Rate of convergence

About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.

Probabilistic approach for granular media equations in the non uniformly convex case

Abstract: We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T 1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.

An Accelerated Proximal Coordinate Gradient Method

Abstract: We develop an accelerated randomized proximal coordinate gradient (APCG) method, for solving a broad class of composite convex optimization problems. In particular, our method achieves faster linear convergence rates for minimizing strongly convex functions than existing randomized proximal coordinate gradient methods. We show how to apply the APCG method to solve the dual of the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method.

Acceleration schemes for ab initio molecular-dynamics simulations and electronic-structure calculations.

Abstract: We study the convergence and the stability of fictitious dynamical methods for electrons. first, we show that a particular damped second-order dynamics has a much faster rate of convergence to the ground state than first-order steepest-descent algorithms while retaining their numerical cost per time step. Our damped dynamics has efficiency comparable to that of conjugate gradient methods in typical electronic minimization problems. Then, we analyze the factors that limit the size of the integration time step in approaches based on plane-wave expansions. The maximum allowed time step is dictated by the highest frequency components of the fictitious electronic dynamics. These can result either from the larger wave vector components of the kinetic energy or from the small wave vector components of the Coulomb potential giving rise to the so called charge sloshing problem. We show how to eliminate large wave vector instabilities by adopting a preconditioning scheme in the context of Car-Parrinello ab initio molecular-dynamics simulations of the ionic motion. We also show how to solve the charge sloshing problem when this is present. We substantiate our theoretical analysis with numerical tests on a number of different silicon and carbon systems having both insulating and metallic character.