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Harald Hofstätter

Publications and source records attributed to Harald Hofstätter.

17 recordsLinked to original sources
arXiv2022

A simple and efficient algorithm for computing the Baker-Campbell-Hausdorff series

A new algorithm for computing coefficients of the Baker--Campbell--Hausdorff series is presented, which can be straightforwardly implemented in any general-purpose programming language or computer algebra system. The algorithm avoids rational arithmetic and performs all its calculations in pure integer arithmetic, allowing for a very efficient implementation. An implementation in the Julia programming language is available.

math.RA↗
arXiv2021

User manual for bch, a program for the fast computation of the Baker-Campbell-Hausdorff and similar series

This manual describes bch, an efficient program written in the C programming language for the fast computation of the Baker-Campbell-Hausdorff (BCH) and similar Lie series. The Lie series can be represented in the Lyndon basis, in the classical Hall basis, or in the right-normed basis of E.S. Chibrikov. In the Lyndon basis, which proves to be particularly efficient for this purpose, the computation of 111013 coefficients for the BCH series up to terms of degree 20 takes less than half a second on an ordinary personal computer and requires negligible 11MB of memory. Up to terms of degree 30, which is the maximum degree the program can handle, the computation of 74248451 coefficients takes 55 hours but still requires only a modest 5.5GB of memory.

cs.MS↗
arXiv2021

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schrödinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effort

math.NA↗
arXiv2020

Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series

In a recent paper the author derived a formula for calculating common denominators for the homogeneous components of the Baker-Campbell-Hausdorff (BCH) series. In the present work it is proved that this formula actually yields the smallest such common denominators. In an appendix a new efficient algorithm for computing coefficients of the BCH series is presented, which is based on these common denominators, and requires only integer arithmetic rather than less efficient rational arithmetic.

math.NT↗
arXiv2020

Denominators of coefficients of the Baker-Campbell-Hausdorff series

For the computation of terms of the Baker-Campbell-Hausdorff series $H = \log(e^Ae^B})$ some a priori knowledge about the denominators of the coefficients of the series can be beneficial. In this paper an explicit formula for the computation of common denominators for the rational coefficients of the homogeneous components of the series is derived. Explicit computations up to degree 30 show that the common denominators obtained by this formula are as small as possible, which suggests that the formula is in a sense optimal. The sequence of integers defined by the formula seems to be interesting also from a number-theoretic point of view. There is, e.g., a connection with the denominators of the Bernoulli numbers and the Bernoulli polynomials.

math.NT↗
arXiv2020

Chaos-induced depletion of a Bose-Einstein condensate

The mean-field limit of a bosonic quantum many-body system is described by (mostly) non-linear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e. by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent $λ$. The question now is whether $λ$ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show, that the number of non-condensed particles is subject to an exponentially fast increase, i.e. depletion. Furthermore, we show that the rate of exponential depletion is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced depletion is accessible experimentally through the visibility of interference fringes in the total density after time of flight, thus opening the possibility to measure $λ$, and with it, the interplay between chaos and non-equilibrium quantum matter, in a real experiment.

cond-mat.quant-gas↗
arXiv2020

A relatively short self-contained proof of the Baker-Campbell-Hausdorff theorem

We give a new purely algebraic proof of the Baker-Campbell-Hausdorff theorem, which states that the homogeneous components of the formal expansion of $\log(e^Ae^B)$ are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given polynomial itself is a Lie polynomial.

math.RA↗
arXiv2020

Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime

Time dependent Schrödinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations originate from the semiclassical parameter, and call for appropriate methods. We propose to employ a combination of asymptotic Zassenhaus splitting with time adaptivity. While the former turns the disadvantage of the semiclassical parameter into an advantage, leading to highly efficient methods with low error constants, the latter enables to choose an optimal time step and to speed up the calculations when the oscillations subside. We support the results with numerical examples.

math.NA↗
arXiv2019

An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and its Application to the Construction of Exponential Integrators

This paper discusses an efficient implementation of the generation of order conditions for the construction of exponential integrators like exponential splitting and Magnus-type methods in the computer algebra system Maple. At the core of this implementation is a new algorithm for the computation of coefficients of words in the formal expansion of the local error of the integrator. The underlying theoretical background including an analysis of the structure of the local error is briefly reviewed. As an application the coefficients of all 8th order self-adjoint commutator-free Magnus-type integrators involving the minimum number of 8 exponentials are computed.

math.NA↗
arXiv2019

Adaptive Exponential Integrators for MCTDHF

We compare exponential-type integrators for the numerical time-propagation of the equations of motion arising in the multi-configuration time-dependent Hartree-Fock method for the approximation of the high-dimensional multi-particle Schr{ö}dinger equation. We find that among the most widely used integrators like Runge-Kutta, exponential splitting, exponential Runge-Kutta, exponential multistep and Lawson methods, exponential Lawson multistep methods with one predictor/corrector step provide optimal stability and accuracy at the least computational cost, taking into account that the evaluation of the nonlocal potential terms is by far the computationally most expensive part of such a calculation. Moreover, the predictor step provides an estimator for the time-stepping error at no additional cost, which enables adaptive time-stepping to reliably control the accuracy of a computation.

math.NA↗
arXiv2019

Order conditions for exponential integrators

This paper provides an algebraic framework for the generation of order conditions for the construction of exponential integrators like splitting and Magnus-type methods for the numerical solution of evolution equations. The generation of order conditions is based on an analysis of the structure of the leading local error term of such an integrator, and on a new algorithm for the computation of coefficients of words in expressions involving exponentials. As an application a new 8th order commutator-free Magnus-type integrator involving only 8 exponentials is derived.

math.NA↗
arXiv2019

Non-satisfiability of a positivity condition for commutator-free exponential integrators of order higher than four

We consider commutator-free exponential integrators as put forward in [Alverman, A., Fehske, H.: High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 230, 5930-5956 (2011)]. For parabolic problems, it is important for the well-definedness that such an integrator satisfies a positivity condition such that essentially it only proceeds forward in time. We prove that this requirement implies maximal convergence order of four for real coefficients, which has been conjectured earlier by other authors.

math.NA↗
arXiv2019

Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations

Prior work on computable defect-based local error estimators for (linear) time-reversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case [W. Auzinger and O. Koch, An improved local error estimator for symmetric time-stepping schemes, Appl.Math.Lett. 82 (2018), pp. 106-110] remain valid, i.e., the modified estimators yield an improved asymptotic order as the step size goes to zero. Typically, the computational effort is only slightly higher than for conventional defect-based estimators, and it may even be lower in some cases. We illustrate this by some examples and present numerical results for evolution equations of Schrödinger type, solved by either time-splitting or Magnus-type integrators. Finally, we demonstrate that adaptive time-stepping schemes can be successfully based on our local error estimators.

math.NA↗
arXiv2017

Long-time expansion of a Bose-Einstein condensate: Can Anderson localization be observed?

We numerically explore the long-time expansion of a one-dimensional Bose-Einstein condensate in a disorder potential employing the Gross-Pitaevskii equation. The goal is to search for unique signatures of Anderson localization in the presence of particle-particle interactions. Using typical experimental parameters we show that the time scale for which the non-equilibrium dynamics of the interacting system begins to diverge from the non-interacting system exceeds the observation times up to now accessible in the experiment. We find evidence that the long-time evolution of the wavepacket is characterized by (sub)diffusive spreading and a growing effective localization length suggesting that interactions destroy Anderson localization.

cond-mat.quant-gas↗
arXiv2016

Setup of Order Conditions for Splitting Methods

This article is based on earlier papers where an approach based on Taylor expansion and the structure of its leading term as an element of a free Lie algebra was described for the setup of a system of order conditions for operator splitting methods. Along with a brief review of these materials and some theoretical background, we discuss the implementation of the ideas from these papers in computer algebra, in particular using Maple 18. A parallel version of such a code is described.

math.NA↗
arXiv2016

Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

math.NA↗