GPU-accelerated simulation of colloidal suspensions with direct hydrodynamic interactions

Solvent-mediated hydrodynamic interactions between colloidal particles can significantly alter their dynamics. We discuss the implementation of Stokesian dynamics in leading approximation for streaming processors as provided by the compute unified device architecture (CUDA) of recent graphics processors (GPUs). Thereby, the simulation of explicit solvent particles is avoided and hydrodynamic interactions can easily be accounted for in already available, highly accelerated molecular dynamics simulations. Special emphasis is put on efficient memory access and numerical stability. The algorithm is applied to the periodic sedimentation of a cluster of four suspended particles. Finally, we investigate the runtime performance of generic memory access patterns of complexity $O(N^2)$ for various GPU algorithms relying on either hardware cache or shared memory.Comment: to appear in a special issue of Eur. Phys. J. Special Topics on "Computer Simulations on GPUs

SqFreeEVAL: An (almost) optimal real-root isolation algorithm

Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds

Intrinsic rotation in tokamaks: theory

Self-consistent equations for intrinsic rotation in tokamaks with small poloidal magnetic field $B_p$ compared to the total magnetic field $B$ are derived. The model gives the momentum redistribution due to turbulence, collisional transport and energy injection. Intrinsic rotation is determined by the balance between the momentum redistribution and the turbulent diffusion and convection. Two different turbulence regimes are considered: turbulence with characteristic perpendicular lengths of the order of the ion gyroradius, $\rho_i$, and turbulence with characteristic lengths of the order of the poloidal gyroradius, $(B/B_p) \rho_i$. Intrinsic rotation driven by gyroradius scale turbulence is mainly due to the effect of neoclassical corrections and of finite orbit widths on turbulent momentum transport, whereas for the intrinsic rotation driven by poloidal gyroradius scale turbulence, the slow variation of turbulence characteristics in the radial and poloidal directions and the turbulent particle acceleration can be become as important as the neoclassical and finite orbit width effects. The magnetic drift is shown to be indispensable for the intrinsic rotation driven by the slow variation of turbulence characteristics and the turbulent particle acceleration. The equations are written in a form conducive to implementation in a flux tube code, and the effect of the radial variation of the turbulence is included in a novel way that does not require a global gyrokinetic formalism.Comment: 88 pages, 4 figure

"On the Structure, Asymptotic Theory and Applications of STAR-GARCH Models"

Non-linear time series models, especially regime-switching models, have become increasingly popular in the economics, finance and financial econometrics literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the models or asymptotic theory. Some structural and statistical properties have recently been established for the Smooth Transition Autoregressive (STAR) - Generalised Autoregresssive Conditional Heteroscedasticity (GARCH), or STAR-GARCH, model, including the necessary and sufficient conditions for the existence of moments, and the sufficient condition for consistency and asymptotic normality of the (Quasi)-Maximum Likelihood Estimator ((Q)MLE). While these moment conditions are straightforward to verify in practice, they may not be satisfied for the GARCH model if the underlying long run persistence is close to unity. A less restrictive condition for consistency and asymptotic normality may alleviate this problem. The paper establishes a weak sufficient, or log-moment, condition for consistency and asymptotic normality of (Q)MLE for STAR-GARCH. This condition can easily be extended to any non-linear conditional mean model with GARCH errors, subject to reasonable regularity conditions. Although the log-moment condition cannot be verified as easily as the second and fourth moment conditions, it allows the long run persistence of the GARCH process to exceed one. Monte Carlo experiments show that the log-moment condition is more reliable in practice than the second and fourh moment conditions when the underlying long run persistence is close to unity. These experiments also show that the correct specification of the conditional mean is crucial in obtaining unbiased estimates for the GARCH component. The sufficient conditions for consistency and asymptotic normality are verified empirically using S&P 500 returns, 3-month US Treasury Bill returns, and exchange rates between Australia and the USA. The effects of outliers and extreme observations on the empirical moment conditions are also analysed in detail.

Estimating Smooth Transition Autoregressive Models with GARCH Errors in the Presence of Extreme Observations and Outliers,

This paper investigates several empirical issues regarding quasimaximum likelihood estimation of Smooth Transition Autoregressive (STAR) models with GARCH errors, specifically STAR-GARCH and STAR-STGARCH. Convergence, the choice of different algorithms for maximising the likelihood function, and the sensitivity of the estimates to outliers and extreme observations, are examined using daily data for S&P 500, Heng Seng and Nikkei 225 for the period January 1986 to April 2000.