and its model updating formulation combines the two together: It is the most comprehensive model updating formulation that includes all possible sources of uncertainty, and it requires the most effort to solve. Selective methodologies Much research has been done to solve uncertainty quantification problems, though a majority of them deal with uncertainty propagation. During the past one to two decades, a number of approaches for inverse uncertainty quantification problems have also been developed and have proved to be useful for most small- to medium-scale problems. Forward propagation Existing uncertainty propagation approaches include probabilistic approaches and non-probabilistic approaches. There are basically six categories of probabilistic approaches for uncertainty propagation: Simulation-based methods: Monte Carlo simulations, importance sampling, adaptive sampling, etc. General surrogate-based methods: In a non-instrusive approach, a surrogate model is learnt in order to replace the experiment or the simulation with a cheap and fast approximation. Surrogate-based methods can also be employed in a fully Bayesian fashion. This approach has proven particularly powerful when the cost of sampling, e.g. computationally expensive simulations, is prohibitively high. Local expansion-based methods: Taylor series, perturbation method, etc. These methods have advantages when dealing with relatively small input variability and outputs that don't express high nonlinearity. These linear or linearized methods are detailed in the article Uncertainty propagation. Functional expansion-based methods: Neumann expansion, orthogonal or Karhunen–Loeve expansions (KLE), with polynomial chaos expansion (PCE) and wavelet expansions as special cases. Most probable point (MPP)-based methods: first-order reliability method (FORM) and second-order reliability method (SORM). Numerical integration-based methods: Full factorial numerical integration (FFNI) and dimension reduction (DR). For non-probabilistic approaches, interval analysis, Fuzzy theory, possibility theory and evidence theory are among the most widely used. The probabilistic approach is considered as the most rigorous approach to uncertainty analysis in engineering design due to its consistency with the theory of decision analysis. Its cornerstone is the calculation of probability density functions for sampling statistics. This can be performed rigorously for random variables that are obtainable as transformations of Gaussian variables, leading to exact confidence intervals. Inverse uncertainty Frequentist In regression analysis and least squares problems, the standard error of parameter estimates is readily available, which can be expanded into a confidence interval. Bayesian Several methodologies for inverse uncertainty quantification exist under the Bayesian framework. The most complicated direction is to aim at solving problems with both bias correction and parameter calibration. The challenges of such problems include not only