0361Alvermann, A., & Fehske, H. (2011). High-order commutator-free exponential time-propagation of driven quantum systems. Journal of Computational Physics, 230(15), 5930-5956. doi:10.1016/j.jcp.2011.04.006Blanes, S., & Moan, P. C. (2000). Splitting methods for the time-dependent Schrödinger equation. Physics Letters A, 265(1-2), 35-42. doi:10.1016/s0375-9601(99)00866-xAuer, N., Einkemmer, L., Kandolf, P., & Ostermann, A. (2018). Magnus integrators on multicore CPUs and GPUs. Computer Physics Communications, 228, 115-122. doi:10.1016/j.cpc.2018.02.019Thalhammer, M. (2006). A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2), 851-864. doi:10.1137/05063042Bader, P., Blanes, S., Ponsoda, E., & Seydaoğlu, M. (2017). Symplectic integrators for the matrix Hill equation. Journal of Computational and Applied Mathematics, 316, 47-59. doi:10.1016/j.cam.2016.09.04