# 2.1.3.1. Forward Propagation¶

The forward propagation provides a probabilistic understanding of output variables by producing sample realizations and statistical moments (mean, standard deviation, skewness, and kurtosis). Currently, only the Monte Carlo Sampling (MCS) method is available in the SimCenterUQ engine and the other sampling methods (Latin hypercube sampling, Surrogate model-based efficient sampling) are available in the Dakota engine.

## Sampling from dataset¶

SimCenterUQ uniquely provides an option to define the distribution of random variables (RVs) directly from the samples of the RVs (See RV: Random Variables). This feature is useful when the user does not know the actual distribution of an RV but has sample realizations. When a dataset is provided, quoFEM either treats the data as a discrete distribution (when `Discrete`

distribution option is selected) or fits a parametric distribution to the data (when another distribution option is selected). For the former case, quoFEM uniformly samples the discrete index of the dataset provided by the user and uses the value corresponding to the index for the forward propagation analysis. The index is sampled either **with** replacement if the data size is smaller than the number of samples to draw or **without** replacement if the data size is larger than the number of samples to draw. Further, when the user provides coupled samples of *correlated* RVs, vector-wise resampling can be performed instead of independent resampling. In particular, a single index is sampled and shared in multiple variables for each realization.

Click the checkbox to import correlated datasets

To enable this feature, the user can explicitly define the group of RVs that will share the index samples in the UQ tab using the `Resample RVs from correlated dataset`

:

The actual datasets of the RVs written in this field should be imported in the RV tab, through the

`parameters`

-`discrete`

option.The RVs inside each group should be provided with the same length of the samples (e.g. in

`figSimSamp3`

, \(w\) and \(wR\) should have the same sample size \(N_1\) and \(alp\) and \(F_y\) should have the same sample size \(N_2\))

Note

Any correlation values for this coupled datasets **additionally specified in the RV tab** will be **ignored**. Note that the correlation of the data is already reflected in the analysis by introducing the same resampling index.

For example, consider the case where two variables \(w\) and \(wR\) are provided as 10 discrete data points in the RV tab as in Fig. 2.1.3.1.1.

Below is an example of 100 realizations of the two variables when they are considered to be *independent*, i.e. without checking the “Resample RVs from correlated dataset” option.

On the other hand, if the two datasets are considered correlated, i.e. if “Resample RVs from correlated dataset” are checked and the group {w,wR} is reported in the field as shown in `figSimSamp3`

, 100 realization pairs of the RVs will be stacked on top of the provided 10 data points.

This feature is especially useful when the user wants to perform a forward UQ analysis directly using the posterior samples obtained from Markov Chain Monte Carlo or other Bayesian sampling approaches.

Tip

Summary of capabilities and limitations

o Run Monte Carlo simulation for 12 different kinds of probability distributions with correlations.

o Use data samples as discrete distribution (especially useful when propagating samples from Bayesian updating)

x Run advanced sampling algorithms including Latin hypercube and surrogate-aided sampling.

**[Available in Dakota engine]**

## Multi-fidelity Monte Carlo¶

SimCenter UQ supports a multi-fidelity Monte Carlo option, where the user can import both high-fidelity and low-fidelity simulation models to estimate the statistics of high-fidelity responses. See more information in the technical manual.

The users should define **Max Computation time** and **Seed**

**Max Computation Time (minutes)**: The target computation time for the analysis. Note that this is a “soft” target, meaning the analysis may not necessarily be completed within the specified time limit. The total number of simulations is decided after a few pilot runs of simulations considering the remaining budgets (time), and the process is not enforced to finish even if the target time is exceeded. Therefore, there could be a few minutes of estimation error in the max computation time. Note that, even if the specific time is exceeded, the analysis will not end until the specified minimum number of simulations (default is 40) is reached.**Seed**: a positive integer that is used to reproduce the exact sequence of the pseudo-random number generator. The seed is used to ensure the reproducibility of the results.

Additionally, advanced options can be selected by checking the option at the bottom

Using the advanced options, the user can define the minimum number of simulations per model and choose to get statistics in log-transformed space

**Minimum # simulations per model**: The number of pilot samples. The pilot samples are used to estimate the correlations between the high- and low-fidelity models and decide the optimal number of simulations given the time limit. See technical manual for more details. By default, the minimum number of simulations is 40.**Perform log-transform**: Estimate the mean and variance of the log-transformed values (natural logarithm or \(log_{e}(\bullet)\)). In EE-UQ, the default is**checked**.

Note

The multiple models should be specified in the SIM and FEM tabs, the lower value of the model index corresponds to a higher fidelity model. Therefore, the high-fidelity model should always be imported as model 1 for both SIM and FEM tabs. The model order in the SIM tab should follow that of the FEM tab. The **Belief** field will be ignored.