Editing Inverse problem (section)

==== Numerical solution of our elementary example ====
Here we make use of the Euclidean norm to quantify the data misfits. As we deal with a linear inverse problem, the objective function is quadratic. For its minimization, it is classical to compute its gradient using the same rationale (as we would to minimize a function of only one variable). At the optimal model <math>p_\text{opt}</math>, this gradient vanishes which can be written as:
<math display="block">\nabla_p \varphi = 2 (F^\mathrm{T} F p_\text{opt} - F^\mathrm{T} d_\text{obs}) = 0 </math>
where ''F''<sup>T</sup> denotes the [[matrix transpose]] of ''F''. This equation simplifies to:
<math display="block">F^\mathrm{T} F p_\text{opt} = F^\mathrm{T} d_\text{obs} </math>

This expression is known as the [https://en.wikipedia.org/?title=Normal_equations&redirect=no normal equation] and gives us a possible solution to the inverse problem. 
In our example matrix <math>F^\mathrm{T} F</math> turns out to be generally full rank so that the equation above makes sense and determines uniquely the model parameters: we do not need integrating additional information for ending up with a unique solution.