The sensitivity analysis, uncertainty propagation, reliability analysis and inverse uncertainty quantification rely on accurate surrogate models to deliver accurate UQ analysis. The relative tolerance is provided for all kinds of surrogate models. For a SPCE (sparse polynomial chaos expansion) model, the relative tolerance is used to terminate the model training process. For an ASPCE (adaptive sparse polynomial chaos expansion) model, the relative tolerance is used to terminate the model training process for each adaptive step combined with the maximum number of model evaluations, which is used to terminate the adaptive procedure. For a GP model, the relative tolerance is used to check against the error estimates. For an AGP model, the relative tolerance combined with the maximum number of model evaluations is used to terminate the adaptive process. For the SPCE model, you can also set the q-norm, which determines the truncation level of the polynomial basis. For the GP and AGP methods, different kinds of covariance and mean functions and different types of optimization methods for finding the maximum error estimates are provided. The covariance and mean functions are used to determine the shape of the GP and AGP models. Further details about the surrogate model are given in the theory section (see Surrogate models).

To use a surrogate model for a UQ study, choose a method from the Surrogate model list. The available methods are: Adaptive sparse polynomial chaos expansion, Sparse polynomial chaos expansion, Adaptive Gaussian process, and Gaussian process. Note that reliability analysis only works with Adaptive Gaussian process. A detailed explanation is given in the theory section (see Reliability Analysis — Efficient Global Reliability Analysis). For models of the polynomial chaos expansion type, both Adaptive sparse polynomial chaos expansion and Sparse polynomial chaos expansion models have a Relative tolerance, which is the leave-one-out cross-validation error that is used to find the best multivariate polynomial basis. If this error is not smaller than the Relative tolerance, a warning is given in the Log window. For Adaptive sparse polynomial chaos expansion, if the leave-one-out cross-validation error for the current adaptive iteration step is larger than the Relative tolerance and the total number of model evaluations is smaller than the Maximum of input points, then more input parameter points will be sampled with the LHS method, and more model evaluations will be computed at these points to create a new polynomial chaos expansion model. The adaptive version only has a setting for the Relative tolerance. Here, the polynomial degree and the q-norm are computed iteratively in each adaptation process. For GP-type models, both Adaptive Gaussian process and Gaussian process models have a Relative tolerance. For Gaussian process, the Relative tolerance is used to check if the error estimation is small enough. For Adaptive Gaussian process, the adaptation process finds the next sample point corresponding to the location of the largest error estimation. The searching procedure is a global optimization where DIRECT and Monte Carlo methods can be used. For the DIRECT method, the optimization is stopped when the Maximum surrogate evaluations for optimization and the Maximum number of optimization iteration are reached. For the Monte Carlo method, the optimization computes surrogate model evaluations on all the Surrogate evaluations for optimization sample points. For both Adaptive Gaussian process and Gaussian process, if this error is not smaller than the Relative tolerance, a warning is given in the Log window. For GP-type or polynomial chaos expansion type models, a Gaussian process function or PCE function is created and trained from the compute action with the quantities of interests through COMSOL model evaluation.