## 4 Experimental Results  
As depicted in Figure 12, the experiment in this part uses the Joule effect: a Chromel wire (nickel-chrome), traversed by a current transmitted by a generator, constitutes the heat source (a). The principle of the method is to insert this Chromel thread into PVC (polyvinyl chloride) material to form a particular geometrical figure. In this experiment, the diameter of the Chromel wire was 200 \(\upmu\)m, and its geometric shape was similar to an \"M\", placed on the rear face of a PVC sample of 1 mm thickness. To avoid thermal losses on the side of the sample where the wire is located, a thickness of mousse has been bonded, thereby fixing the heating wire (b). The Chromel wire lies on a small area of the PVC sample, as illustrated in (c); therefore it can be considered semi-infinite along the \(x\) and \(y\) directions. Figure 11: The results of the heat source estimation with a SNR = 100. The estimated source fields are for the following: without any regularization (a), with Tikhonov regularization (b) and with \(L_{1}\) regularization (c). The Chromel wire is bonded to the back of the PVC sample. To estimate the whole 3D effect, the method of the images is used: the sample is then considered to be 2 mm thick with the wire located not on its backside but on the middle of the plane. Thus, at the time of reconstruction, one should find the Chromel wire centred on the middle plane of the sample along the \(z\) direction. The PVC is supposed to be isotropic, with a density of \(\rho=1180\) kg\(\cdot\)m\({}^{-3}\), a heat capacity of \(C_{p}=1000\) J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), and diffusivity \(a=1.2\times 10^{-7}\) m\({}^{2}\cdot\)s\({}^{-1}\). The voltage of the generator was \(U=3.5\) V with an intensity \(I=1.13\) A. The measures were carried out by an IR camera, with an acquisition frequency of 200 Hz, and the size of the pixel at the surface was 290 \(\upmu\)m \(\times\) 290 \(\upmu\)m. In this experiment, the wire is traversed by the electric current for a duration of approximately 7 s. The area of study on which the temperature is measured as a function of time (with the IR camera) is a rectangle of dimensions \(3.65\times 2\) cm\({}^{2}\), divided into \(126\times 72\) pixels. The resulting field (_Digital Level_) at the surface is presented in Figure 13 at \(t=0\) s when the power supply is turned on (a), after 2 s (b), after 7.2 s, where the maximum of the temperature is measured (c), and after 70 s (d). Figure 12: Picture of the experimental setup (a), schema of the home-made reference sample (b) and picture of the sample during the experiment (c). Figure 13: Measured temperature fields at the surface of the sample (PVC side): at \(t=0\) s when the power supply is turned on (a), after 2 s (b), after 7.2 s (c) and after 70 s (d). As explained in Section 3.1, the choice of the time interval (of which the Fourier number depends) is very important in order to be able to reconstruct the thermal sources as well as possible. In this experiment, we choose \(t_{f}=2\) s, allowing to have 400 time steps and a corresponding Fourier number \(Fo_{z}=0.2\). Since the excitation time of the generator is of the order of 7 s, the temperature response between \(t=0\) s and \(t=2\) s is not a Dirac response but a Heaviside response. According to the criterion depicted in Figure 9 the maximum number of discretizations along \(z\) for the selected Fourier number is approximately 20. The number of discretization points taken in \(z\) (depth direction) is here set to 16 in order to have a number of points close to this limit while remaining low enough to limit the computation time of the inverse problem. The parameters of the model are then \(n_{t}=400\), \(n_{y}=72\), \(n_{x}=126\) and \(n_{z}=16\). The size of the operator matrix \(\mathbf{I}\) to be inverted would therefore be \(n_{t}n_{y}n_{x}\times n_{z}n_{y}n_{x}\approx 10^{7}\times 10^{5}\).