Overview & Terminology¶
![Schematic of different UQ problems, taken from [Rezaeiravesh18]_.](../_images/uqFrame.png)
Schematic of different UQ problems, taken from [Rezaeiravesh18].¶
Overview¶
A computational model or code depends on different types of inputs and parameters which according to [Santner03] can be categorized as controlled, environmental and uncertain. The focus of the uncertainty quantification (UQ) techniques is mainly on the last two.
UQit is designed mainly based on the needs for UQ in the CFD (computational fluid dynamics) community.
Its connection with the CFD solvers is non-intrusive where the CFD code is treated as a blackbox.
As a result, we always deal with discrete data which comprises of parameter samples and corresponding responses acquired by running the simulator.
A good overview over different UQ approaches can be found for instance in [Smith13] and [Ghanem17].
Moreover, the terminology and some of materials in this documentation are taken from [Rezaeiravesh20].
Below, we list some of the main features of UQit.
Uncertainty propagation or UQ forward problem:
Estimates how the known uncertainties in the inputs and parameters propagate into the quantities of interest (QoIs).
These problems can be efficiently handled using non-intrusive generalized polynomial chaos expansion (PCE), see [Xiu02], [Xiu07].
In UQit, for constructing PCE both regression and projection methods are implemented.
Using compressed sensing method, PCE can be constructed using a small number of training samples.
Samples from the parameter space can be taken using different methods implemented in Sampling module.
See the details in UQ Forward Problem.
Global sensitivity analysis (GSA):
GSA is performed to quantify the sensitivity of the QoIs to the simultaneous variation of the inputs/parameters.
Contrary to local sensitivity analysis (LSA), in GSA all parameters are allowed to vary simultaneously and no linearization is involved in computing sensitivities.
In UQit, GSA is performed by Sobol Sensitivity Indices [Sobol01].
Surrogates:
UQit uses different approaches including Lagrange interpolation, polynomial chaos expansion and more importantly Gaussian process regression [Rasmussen05], [Gramacy20] to construct Surrogates which connect the QoIs to the inputs/parameters.
Surrogates are the pillars for conducting computer experiments [Santner03].
In particular, highest possible flexibility in constructing GPR surrogates have been considered when it comes to incorporating the observational uncertainties.
Nomenclature¶
Throughout this documentation, we adopt the terminologies and nomenclature from [Rezaeiravesh20], as summarized in the following table.
Symbol |
Definition |
QoI |
Quantity of Interest |
\(f(\cdot)\) |
Model function or simulator |
\(\tilde{f}(\cdot)\) |
Surrogate |
\(\chi\) |
Controlled parameter |
\(q_i\) |
i-th uncertain parameter (single-variate) |
\(\mathbf{q}\) |
Multivariate uncertain parameter |
\(\mathbf{q}^{(j)}\) |
j-th sample of \(\mathbf{q}\) |
\(p\) |
Dimension of \(\mathbf{q}\) |
\(\mathbb{Q}\) |
Admissible space of \(\mathbf{q}\) |
\(\mathbb{Q}_i\) |
Admissible space of \(q_i\) |
\(r\) |
Model response, output or QoI |
\(\bigotimes\) |
Tensor product |
\(\mathcal{U}\) |
Uniform distribution |
\(\mathcal{N}\) |
Normal (Gaussian) distribution |