Formal series and numerical integrators: some history and some new techniques

This paper provides a brief history of B-series and the associated Butcher group and presents the new theory of word series and extended word series. B-series (Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees. They greatly simplify the study of Runge-Kutta schemes and other numerical integrators. We examine the problems that led to the introduction of B-series and survey a number of more recent developments, including applications outside numerical mathematics. Word series (series of functions parameterized by words from an alphabet) provide in some cases a very convenient alternative to B-series. Associated with word series is a group G of coefficients with a composition rule simpler than the corresponding rule in the Butcher group. From a more mathematical point of view, integrators, like Runge-Kutta schemes, that are affine equivariant are represented by elements of the Butcher group, integrators that are equivariant with respect to arbitrary changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552

A Stroboscopic Numerical Method for Highly Oscillatory Problems

International audienceWe suggest a method for the integration of highly oscillatory systems with a single high frequency. The new method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging. The technique may be easily implemented in combination with standard software and may be applied with variable step sizes. Numerical experiments show that the suggested algorithms may be substantially more efficient than standard numerical integrators

Higher-order averaging, formal series and numerical integration II: the quasi-periodic case

International audienceThe paper considers non-autonomous oscillatory systems of ordinary differential equations with d>1 non-resonant constant frequencies. Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all. The new method may be applied to obtain a variety of averaged systems. In particular we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (resp. of a divergence-free) system results in a canonical (resp. in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies

A formal series approach to averaging: exponentially small error estimates

International audienceThe techniques, based on formal series and combinatorics, used nowadays to analyze numerical integrators may be applied to perform high-order averaging in oscillatory periodic or quasi-periodic dynamical systems. When this approach is employed, the averaged system may be written in terms of (i) scalar coefficients that are universal, i.e. independent of the system under consideration and (ii) basis functions that may be written in an explicit, systematic way in terms of the derivatives of the Fourier coefficients of the vector field being averaged. The coefficients may be recursively computed in a simple fashion. We show that this approach may be used to obtain exponentially small error estimates, as those first derived by Neishtadt. All the constants that feature in the estimates have a simple explicit expression

Numerical stroboscopic averaging for ODEs and DAEs

International audienceThe stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt = f(y; t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt = F(Y ) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LISP algorithms of Kirchgraber and multirevolution methods

Optimal tuning of the hybrid Monte Carlo algorithm

We investigate the properties of the hybrid Monte Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible with respect to a given target distribution Π using separable Hamiltonian dynamics with potential −logΠ−log⁡Π. The additional momentum variables are chosen at random from the Boltzmann distribution, and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis–Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an O(1) acceptance probability as the dimension dd of the state space tends to ∞, the leapfrog step size hh should be scaled as h=l×d^(−1/4). Therefore, in high dimensions, HMC requires O(d^(1/4)) steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This value optimally balances the cost of generating a proposal, which decreases as l increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as l increases

Equilibria of Runge-Kutta methods

It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods with q<=1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist