Definition

Figure 2.1: Definition of an inversion problem

Physical properties are usually measured through a scientific experiment. For instance, Torricelli invented the mercury barometer to measure the atmospheric pressure $P_{atm}$ . $P_{atm}$ is estimated by comparing the height of the mercury column ($h_{merc}$ ) with a graduated scale. If $P_{atm}$ is known, $h_{merc}$ can be calculated with a simple linear relationship involving the density of mercury.

$$h_{merc}=\frac{P_{atm}}{13.5}$$ (2.1)

This is the forward problem. However, during the scientific experiment, the observable is not the pressure but the height of mercury. Hence equation (2.1) must be inverted to calculate the model parameter $P_{atm}$ from the observable quantity $h_{merc}$ . The inverse problem is solved with the following equation

$$P_{atm}=13.5 \: h_{merc}$$ (2.2)

This is an example of the inversion of a linear problem which is pretty simple and always solved analytically. In this case, the number of unknowns is one as well as the number of observable (or data). Scientific models are generally completely described by the means of more than one parameter. With these models, it is also possible to calculate various theoretical characteristics. For instance, a characteristic of the model may be a curve which is numerically represented by a vector of $n_{obs}$ components. Hence the forward problem is a function that transforms a parameter space of dimension $n_{param}$ (number of involved parameters) into the observable space of dimension $n_{obs}$ .

If the function is linear, the linear algebra is used to solved the inversion problem. In this case, there is no absolute limit for the number of observables and the number of parameters. If $n_{obs}$ is less than $n_{param}$ , there is an infinite number of solutions for the parameter vector. On the contrary, if $n_{obs}$ is greater than $n_{param}$ , a least-square method is generally used to find the best set of parameters.

However, in most situations, the relationship is not linear and even more, the forward problem cannot be solved analytically. Even if the forward problem has an analytical expression, there are very few special cases where the inversion problem is also analytical. Hence, in most cases, an inversion method is necessary to calculate the set of parameters corresponding to the observables. The number of solutions of the inverse problem is generally a complex issue. For instance, if the forward function is simply $y=x^2$ between two one-dimensional spaces, the inverse problem may have zero, one or two solutions. The non-uniqueness is hence specific to each problem and has to be studied on a case-by-case basis.

All scientific observables are measured with a certain degree of error, even if it is not explicitly quantified. In Torricelli's experiment, the height of mercury can be measured for instance down to a 0.5 mm precision. In this one-dimensional linear example, the error on $P_{atm}$ is easily deduced. For multi-dimensional linear problems the error propagation is also possible. But for non linear and multi-dimensional problems, calculating the errors on the model parameters from the errors on the acquired measurements is not straightforward.