1 Introduction to Spectral Methods for Uncertainty Quantification
3
will try to summarise the different possible applications, and we will provide the
reader with some thoughtful examples.
The most simple example consists in characterising a quantity of interest (QoI)
within a given process. Roughly speaking, very often engineers are challenged with
designing complex systems that are supposed to behave in a certain manner and
under nominal conditions. One may think, for instance, of an aircraft that flies at
a fixed altitude, with a well-defined cruise speed and with a given payload. The
aircraft thus represents a system which depends on three inputs (air density, speed
and payload), and a QoI may be the fuel consumption rate.
In real world applications, the atmosphere is not a homogeneous continuum.
Turbulence, wind, clouds, and any meteorological phenomena contribute to modify
the atmosphere through which the aircraft is flying, causing local fluctuations.
Moreover, different payloads may be carried on board, possibly due to a different
number of passengers or to a different mission scope.
Engineers are then concerned with the fact that the aircraft they are designing
must face a wide range of different operating conditions. The operating conditions
are not known a priori, if not as just parameter ranges or desired flight envelope.
The goal of engineers is to come up with an aircraft which is able to safely
accomplish the appointed missions, thus tolerating considerable variation of the
operating conditions.
Therefore, engineers may want to understand how the uncertainties on the
nominal flight conditions affect aircraft performances. They may, for instance,
assess how a small variation in the cruising speed affects the efficiency and thus
the fuel consumption. Uncertainty quantification techniques can indeed be used to
propagate uncertainties through a computational fluid dynamics model and help
characterising the QoI. Engineers may estimate the mean fuel consumption rate
and its variance with respect of uncertainties on the value of speed, air density and
payload (Passagers and cargo). This is known as forward uncertainty propagation.
Moreover, UQ techniques may be exploited to carry out sensitivity analysis.
Sensitivity analysis is very useful when one is trying to assess the contribution of
every single source of uncertainty to the variance of the QoI.
With reference to the aircraft example, a sensitivity analysis may be carried out
to understand if the largest variations of fuel burning rate are related more to a
variation in the payload than to the cruising speed. This information drives engineers
during the design phase of the aircraft and allows them to achieve a more robust
configuration.
Nonetheless, UQ techniques have a broad range of applications, and they can
be exploited to compute the reliability of a device (or, more in general, a system).
As known from the control theory, each device can be decomposed (up to a certain
limit) into a collection of interacting sub-systems. A sub-system may be seen as
an independent block that exchanges information through the connections from and
towards other blocks (inputs and outputs). As such, each sub-system may undergo
failures during its lifetime.
Due to the gradual deterioration of its components, the reliability of a device
can be investigated as a time-dependent function, the survival function. This latter
3
will try to summarise the different possible applications, and we will provide the
reader with some thoughtful examples.
The most simple example consists in characterising a quantity of interest (QoI)
within a given process. Roughly speaking, very often engineers are challenged with
designing complex systems that are supposed to behave in a certain manner and
under nominal conditions. One may think, for instance, of an aircraft that flies at
a fixed altitude, with a well-defined cruise speed and with a given payload. The
aircraft thus represents a system which depends on three inputs (air density, speed
and payload), and a QoI may be the fuel consumption rate.
In real world applications, the atmosphere is not a homogeneous continuum.
Turbulence, wind, clouds, and any meteorological phenomena contribute to modify
the atmosphere through which the aircraft is flying, causing local fluctuations.
Moreover, different payloads may be carried on board, possibly due to a different
number of passengers or to a different mission scope.
Engineers are then concerned with the fact that the aircraft they are designing
must face a wide range of different operating conditions. The operating conditions
are not known a priori, if not as just parameter ranges or desired flight envelope.
The goal of engineers is to come up with an aircraft which is able to safely
accomplish the appointed missions, thus tolerating considerable variation of the
operating conditions.
Therefore, engineers may want to understand how the uncertainties on the
nominal flight conditions affect aircraft performances. They may, for instance,
assess how a small variation in the cruising speed affects the efficiency and thus
the fuel consumption. Uncertainty quantification techniques can indeed be used to
propagate uncertainties through a computational fluid dynamics model and help
characterising the QoI. Engineers may estimate the mean fuel consumption rate
and its variance with respect of uncertainties on the value of speed, air density and
payload (Passagers and cargo). This is known as forward uncertainty propagation.
Moreover, UQ techniques may be exploited to carry out sensitivity analysis.
Sensitivity analysis is very useful when one is trying to assess the contribution of
every single source of uncertainty to the variance of the QoI.
With reference to the aircraft example, a sensitivity analysis may be carried out
to understand if the largest variations of fuel burning rate are related more to a
variation in the payload than to the cruising speed. This information drives engineers
during the design phase of the aircraft and allows them to achieve a more robust
configuration.
Nonetheless, UQ techniques have a broad range of applications, and they can
be exploited to compute the reliability of a device (or, more in general, a system).
As known from the control theory, each device can be decomposed (up to a certain
limit) into a collection of interacting sub-systems. A sub-system may be seen as
an independent block that exchanges information through the connections from and
towards other blocks (inputs and outputs). As such, each sub-system may undergo
failures during its lifetime.
Due to the gradual deterioration of its components, the reliability of a device
can be investigated as a time-dependent function, the survival function. This latter