Users of locally-adaptive software for initial value ordinary differential equations are likely to be concerned with global errors. At the cost of extra computation, global error estimation is possible. Zadunaisky's method and 'solving for the error estimate' are two techniques that have been successfully incorporated into Runge-Kutta algorithms. The standard error analysis for these techniques...

We consider, for the first time, general diminishing stepsize methods for nonconvex, constrained optimization problems. We show that by using directions obtained in an SQP-like fashion convergence to generalized stationary points can be proved. In order to do so, we make use of classical penalty functions in an unconventional way. In particular, penalty functions only enter in the theoretical a...

Abstract We study the use of inverse harmonic Rayleigh quotients with target for stepsize selection in gradient methods nonlinear unconstrained optimization problems. This not only provides an elegant and flexible framework to parametrize reinterpret existing schemes, but it also gives inspiration new tunable families steplengths. In particular, we analyze extend adaptive Barzilai–Borwein metho...

Abstract Xiao and Qin (Comput Phys Commun 265:107981, 2021) recently proposed a remarkably simple modification of the Boris algorithm to compute guiding centre highly oscillatory motion charged particle with step sizes that are much larger than period gyrorotations. They gave strong numerical evidence but no error analysis. This paper provides an analysis large-stepsize modified method in setti...

In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a bounded gradient condition, or without quadratic functional growth property. These models include most well-known classes of functions analyzed in literature: nonsmooth Lipschitz and composition (potentially) smooth function, strong convexity. Based on flexibility offered by our...

A class of explicit symmetric multistep methods is proposed for integrating the equations of motion of charged particles in an electro-magnetic field. The magnetic forces are built into these methods in a special way that respects the Lagrangian structure of the problem. It is shown that such methods approximately preserve energy and momentum over very long times, proportional to a high power o...

Predictor-corrector (PC) methods for the numerical solution of stiff ODEs can be extended to include the second derivative of the solution. In this paper, we consider second derivative PC methods with the three-step second derivative Adams-Bashforth as predictor and two-step second derivative Adams-Moulton as corrector which both methods have order six. Implementation of the proposed PC method ...

Abstract The numerical treatment of an atmospheric chemical scheme, which contains 56 species, is discussed in this paper. This scheme often used studies air pollution levels different domains, as, for example, Europe, by large-scale environmental models containing additionally two other important physical processes—transport pollutants the atmosphere (advection) and diffusion phenomena. We sha...

This paper propose a new complex adaptive notch Þlter (ANF) structure based on the Steiglitz-McBride (SM) method. Recursive least square (RLS) algorithm is applied to the proposed ANF with optimized stepsize. Simulations show that RLS-SM ANF converges fast and requires less computational complexity than the conventional ANF using recursive prediction error (RPE) algorithm.