When the stepsize in non-stii ODE codes is restricted by stability, an uneven pattern of stepsizes with many step rejections is frequently observed. Results analysing this behaviour have been obtained for Runge-Kutta methods, leading to several papers attempting to improve stepsize control. It is shown here that a similar analysis can be carried out for mul-tistep methods. The explicit Adams 2-...

Motivated by machine learning problems over large data sets and distributed optimization over networks, we develop and analyze a new method called incremental Newton method for minimizing the sum of a large number of strongly convex functions. We show that our method is globally convergent for a variable stepsize rule. We further show that under a gradient growth condition, convergence rate is ...

Previous studies of the stability of the second-order backward differentiation formula have concluded that stability is possible only if restrictions are placed on the stepsize ratios, for example, limiting the ratio to some value less than 1 + fi. However, actual implementations of the BDFs differ from the usual theoretical models of such methods; in particular, practical codes use scaled deri...

Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in (0, (1 + √ 5)/2) have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to (1+ √ 5)/2. As...

A minimax method for finding multiple critical points in Banach spaces is successfully developed in [12] by using a projected pseudo-gradient as a search direction. Since several different techniques can be used to compute a projected pseudo-gradient, the uniform stepsize and the continuity of a search direction, two key properties for the convergence results in [5], get lost. In this paper, in...

In this paper we present a meta-algorithm which can symmetrize any one-step timeintegration scheme. The meta-algorithm creates time symmetry, even for those algorithms which are intrinsically non-symmetric. We also present the way to use an adaptive stepsize control with this meta-algorithm. It is shown that the adaptive stepsize control can be implemented naturally and easily. We demonstrate h...

This paper develops a Bregman operator splitting algorithm with variable stepsize (BOSVS) for solving problems of the form min{φ(Bu)+ 1/2‖Au− f ‖2}, where φ may be nonsmooth. The original Bregman Operator Splitting (BOS) algorithm employed a fixed stepsize, while BOSVS uses a line search to achieve better efficiency. These schemes are applicable to total variation (TV)-based image reconstructio...

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call delay, is the number of times it has been updated since its creation. Almos...

The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charm...

Overview It has long been known that the solutions produced by adaptive solvers for ordinary differential (ODE) and differential algebraic (DAE) systems, while generally reliable, are not smooth with respect to perturbations in initial conditions or other problem parameters. Söderlind and Wang [12, 13] have recently developed a digital filter stepsize controller that has a theoretical basis fro...