Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

Assessment for learning - Why the theory needs the practice

Assessment for Learning is a pedagogical practice with anticipated gains of increased student motivation, mastery and autonomy as learners develop their capacity to monitor and plan their own learning progress. Assessment for Learning (AfL) differs from Assessment of learning in its timing, occurring within the regular flow of learning rather than end point, in its purpose of improving student learning rather than summative grading and in the ownership of the learning where the student voice is heard in judging quality. Since Black and Wiliam (1998) highlighted the achievement gains that AfL practices seem to bring to all learners in classrooms, it has become part of current educational policy discourse in Australia, yet teacher adoption of the practices is not a straightforward implementation of techniques within an existing classroom repertoire. As can be seen from the following meta-analysis, recent research highlights a more complex interrelationship between teacher and student beliefs about learning and assessment, and the social and cultural interactions in and contexts of the classroom. More research is needed from a sociocultural perspective that allows meaning to emerge from practice. Before another policy push, we need to understand better the many factors within the assessment relationship. We need to hear from teachers and students through long-term AfL case studies both to inform AfL theory and to shed light on the complexities of pedagogical change for enhancing learner autonomy.