Modulation of the refractive index of glass using ion exchange is a common fabrication method for various micro-optics components, such as waveguide devices, microlenses, and diiractive elements. The design of elements with complicated optical functions has increased the demand for fast numerical simulation of the nonlinear ion-exchange process of two monovalent ions. In this paper, a linear im...

In this paper, we focus on the convergence analysis for the application of the PeacemanRachford splitting method to a convex minimization model whose objective function is the sum of a smooth and nonsmooth convex functions. The sublinear convergence rate in term of the worst-case O(1/t) iteration complexity is established if the gradient of the smooth objective function is assumed to be Lipschi...

The Peaceman–Rachford scheme is a commonly used splitting method for discretizing semilinear evolution equations, where the vector fields are given by the sum of one linear and one nonlinear dissipative operator. Typical examples of such equations are reaction-diffusion systems and the damped wave equation. In this paper we conduct a convergence analysis for the Peaceman– Rachford scheme in the...

In this paper, we propose a couple of new Stochastic Strictly Contractive PeacemanRachford Splitting Method (SCPRSM), called Stochastic SCPRSM (SS-PRSM) and Stochastic Conjugate Gradient SCPRSM (SCG-PRSM) for large-scale optimization problems. The two types of Stochastic PRSM algorithms respectively incorporate stochastic variance reduced gradient (SVRG) and conjugate gradient method. Stochasti...

In this lecture, we discuss splitting algorithms for convex minimization problems with objective given by the sum of two nonsmooth functions. We start with the fixed point property of such problems and derive a general scheme of splitting algorithm based on fixed point iteration. This covers Douglas-Rachford splitting and Peaceman-Rachford splitting algorithms. We also discuss the convergence r...

In this paper we are concerned with the convergence analysis of splitting methods for nonautonomous abstract evolution equations. We introduce a framework that allows us to analyze the popular Lie, Peaceman– Rachford and Strang splittings for time dependent operators. Our framework is in particular suited for analyzing dimension splittings. The influence of boundary conditions is discussed.

The alternating direction method of Peaceman and Rachford is considered for elliptic difference schemes of second order and with two independent variables. An earlier result of the author's on the rapid convergence of multi-parameter noncommutative problems is described and a connection is established between that result and theorems on optimal scaling of band matrices. Simple algorithms to dec...