In this paper, we present variable-stepsize explicit parallel peer methods grounded in the interpolation idea. Approximation, stability, and convergence are studied in detail. In particular, we prove that some interpolating peer methods are stable on any variable mesh in practice. Double quasi consistency is utilized to introduce an efficient global error estimation formula in the numerical met...

Abstract. An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing φ(Bu) + H(u) where H and φ are convex functions, and φ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is an...

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedić and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109–138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The s...

The split feasibility problem (SFP) is finding a point [Formula: see text] such that [Formula: see text], where C and Q are nonempty closed convex subsets of Hilbert spaces [Formula: see text] and [Formula: see text], and [Formula: see text] is a bounded linear operator. Byrne's CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute [Formula: see text], and sometimes [...

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of v...

The numerical treatment of linear-quadratic regulator problems on finite time horizons for parabolic partial differential equations requires the solution of large-scale differential Riccati equations (DREs). Typically the coefficient matrices of the resulting DRE have a given structure (e.g. sparse, symmetric or low rank). Here we discuss numerical methods for solving DREs capable of exploiting...

Previously, Bottou and LeCun [1] established that the second-order stochastic gradient descent (SGD) method can potentially achieve generalization performance as well as empirical optimum in a single pass through the training examples. However, second-order SGD requires computing the inverse of the Hessian matrix of the loss function, which is usually prohibitively expensive. Recently, we inven...

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing φ(Bu) + H(u), where H and φ are convex functions, and φ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is analyzed. W...

In this paper we deal with several issues concerning variablestepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with tim...