Monte Carlo
Description
Monte Carlo is a simulation technique based on generating large number of random samples of the input variables according to their specified probability distributions, and then evaluating the system at each sample point. The statistical properties of the model output, such as mean or variance, can then be estimated from the results of model evaluations. The Monte Carlo sampling method is a popular method for uncertainty quantification, and is often used to compute sensitivity indices such as Sobol indices.
In the mUQSA portal, a variation of Monte Carlo method, called Quasi Monte Carlo, is used. Quasi Monte Carlo uses Saltelli sampling plan which is a specific type of sampling method that can be used to estimate variance-based sensitivity indices for the input variables and their interactions. Unlike general Monte Carlo sampling, which generates random samples, the Saltelli sampling plan generates a set of quasi-random input samples that allow to estimate sensitivity indices with a minimal number of model evaluations.
Advantages
Monte Carlo sampling has several advantages over other methods. One advantage is that Monte Carlo sampling does not require any assumptions about the underlying probability distribution of the input variables, whereas the effectiveness of methods like Stochastic Collocation and Polynomial Chaos Expansion is strictly dependent on these distributions. This makes Monte Carlo sampling more flexible and robust in situations where the distribution of the input variables is not well-known or is difficult to model. Monte Carlo can also be more efficient than Stochastic Collocation and Polynomial Chaos Expansion when the number of input variables is large.
Disadvantages
Monte Carlo sampling can require a large number of samples to achieve a desired level of accuracy, thus making this method computationally expensive.
Number of produced samples
$$ (d + 2) * N $$
Where $d$ is the number of uncertain parameters, and $N$ is the (estimated) number of samples for the Monte Carlo method.