Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat Comput 27(4):1065–1082, 2017. https://doi.org/10.1007/s11222-016-9671-0), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.
Sexual health education for young people is crucial not only for development of norms but also for protection against vulnerabilities during this stage in life. Although several systematic reviews have examined the effectiveness of peer-led sex education, none have focused on the extent of peer participation. The purpose of this review was to evaluate peer-led sexual health education interventions in more developed countries (MDCs). Electronic and manual searches across five social science, education, and medical databases were conducted. Fifteen articles were selected in total. Most (10 of 15) studies gave low or no responsibility to peers. The majority of articles found improvements in sexual health knowledge (13 of 14) and attitudes (11 of 15) at postintervention stages. Two studies showed improved self-efficacy, and three showed behavioral changes. A preliminary synthesis of effectiveness and level of participation was done. Meta-analysis revealed a large effect on knowledge change (Hedges’ g = 0.84, 95% confidence interval [CI]: 0.43 to 1.25) and a medium effect on attitude change (Hedges’ g = 0.49, 95% CI: 0.19 to 0.80). Peer-led interventions could be a powerful tool. This review shows that this approach is effective in changing knowledge and attitudes but not behaviors. Further research and action are needed to understand optimal implementation. 相似文献
