# Introduction to Uncertainty Quantification and Management¶

## Operate in an uncertain world¶

Uncertainty Quantification and Management (UQ&M) is a field of engineering on the rise where several questions arise for the user:

• uncertainty quantification: how to represent the sources of uncertainties, whether by expert opinion or via data?

• uncertainty propagation: how to propagate these uncertainties through a model or a system of coupled models?

• uncertainty quantification (again!): how to represent the resulting uncertainty about the output quantity of interest?

• sensitivity analysis: how to explain this output uncertainty from the input ones? Are there non-influential sources? Can the others be ordered?

• reliability: what is the probability that the quantity of interest exceeds a threshold? Conversely, what is the guaranteed threshold for a given confidence level?

• robust optimization: what would be a good design solution in terms of performance (or cost) and constraints satisfaction, in an uncertain world? Rather than looking for the best solution in the worst case scenario, which would lead to a very conservative solution, why not relax the constraints by guaranteeing them in 99% of cases while maximizing an average performance (or minimizing an average cost)?

GEMSEO implements several UQ&M key concepts through a dedicated package. Moreover, its class ParameterSpace extends the notion of DesignSpace by defining both deterministic and uncertain variables. It can already be used in a DOEScenario to sample a multidisciplinary system. Moreover, the GEMSEO community is currently working on extending its use to any kind of Scenario for robust MDO purposes (see Roadmap) by means of dedicated MDO formulations.

## The package uncertainty¶

Uncertainty quantification and management.

The package uncertainty provides several functionalities to quantify and manage uncertainties. Most of them can be used from the dedicated API.

The sub-package distributions offers an abstract level for probability distributions as well as interfaces to the OpenTURNS and SciPy ones. It is also possible to fit a probability distribution from data or select the most likely one from a list of candidates. These distributions can be used to define random variables in a ParameterSpace before propagating these uncertainties through a system of MDODiscipline, by means of a DOEScenario.

The sub-package sensitivity offers an abstract level for sensitivity analysis, as well as concrete features. These sensitivity analyses compute indices by means of various methods: correlation measures, Morris technique and Sobol’ variance decomposition. This sub-package is based in particular on OpenTURNS.

The sub-package statistics offers an abstract level for statistics, as well as parametric and empirical versions. Empirical statistics are estimated from a Dataset while parametric statistics are analytical properties of a Distribution fitted from a Dataset.