Uncertainty quantification (UQ) is an active area of research, and an
essential technique used in all fields of science and engineering. The most
common methods for UQ are Monte Carlo and surrogate-modelling. The former
method is dimensionality independent but has slow convergence, while the latter
method has been shown to yield large computational speedups with respect to
Monte Carlo. However, surrogate models suffer from the so-called curse of
dimensionality, and become costly to train for high-dimensional problems, where
UQ might become computationally prohibitive. In this paper we present a new
technique, Lasso Monte Carlo (LMC), which combines surrogate models and the
multilevel Monte Carlo technique, in order to perform UQ in high-dimensional
settings, at a reduced computational cost. We provide mathematical guarantees
for the unbiasedness of the method, and show that LMC can converge faster than
simple Monte Carlo. The theory is numerically tested with benchmarks on toy
problems, as well as on a real example of UQ from the field of nuclear
engineering. In all presented examples LMC converges faster than simple Monte
Carlo, and computational costs are reduced by more than a factor of 5 in some
cases