From Averaging to Acceleration, There is Only a Step-size

We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of iterations. We provide a detailed analysis of the eigenvalues of the corresponding linear dynamical system , showing various oscillatory and non-oscillatory behaviors, together with a sharp stability result with explicit constants. We also consider the situation where noisy gradients are available, where we extend our general convergence result, which suggests an alternative algorithm (i.e., with different step sizes) that exhibits the good aspects of both averaging and acceleration

DNS of vertical plane channel flow with finite-size particles: Voronoi analysis, acceleration statistics and particle-conditioned averaging

We have performed a direct numerical simulation of dilute turbulent particulate flow in a vertical plane channel, fully resolving the phase interfaces. The flow conditions are the same as those in the main case of "Uhlmann, M., Phys. Fluids, vol. 20, 2008, 053305", with the exception of the computational domain length which has been doubled in the present study. The statistics of flow and particle motion are not significantly altered by the elongation of the domain. The large-scale columnar-like structures which had previously been identified do persist and they are still only marginally decorrelated in the prolonged domain. Voronoi analysis of the spatial particle distribution shows that the state of the dispersed phase can be characterized as slightly more ordered than random tending towards a homogeneous spatial distribution. It is also found that the p.d.f.'s of Lagrangian particle accelerations for wall-normal and spanwise directions follow a lognormal distribution as observed in previous experiments of homogeneous flows. The streamwise component deviates from this law presenting significant skewness. Finally, a statistical analysis of the flow in the near field around the particles reveals that particle wakes present two regions, a near wake where the velocity deficit decays as 1/x and a far wake with a decay of approximately 1/(x*x).Comment: accepted for publication in Int. J. Multiphase Flo

Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods

This paper is concerned with the comparison of semi-analytical and non-averaged propagation methods for Earth satellite orbits. We analyse the total integration error for semi-analytical methods and propose a novel decomposition into dynamical, model truncation, short-periodic, and numerical error components. The first three are attributable to distinct approximations required by the method of averaging, which fundamentally limit the attainable accuracy. In contrast, numerical error, the only component present in non-averaged methods, can be significantly mitigated by employing adaptive numerical algorithms and regularized formulations of the equations of motion. We present a collection of non-averaged methods based on the integration of existing regularized formulations of the equations of motion through an adaptive solver. We implemented the collection in the orbit propagation code THALASSA, which we make publicly available, and we compared the non-averaged methods to the semi-analytical method implemented in the orbit propagation tool STELA through numerical tests involving long-term propagations (on the order of decades) of LEO, GTO, and high-altitude HEO orbits. For the test cases considered, regularized non-averaged methods were found to be up to two times slower than semi-analytical for the LEO orbit, to have comparable speed for the GTO, and to be ten times as fast for the HEO (for the same accuracy). We show for the first time that efficient implementations of non-averaged regularized formulations of the equations of motion, and especially of non-singular element methods, are attractive candidates for the long-term study of high-altitude and highly elliptical Earth satellite orbits.Comment: 33 pages, 10 figures, 7 tables. Part of the CMDA Topical Collection on "50 years of Celestial Mechanics and Dynamical Astronomy". Comments and feedback are encourage

Attitude and orbit coupling of planar helio-stable solar sails

The coupled attitude and orbit dynamics of solar sails is studied. The shape of the sail is a simplified quasi-rhombic-pyramid that provides the structure helio-stablility properties. After adimensionalisation, the system is put in the form of a fast-slow dynamical system where the different time scales are explicitely related to the physical parameters of the system. The orientation of the body frame with respect to the inertial orbit frame is a fast phase that can be averaged out. This gives rise to a simplified formulation that only consists of the orbit dynamics perturbed by a flat sail with fixed attitude perpendicular to the direction of the sunlight. The results are exemplified using numerical simulations.Comment: 36 pages, 31 figure

Errors in particle tracking velocimetry with high-speed cameras

Velocity errors in particle tracking velocimetry (PTV) are studied. When using high-speed video cameras, the velocity error may increase at a high camera frame rate. This increase in velocity error is due to particle-position uncertainty, which is one of two sources of velocity errors studied here. The other source of error is particle acceleration, which has the opposite trend of diminishing at higher frame rates. Both kinds of errors can propagate into quantities calculated from velocity, such as the kinetic temperature of particles or correlation functions. As demonstrated in a dusty plasma experiment, the kinetic temperature of particles has no unique value when measured using PTV, but depends on the sampling time interval or frame rate. It is also shown that an artifact appears in an autocorrelation function computed from particle positions and velocities, and it becomes more severe when a small sampling-time interval is used. Schemes to reduce these errors are demonstrated.Comment: 6 pages, 5 figures, Review of Scientific Instruments, 2011 (In Press