Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Thermal biofeedback, locus of control and precompetitive anxiety in young athletes

Small groups of athletes (maximum size 8) were taught to voluntarily control their finger temperature, in a test of the feasibility of thermal biofeedback as a tool for coaches. The objective was to decrease precompetitive anxiety among the 140 young, competitive athletes (track and field, N=61; swimming, N=79), 66 females and 74 males, mean age 14.8 years, age range 8.9-20.5 years, from local high schools and swimming clubs. The biofeedback (visual and auditory) was provided by small, battery-powered devices that were connected to thermistors attached to the middle finger of the dominant hand. An easily readable digital LCD display, in 0.01 degrees C increments, provided visual feedback, while a musical tone, which descended in pitch with increased finger temperature, provided the audio component via small headphones. Eight twenty minute sessions were scheduled, with 48 hours between sessions. The measures employed in this prestest-posttest study were Levenson's locus of control scale (IPC), and the Competitive Sport Anxiety Inventory (CSAI-2). The results indicated that, while significant control of finger temperature was achieved, F(1, 160)=5.30, p