# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 2 Formulation of the Inverse Problem  
This section is devoted to explain, from a general point of view, the nature of the inverse problem, as well as the topological sensitive method, in order to apply to the specific distinct examples discussed in next sections. In mathematical terms, considering the simplest _discrete formulation_, the outcomes in the sensors for the direct problem are given by  
\[\mathbf{y}=\mathbf{\Gamma}(\mathbf{\alpha}), \tag{1}\]  
where the _sensor vector_\(\mathbf{y}\) collects the obtained data in the sensors that somewhat measure (partially) the response of the system (f.i. the produced sound in the case of the drum mentioned above), and the vector \(\mathbf{\alpha}\) gives the properties of the system that are intended to be found (the drum shape). In principle, the non-linear vector function \(\mathbf{F}\) could be computed using a _black-box solver_, with little or no knowledge of the internal details of the solver. The full inverse problem will consist in solving the system of equations (1), to compute \(\mathbf{\alpha}\) from a known sensor vector \(\mathbf{y}\). In the perturbed formulation, \(\mathbf{\alpha}\) is perturbed around its known healthy, default value, denoted as \(\mathbf{\alpha}_{0}\), in the form  
\[\mathbf{\alpha}=\mathbf{\alpha}_{0}+\tilde{\mathbf{\alpha}},\quad\text{with }\|\tilde{\mathbf{ \alpha}}\|\ll\|\mathbf{\alpha}_{0}\|, \tag{2}\]  
for some appropriate norm \(\|\cdot\|\) (e.g., the Euclidean norm). The perturbation \(\tilde{\mathbf{\alpha}}\) can be called _degradation vector_ because it measures the small degradation of the system from its healthy state \(\mathbf{\alpha}_{0}\). Linearizing the direct problem around the healthy condition, taking advantage of the fact that the perturbation is small, yields  
\[\mathbf{F}(\mathbf{\alpha}_{0}+\tilde{\mathbf{\alpha}})\simeq\mathbf{F}(\mathbf{\alpha}_{0})+ \mathcal{L}_{0}\tilde{\mathbf{\alpha}}, \tag{3}\]  
where \(\mathcal{L}_{0}\) is the _Jacobian_ of \(\mathbf{F}\) at the healthy state. The aim is to compute \(\bar{\mathbf{\kappa}}\) using measurements of the response of the system at the perturbed state, denoted as \(\mathbf{y}^{\text{measured}}\), which are intended to correspond to the perturbed properties of the system, namely  
\[\mathbf{y}^{\text{measured}}\simeq\mathbf{F}(\mathbf{\alpha}_{0}+\bar{\mathbf{\kappa}}). \tag{4}\]  
This (generally non-linear) problem could be solved via an iterative method, which is usually quite computationally expensive, specially because this problem is usually ill-conditioned. An alternative formulation consists of minimizing the _root mean square_ (RMS) of the difference between the two sides of (4), thus considering the objective function  
\[\mathcal{H}(\bar{\mathbf{\kappa}})=\frac{1}{2}\Big{[}\mathbf{y}^{\text{measured}}-\mathbf{ F}(\mathbf{\alpha}_{0}+\bar{\mathbf{\kappa}})\Big{]}^{\top}\Big{[}\mathbf{y}^{\text{ measured}}-\mathbf{F}(\mathbf{\alpha}_{0}+\bar{\mathbf{\kappa}})\Big{]}, \tag{5}\]  
where \({}^{\top}\) stands for the transpose. Again, this non-linear optimization problem could be solved using appropriate optimization algorithms [76; 77]. However, as it happened with the former approach, this optimization problem is usually very computationally expensive, specially because the Hessian of the objective function is usually ill-conditioned. Instead, in the _topological sensitivity_ approximation, the (linear) _sensitivity_ of the objective function \(\mathcal{H}\) to the small perturbation \(\bar{\mathbf{\kappa}}\) is considered.