In this paper, we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the stepsize is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the stepsize tends to zero is in fact the st...

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-...

Recently, Kulikov [1] presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely , a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods. In ...

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...

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 aim of this talk is to present highly stable collocation based numerical methods for Volterra Integral Equations (VIEs). As it is well known, a collocation method is based on the idea of approximating the exact solution of a given integral equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the integral ...

Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in doubly quasi-consistent numerical methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme ...

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...