Dear Ali

The *leave-one-out* (LOO) error \varepsilon_{LOO} is an estimate of the mean-square error between the original model \mathcal M and the surrogate model \hat{\mathcal M}, that is \varepsilon = \mathbb{E} \left[ \left( {\mathcal M}(\boldsymbol{x}) - \hat{\mathcal M}(\boldsymbol{x})\right)^2\right]. As such, it is an average over the parameters input domain.

When the surrogate is used for uncertainty quantification, i.e. when the input is considered as a random vector \boldsymbol{X} with probability density function f_{\boldsymbol{X} }, and when moments or sensitivity indices of the output are of interest, then good results may be obtained with a rather large LOO error.

Based on my experience I consider that \varepsilon_{LOO} =\le 10^{-2} is sufficient for this purpose, especially to compute Sobol’ sensitivity indices. This is especially true when using a polynomial chaos expansion (PCE) for sensitivity analysis (See the UQLab user manual on sensitivity analysis, Section 1.5.3). I even have examples where the screening of important parameters, leading to 2-digit accurate Sobol indices, is obtained with a PCE of (rather limited) accuracy \varepsilon_{LOO} \approx 0.05 - 0.1.

However, the above mean square error does not guarantee that, for any \boldsymbol x_0 in your input space, the pointwise error \left| {\mathcal M}(\boldsymbol{x_0}) - \hat{\mathcal M}(\boldsymbol{x_0})\right| is as small as \varepsilon. Although the mean-square and maximum pointwise error are somehow linked, there is no general results. If you want to properly estimate this maximal error, you need an independent validation set (what we tend to avoid in surrogate modelling, as each point usually results from a costly simulation). Getting a LOO error smaller than 10^{-4} is in general good enough to have pointwise errors less than 1%, *but this is highly problem-dependent !*

**As a conclusion:** yes, the LOO error is a good estimator of the quality of your surrogate, and the thresholds mentioned above can serve as guidelines. Our experience in structural mechanics is that the finite element models are usually rather smooth functions of the input parameters (e.g. they depend strictly linearly on the load parameters in the case of elastic analysis !) so that the threshold mentioned above are often easily achieved.

If you have a reasonable number of input parameters (say d=5-30), using an LHS experimental design (ED) of size N = 10 \times d\; is a good starting point. You can always enrich your experimental design later on (see the UQLab commands `uq_enrichLHS`

or `uq_lhsify`

if the accuracy of the surrogate obtained with this first ED is not sufficient.

Best regards

Bruno