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.
The role of voluntary and community sector organizations in the delivery of public services is increasing and these changes
bring new responsibilities and benefits to organizations that have the capacity to participate. There are concerns within
the sector about the implications for citizenship and participation. The sector is highly dependent on volunteers yet little
is known about how organizational change in response to new relationships with the statutory sector impact upon the commitment
and well-being of people who volunteer. This paper addresses that gap in knowledge for older volunteers. Drawing upon collaborative
research with a voluntary organization in the north of England, the authors explore the meanings and aspirations of volunteering
for older people, and explain how and why changes associated with closer engagement with public service delivery and less
grant dependency can be disempowering for them.