Method Evaluation

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 43

where κ(K) is the condition number, kKk₂ is the L₂ norm of the matrix and K⁺ denotes the Moore-Penrose pseudo-inverse matrix of K, kδKk2 = eKkKk2 6= 0 and kδfk = e_fkvk 6= 0. In the case of an approaching zero sampling step, the sampling error is negligible and (3.2.25) becomes

kf⁰−¯fk

kf⁰k ≤ e_fκ(K) 1−ef

. (3.2.26)

Clearly, from (3.2.26), the relative numerical solution error is bounded when the discretization error is greater than zero, with 0 < e_f 1 and κ(K) is bounded, a condition that is satisfied, since the numerical system of equations is an approximation to the line integrals. Naturally, a very dense discretization of the volume will lead to an unbounded condition number of the system matrix, a rank deficient system and an unbounded solution error.

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 44

angular errors (AE) are estimated for each discrete tile and shown in slices of the z- axis, for easier inspection (Figs. 3.4(iii),(iv)-(a):(h)). The histograms of the two kinds of errors are also depicted in Figs. 3.4(v),(vi), respectively. As evidenced, the mass of magnitude errors lie below 5%, while most angular errors lie below 3^◦. Although both error types remain very low, partly because of the choice of a relatively small sampling step ∆r, the highest divergences are observed close to the boundaries of the domain;

in particular, in the tiles that lie closest to and farthest from the electric source. If the reconstruction is performed using the same settings, but ∆ris increased from 0.1 to 1, then the total RME and AE, averaged over the all the tiles are increased from 2.04%

to 9.9%, and from 1.5^◦ to 5.4^◦, respectively, but the condition number of the system matrix declines from 67.85 to 15.48. In order to define the range of the values of ∆r that give good solutions a numerical test where ∆r was set to vary from 2 down to 10⁻⁶ was employed and the results are shown in Fig. 3.5. Fig. 3.5(i) shows the mean relative error in the magnitude of the vector field, Fig. 3.5(ii) shows the orientation error, and Fig. 3.5(iii) presents the condition number of the system. Clearly, as

∆r→0, the solution error increases since the condition κ(K) might tend to infinity (see (30)). As it is apparent from Fig. 3.5, both errors remain low within the range of ∆r ∈ [10⁻¹ −5×10⁻³]. The point where the minimum error lies corresponds to 5×10⁻⁴, yet, the condition number is quite high. Further decrease of ∆r increases the corresponding errors as was expected and the system becomes too sensitive to perturbations.

The results of a second reconstruction are presented in Fig. 3.6. In this case, where the vector field is created by a larger set of external sources, placed along the circle with radius 12, the proposed method performs adequately, providing a solution that is the exact image of the theoretical field, with low errors, i.e., mean RME is equal to 2.87%, while mean AE is equal to 3.24^◦. In agreement with Fig. 3.4, Fig. 3.6 depicts the reconstructed field (Fig. 3.6(ii)) juxtaposed with the corresponding theoretical field (Fig. 3.6(i)), the RME, AE, organized in slices of the z-axis (Figs. 3.6(iii),(iv)- (a):(h), respectively) and their distributions (Figs. 3.6(v),(vi)). Analogous to the first reconstruction conclusions can be drawn following a change in the value of ∆r. After a series of simulations with sources placed outside the domain, it has been noted that

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 45

the solution errors increase when the monopoles approach the domain boundaries.

For example, if the sources of Fig. 3.6(i) move from the circle with radius equal to 12, to the circle with radius equal to 6, the mean RME, AE increase to 19.25% and 12.78^◦, respectively.

An example of a simulation where two point sources are located within the bounded volume appears in Fig. 3.7, following the presentation structure of Figs 3.4, 3.6. In this case, the mean RME, AE from all recovery points are equal to 28.68%, 10.18^◦, respectively. When the point sources are placed inside the domain the method is less reliable, as high magnitude errors are produced, especially on the reconstruc- tion points that lie closer to the sources. The error in the orientation of the field is lower, therefore still indicating the locations of the sources. Moreover, in spite of the high magnitude errors, the maxima of the field lie around the monopoles, correctly indicating the locations of the sources.

3.3.2 Noise Effect Assessment

In a real-world application, the values measured by the sensors cannot be expected to be perfectly accurate. Depending on the application, another natural source of noise can be the displacement of the sensors. In order to account for these possible inaccuracies, a set of noise-perturbed simulations was performed. To address the case of noise at the measurements, a 0-7% fraction of random sign of the exact values, as estimated from Coulomb’s law, was added to the measurements (simulation #1).

Then, the positions of the sensors were considered to be misplaced by a randomly signed percentage of 0-7% of their regular positions (simulation #2). In the third simulation, the two kinds of perturbations were considered simultaneously (simulation

#3). In all cases, noise was added gradually to 25%, 50%, 75% and all sensors, for a sampling step ∆r ranging from 0.1 to 3. The inclusion of a ∆r variation in the experiments is justified by the preceding analysis, indicating that decreases in ∆r increase the ill-conditioning of the system, turning the solution sensitive to perturbations. In the simulations performed, the edge of the cube was equal to 6, and the bounded field was created by a point source of strength q = −1 in location with coordinates (20, 20, 20).

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 46

The relative magnitude and angular errors from these simulations are shown in Fig. 3.8-3.10. The columns of Figures 3.8(i),(ii)-(a):(d) show the mean RME and AE of the recovered solution for simulation #1 as a function of both the sampling step and the percentage of noise added, while the rows correspond to different percentages of sensors perturbed. The sensitivity of the method to the additive noise is high, especially when ∆r is small, verifying the intrinsic instability in this case. As the value of ∆rincreases, both types of errors drop significantly. Although the magnitude errors are not very low, the maxima of the field the bigger ∆r help determine the location of the source.

On the other hand, the field reconstructions are more robust when we displace the sensors (simulation #2), as, even in the poorest recovery, the errors remain low, as shown in Figs. 3.9(i),(ii)-(a):(d). The errors here increase smoothly with the rise of the spatial noise percentage, and, contrary to the first experiment, the low values of

∆r seem to affect the angular errors more than the magnitude ones. Higher ∆r does not produce lower errors either, in particular, using a medium sampling step yields the best results.

Finally, as depicted in Figs. 3.10(i),(ii)-(a):(d), in the case of perturbations in both the measurements and the positions of the sensors (simulation #3), the solution errors seem to be accumulative of the errors of the two previous cases. Naturally, the errors are high due to the system sensitivity to additive noise to the sensor measurements.

3.3.3 Comparison with Alternative Methods

ART is a traditional method in VFT that was proposed for the complete reconstruc- tion of vector fields, assuming that the divergence of the field is known. It has been applied in the literature mainly for the recovery of incompressible flows. However, the model equations used can be applied for the recovery of irrotational fields and in the framework of the geometrical model described in this work. A more detailed description of the implementation of ART for its comparison to 3D-VFT can be found in Appendix A.

Then, a well-established methodology in electrostatics, commonly employed to

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 47

determine the electric potential is the second order Poisson’s partial differential equa- tion (PDE) [73]. In free space the equation is simplified to its reduced form, Laplace’s equation, which is the second method to which 3D-VFT is compared here. Details about the implementation of Laplace’s equation to recover the electric field within the bounded space used for 3D-VFT can be found in Appendix B.

Instances of comparisons among the three methods, using a discretized bounded volume of U=4 and P = 1, are presented in Fig. 3.11. In particular, Fig. 3.11(i) corresponds to the reconstruction achieved by the proposed 3D-VFT method, 3.11(i) shows the ART-based recovery, while Fig. 3.11(iii) corresponds to the reconstruc- tion using Laplace’s equation. Two series of simulations were implemented, which include cases of fields caused by point sources lying outside and inside the bounded domain, shown in Figs. 3.11(i):(iii)-(a),(b), respectively. In order to facilitate the visual comparison, only the z-slice that lies closer to the source is shown. Figures 3.11(i):(iii)-(c),(d) depict the mean RME (%) and AE (^◦) calculated along the same z-slice for each subcase, respectively, whereas Figs. 3.11(i):(iii)-(e),(f) present the distribution of errors resulting from the three reconstruction methods for the whole volume of the bounded domain, accordingly. The use of a non-uniform colormap or histogram scale was preferred, in order to better show the different ranges of errors in each case. As illustrated in Figs. 3.11(i):(iii)-(b), in the case of a point source placed outside the volume of interest at location (10, 10, 10), the solutions from all three recovery methods have small errors. Although the reconstructions seem visu- ally indistinguishable, the errors in the solutions of 3D-VFT and ART-based method are larger; in particular, the mean RME and AE are equal to 2.7% and 2.5^◦, respec- tively, in the 3D-VFT reconstruction, 3% and 1.6^◦, respectively, in the ART-based reconstruction, when the corresponding Laplace’s equation errors are only 0.2% and 0.17^◦. A larger difference is observed when the point source is placed inside the domain, at location (1, 1, 1) (Figs. 3.11(i):(iii)-(b)). Although the performance of all methods deteriorates, 3D-VFT and the ART-based method outperform Laplace’s equation solution by more than 20% and 25^◦ on the mean magnitude and orientation errors, respectively. It should be noted that because the source is inside the domain, Poisson’s PDE should be used to model the electric field; however, we compare with

CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 48

Laplace’s PDE because we want to reconstruct the field without using any source modeling and relying only on the sensor’s measurements. After 20 iterations, when the Kaczmarz method has converged to the minimum norm solution, the 3D-VFT and ART-based solutions are comparable, since in this case the 3D-VFT approxima- tion performs better by less than 1% and 3^◦. Hence, 3D-VFT regularizes the ill-posed problem by discretization and can yield a solution by applying a direct solver, instead of solving it through an iterative method.