Extrapolative Approach Research Articles

NOTE DE RECHERCHE: TRADITIONS NATIONALES ET GESTION D'ENTREPRISE: PORTÉE ET LIMITES DE L'APPROCHE D'IRIBARNIENNE DANS L'ANALYSE DE LA CULTURE MAROCAINE

This article presents and discusses the extrapolative analytical approach used by Philippe d'Iribarne and his team to study corporate management and national traditions. d'Iribarne's study of SGS-Thomson in Casablanca and contextualization in Moroccan culture of representations made by individuals and groups of their experience working in the enterprise. Other explanations for the behaviors analyzed are proposed, and it is concluded that, although still valid overall, d'Iribarne's framework would benefit from being adjusted to avoid hasty extrapolations.

Masking Resonance Artifacts in Force-Splitting Methods for Biomolecular Simulations by Extrapolative Langevin Dynamics

Numerical resonance artifacts have become recognized recently as a limiting factor to increasing the timestep in multiple-timestep (MTS) biomolecular dynamics simulations. At certain timesteps correlated to internal motions (e.g., 5 fs, around half the period of the fastest bond stretch,Tmin), visible inaccuracies or instabilities can occur. Impulse-MTS schemes are vulnerable to these resonance errors since large energy pulses are introduced to the governing dynamics equations when the slow forces are evaluated. We recently showed that such resonance artifacts can be masked significantly by applying extrapolative splitting to stochastic dynamics. Theoretical and numerical analyses of force-splitting integrators based on the Verlet discretization are reported here for linear models to explain these observations and to suggest how to construct effective integrators for biomolecular dynamics that balance stability with accuracy. Analyses for Newtonian dynamics demonstrate the severe resonance patterns of the Impulse splitting, with this severity worsening with the outer timestep, Δt; Constant Extrapolation is generally unstable, but the disturbances do not grow with Δt. Thus, the stochastic extrapolative combination can counteract generic instabilities and largely alleviate resonances with a sufficiently strong Langevin heat-bath coupling (γ), estimates for which are derived here based on the fastest and slowest motion periods. These resonance results generally hold for nonlinear test systems: a water tetramer and solvated protein. Proposed related approaches such as Extrapolation/Correction and Midpoint Extrapolation work better than Constant Extrapolation only for timesteps less thanTmin/2. An effective extrapolative stochastic approach for biomolecules that balances long-timestep stability with good accuracy for the fast subsystem is then applied to a biomolecule using a three-class partitioning: the medium forces are treated byMidpoint Extrapolationvia position Verlet, and the slow forces are incorporated byConstant Extrapolation. The resulting algorithm (LN) performs well on a solvated protein system in terms of thermodynamic properties and yields an order of magnitude speedup with respect to single-timestep Langevin trajectories. Computed spectral density functions also show how the Newtonian modes can be approximated by using a small γ in the range of 5–20 ps−1.

Extrapolative approaches to Brillouin-zone integration

A highly efficient extrapolative Brillouin-zone integration scheme is presented that requires a very low k-point sampling density for spectral integrations. It is important to use an extrapolative approach, since at low sampling densities interpolative schemes are hindered by problems associated with band crossing, which introduce spurious singularities in the density of states (DOS). The information for the extrapolation is obtained using second-order $\mathbf{k}\mathbf{\ensuremath{\cdot}}\mathbf{p}$ perturbation theory within a set of subcells of the Brillouin zone, which can be chosen to make full use of symmetry. The resulting piecewise quadratic representation of the band structure is converted directly to a DOS using an analytic approach. It is also shown that this method can be successfully applied even in the linear extrapolative case.