  Engineering and applied sciences use models of increasing complexity to
simulate the behaviour of manufactured and physical systems. Propagation of
uncertainties from the input to a response quantity of interest through such
models may become intractable in cases when a single simulation is time
demanding. Particularly challenging is the reliability analysis of systems
represented by computationally costly models, because of the large number of
model evaluations that are typically required to estimate small probabilities
of failure. In this paper, we demonstrate the potential of a newly emerged
meta-modelling technique known as low-rank tensor approximations to address
this limitation. This technique is especially promising for high-dimensional
problems because: (i) the number of unknowns in the generic functional form of
the meta-model grows only linearly with the input dimension and (ii) such
approximations can be constructed by relying on a series of minimization
problems of small size independent of the input dimension. In example
applications involving finite-element models pertinent to structural mechanics
and heat conduction, low-rank tensor approximations built with polynomial bases
are found to outperform the popular sparse polynomial chaos expansions in the
estimation of tail probabilities when small experimental designs are used. It
should be emphasized that contrary to methods particularly targeted to
reliability analysis, the meta-modelling approach also provides a full
probabilistic description of the model response, which can be used to estimate
any statistical measure of interest.
