  Mathematical models are essential for understanding and making predictions
about systems arising in nature and engineering. Yet, mathematical models are a
simplification of true phenomena, thus making predictions subject to
uncertainty. Hence, the ability to quantify uncertainties is essential to any
modelling framework, enabling the user to assess the importance of certain
parameters on quantities of interest and have control over the quality of the
model output by providing a rigorous understanding of uncertainty. Peridynamic
models are a particular class of mathematical models that have proven to be
remarkably accurate and robust for a large class of material failure problems.
However, the high computational expense of peridynamic models remains a major
limitation, hindering outer-loop applications that require a large number of
simulations, for example, uncertainty quantification. This contribution
provides a framework to make such computations feasible. By employing a
Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are
performed using a coarse mesh, and performing relatively few simulations using
a fine mesh, a significant reduction in computational cost can be realised, and
statistics of structural failure can be estimated. The results show a speed-up
factor of 16x over a standard Monte Carlo estimator, enabling the forward
propagation of uncertain parameters in a computationally expensive peridynamic
model. Furthermore, the multilevel method provides an estimate of both the
discretisation error and sampling error, thus improving the confidence in
numerical predictions. The performance of the approach is demonstrated through
an examination of the statistical size effect in quasi-brittle materials.
