Magnus Expansions Research Articles

Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(n,R), which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.

The Floquet–Magnus and Fer Expansions: Application to Control the Spin Dynamics During the Phase Modulated Lee–Goldburg Radiation in Solid-State NMR

Floquet–Magnus and Fer expansion (FE) schemes are used in nuclear magnetic resonance (NMR) for the calculation of effective Hamiltonians and propagators. This work focuses on applying the Floquet–Magnus and FE approaches to control the spin system evolution during the phase modulate Lee–Goldburg radiation experiment. Until now, the Frequency switched Lee–Goldburg and its variant called the Phase module Lee–Goldburg have been treated by only the average Hamiltonian theory and the bimodal Floquet approach. In this paper, we use the two developing expansion schemes in solid-state NMR for the calculation of the effective Hamiltonian and propagator during the spin dynamics. Our work unifies and generalizes existing results of the Floquet–Magnus and FEs and delivers illustrations of novel springs that boost previous applications that are based on the classical information. The generality of this work points to potential applications in problems related to solid-state NMR and theoretical developments of spectroscopy as well as interdisciplinary research areas whenever they include spin dynamics concepts.

Closed form parametrisation of 3D clothoids by arclength with both linear varying curvature and torsion

The extension from the planar case to three dimensions of the clothoid curve (Euler spiral) is herein presented, that is, a curve parametrised by arc length, whose curvature and torsion are linear (affine) functions of the arc length. The problem is modelled as a linear time variant system and its stability is studied with Lyapunov techniques. Closed form solutions in terms of the standard Fresnel integrals are for the first time presented and they are valid for a wide family of clothoids; for the remaining cases, numerical methods of order four, based on Lie algebra, Magnus Expansions and Commutator-Free Expansions are provided. These geometric integrators have been optimised to be easily implemented with few lines of code and require only matrix-vector products (avoiding explicit matrix exponentials at each iteration). The achieved computational efficiency has a dramatic impact for several real-time applications, especially for designing trajectories for flying vehicles, as previous approaches were fully numeric and could not take advantage of the present closed form solutions, which render the computation of the 3D clothoid as computationally expensive as its planar companion.

Unified signature cumulants and generalized Magnus expansions

AbstractThe signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard its expectation, theexpected signature, as a path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines thesignature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions with motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.

The Magnus expansion and post-Lie algebras

We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows us to place the classical Magnus expansion in the context of Lie group integrators.

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

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 Schrödinger-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 Schrödinger 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.

Continuous changes of variables and the Magnus expansion

In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by operators. As an application of the simple formulas so-obtained, we explicitly compute the first terms of the Floquet-Magnus expansion for the Van der Pol oscillator and the nonlinear Schrödinger equation on the torus.

On Pre-Lie Magnus Expansion

In this paper, the classical and pre-Lie Magnus expansions have been studied, discussing how one can find a recursion for the pre-Lie case which already incorporates the pre-Lie identity. A combinatorial vision of a numerical method proposed by S. Blanes, F. Casas, and J. Ros in [4], has been given on a writing of the classical Magnus expansion in a free Lie algebra, using a pre-Lie structure.

Eigenvalue Computation of Regular 4th Order Sturm-Liouville Problems

In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.

Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of \begin{document}$ SU(2) $\end{document} . We then consider variational problems for evolving curves which are invariant under the projective action of \begin{document}$ SL(2) $\end{document} and finally the standard affine action of \begin{document}$ SE(2) $\end{document} .

Application of the Magnus and Fer expansions to a perturbative study of the time dependent Jaynes Cummings Model

We apply the Magnus and Fer perturbation methods to Jaynes Cummings Model (JCM) with linear time dependence and show that the perturbation methods yield results which are appreciably close to the results obtained by Wei-Norman method, leading to exact solution when the Hamiltonian under consideration is an element of su(2), thus making the perturbation methods important for Hamiltonians for which a method of exact solution is not available.

Solving Schrödinger equation in semiclassical regime with highly oscillatory time-dependent potentials

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic and molecular scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of exponential splitting schemes that allow us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions. These are derived by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. The efficacy of these methods is demonstrated through 1D, 2D and 3D numerical examples.

Applications of Picard and Magnus expansions to the Rabi model

We apply the Picard and Magnus expansions to both the semiclassical and the quantum Rabi model, with a switchable matter-field coupling. The case of the quantum Rabi model ia a paradigmatic example of finite-time quantum electrodynamics (QED), and in this case we build an intuitive diagrammatic representation of the Picard series. In particular, we show that regular oscillations in the mean number of photons, ascribed to the dynamical Casimir effect (DCE) for the the generation of photons and to the anti-DCE for their destruction, take place at twice the resonator frequency $\omega$. Such oscillations, which are a clear dynamical "smoking gun" of the ultrastrong coupling regime, can be predicted by first-order Picard expansion. We also show that the Magnus expansion can be used, through concatenation, as an efficient numerical integrator for both the semiclassical and the quantum Rabi model. In the first case, we find distinctive features in the Fourier spectrum of motion, with a single peak at the Rabi frequency $\Omega$ and doublets at frequencies $2n\omega\pm\Omega$, with $n$ positive integer. We explain these doublets, which are a feature beyond the rotating wave approximation (RWA), on the basis of the Picard series.

Probability distribution for neutron density of two energy group point kinetics system

The probability distribution of the number of neutrons and delayed neutron precursors in a multiplying assembly of various types of reactivities is developed for two energy groups. The space independent point reactor kinetics model for six precursors group of delayed neutrons is considered. The problem is formulated in terms of the probability distribution, generating function which satisfies a partial differential equation, derived in this paper. The probability distribution of two energy group delayed neutrons and the density of the precursors are obtained by solving this system of reactor kinetics model by adopting the mathematical methods. At the point, when the system is formulated in an operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the kinetics system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as time-dependent exponential perturbation theory. Every Magnus approximate corresponds in perturbation theory to a partial re-summation of infinite terms with the important additional property of preserving, in any order, certain symmetries of the exact solution. The first, second and third Magnus expansions are described and used to predict the first moment of fast, thermal and multi-group of delayed neutrons precursor for the two-energy point kinetics reactor system. The validity of the presented method is tested with the aid of the eigenvectors and eigenvalues of the kinetics system in the matrix form by comparing with the conventional methods.

Applications of Magnus expansions and pseudospectra to Markov processes

New directions in Markov processes and research on master equations are showcased by example. The utility of Magnus expansions for handling time-varying rates is demonstrated. The useful notion in applied mathematics often turns out to be the pseudospectra and not simply the eigenvalues. We highlight that general principle with our own examples of Markov processes where exact eigenvalues are found and contrasted with the large errors produced by standard numerical methods. As a motivating application, isomerisation provides a running example and an illustration of our approaches to chemical kinetics. We also present a brief example of a totally asymmetric exclusion process.

Fourier-splitting methods for the dynamics of rotating Bose–Einstein condensates

We present a new method to propagate rotating Bose–Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose–Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can be adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.

Equivalence of the Floquet-Magnus and Fer Expansions to Investigate the Dynamics of a Spin System in the Three-Level System.

In this work, we investigated the orders to which the Floquet-Magnus expansion (FME) and Fer expansion (FE) are equivalent or different for the three-level system. Specifically, we performed the third-order calculations of both approaches based on elegant integrations formalism. We present an important close relationship between the Floquet-Magnus and Fer expansions. As the propagator from the FME takes the form of the evolution operator, which removes the constraint of a stroboscopic observation, we appreciated the effects of time-evolution under Hamiltonians with different orders separately. Our work unifies and generalizes existing results of Floquet-Magnus and Fer approaches and delivers illustrations of novel springs that boost previous applications that are based on the classical information. Due to the lack of an unequivocal relationship between the FME and FE, some disagreements between the results produced by these theories will be found, especially in NMR experiments. Our results can find applications in the optimization of NMR spectroscopy, quantum computation, quantum optical control, and coherence in optics and might bear new awareness in fundamental perusals of quantum spin dynamics. This work is an important theoretical and numerical contribution in the general field of spin dynamics.

Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) Φ(t,α), associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix Q(t,α) and a time-invariant matrix R(α) containing a set of symbolic system parameters α. Computation of Q(t,α) and R(α) in symbolic form as a function of α is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the Q(t,α) matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of Q(t,α) and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, R(α) is computed using a Gaussian quadrature integral formula. Finally, Q(t,α) is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Efficient numerical integration of [formula omitted]th-order non-autonomous linear differential equations

We consider the numerical integration of high-order linear non-homogeneous differential equations, written as first order homogeneous linear equations, and using exponential methods. Integrators like Magnus expansions or commutator-free methods belong to the class of exponential methods showing high accuracy on stiff or oscillatory problems, but the computation of the exponentials or their action on vectors can be computationally costly. The first order differential equations to be solved present a special algebraic structure (associated with the companion matrix) which allows to build new methods (hybrid methods between Magnus and commutator-free methods). The new methods are of similar accuracy as standard exponential methods with a reduced complexity. Additional parameters can be included into the scheme for optimization purposes. We illustrate how these methods can be obtained and present several sixth-order methods which are tested in several numerical experiments.

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.