2
J. F. Reis et al.
Engineers pledge their efforts to design devices to make our lives easier, and, to
achieve this goal, they must predict the behaviour of the system they are creating.
Predicting the outcome of a process is a very challenging goal which in general
implies the definition of a mathematical model. The equations included in the model
describe the behaviour of the system under investigation (where the word system is
entailed with its most general meaning), and their solution will yield to a prediction
of the underlying outcome. To provide an example, this process is very much like
translating a book from one language to another. Translation and modelling are so
much alike that they share a common issue: information is lost. Indeed, it might
be difficult to translate an English saying into Chinese. This is due to a different
language structure and, even more subtle, to the fact that a saying is usually strongly
related to the specific culture, which in principle is different from country to country.
Therefore, information is often lost during the translation process.
In the same way, modelling a physical phenomenon is challenging, especially
when its complexity grows. The mathematical equations lack information about the
phenomenon they model, and, rather than an exact, they become an approximate
description of physics.
The discrepancies found between model predictions and the actual phenomenon
are referred to as the model error. The model error can be divided in two types:
the aleatory and the epistemic error. The aleatory error is related to the randomness
of physics under which the phenomenon develops. As exact physical conditions
are impossible to measure, exact predictions of the outcome are also impossible to
obtain. The epistemic error is instead related to a lack of knowledge regarding the
real physics of the event, i.e., it is due to things one could in principle know but
doesn’t in practice. As a consequence, the equations included in the model may
not be suitable to represent reality in a general sense. To make an example, when
Newton wrote his famous equations for classical mechanics, he was not aware of the
relativistic effects. His equations work perfectly in many cases, but they fail when
the system under investigation consists of an object traveling at a velocity close to
the light speed or when the object has a very large mass.
This is due to an epistemic uncertainty; the relativistic effects are not modelled;
that affects Newton’s dynamics, and it was not until Einstein fulfilled this deficiency
that predictions about astronomical phenomena could be accurately made. Of
course, the accuracy gained by modelling relativistic effects would not be worth
the growth in complexity of the model itself in the limit of Newton’s physics.
1.1.1 Typical UQ Questions
In real world applications, there exist many different questions that the methodologies presented in this chapter help to address.
Starting from the computation of statistical quantities, for instance, the mean and
the variance of the output of a stochastic process, uncertainty quantification (UQ)
techniques span from inference problems to data analysis. In this subsection, we
J. F. Reis et al.
Engineers pledge their efforts to design devices to make our lives easier, and, to
achieve this goal, they must predict the behaviour of the system they are creating.
Predicting the outcome of a process is a very challenging goal which in general
implies the definition of a mathematical model. The equations included in the model
describe the behaviour of the system under investigation (where the word system is
entailed with its most general meaning), and their solution will yield to a prediction
of the underlying outcome. To provide an example, this process is very much like
translating a book from one language to another. Translation and modelling are so
much alike that they share a common issue: information is lost. Indeed, it might
be difficult to translate an English saying into Chinese. This is due to a different
language structure and, even more subtle, to the fact that a saying is usually strongly
related to the specific culture, which in principle is different from country to country.
Therefore, information is often lost during the translation process.
In the same way, modelling a physical phenomenon is challenging, especially
when its complexity grows. The mathematical equations lack information about the
phenomenon they model, and, rather than an exact, they become an approximate
description of physics.
The discrepancies found between model predictions and the actual phenomenon
are referred to as the model error. The model error can be divided in two types:
the aleatory and the epistemic error. The aleatory error is related to the randomness
of physics under which the phenomenon develops. As exact physical conditions
are impossible to measure, exact predictions of the outcome are also impossible to
obtain. The epistemic error is instead related to a lack of knowledge regarding the
real physics of the event, i.e., it is due to things one could in principle know but
doesn’t in practice. As a consequence, the equations included in the model may
not be suitable to represent reality in a general sense. To make an example, when
Newton wrote his famous equations for classical mechanics, he was not aware of the
relativistic effects. His equations work perfectly in many cases, but they fail when
the system under investigation consists of an object traveling at a velocity close to
the light speed or when the object has a very large mass.
This is due to an epistemic uncertainty; the relativistic effects are not modelled;
that affects Newton’s dynamics, and it was not until Einstein fulfilled this deficiency
that predictions about astronomical phenomena could be accurately made. Of
course, the accuracy gained by modelling relativistic effects would not be worth
the growth in complexity of the model itself in the limit of Newton’s physics.
1.1.1 Typical UQ Questions
In real world applications, there exist many different questions that the methodologies presented in this chapter help to address.
Starting from the computation of statistical quantities, for instance, the mean and
the variance of the output of a stochastic process, uncertainty quantification (UQ)
techniques span from inference problems to data analysis. In this subsection, we