Tomographic reconstructions of the fast-ion phase space using imaging neutral particle analyser measurements

In this paper we demonstrate how the inversion, in energy and major radius (E, R) coordinates, of imaging neutral particle analyser (INPA) measurements can be used to obtain the fast-ion distribution. The INPA is most sensitive to passing ions with energies in the range (20–150) keV and pitches near 0.5 in the core and 0.7 near the plasma edge. Inversion of synthetic signals, via 0th-order Tikhonov and Elastic Net regularization, were performed to demonstrate the capability of recovering the ground truth fast-ion 2D phase-space distribution resolved in major radius and energy, even in the presence of moderate noise levels (10%). Finally, we apply our method to measure the 2D phase-space distribution in an MHD quiescent plasma at ASDEX Upgrade and find good agreement with the slowing down fast-ion distribution predicted by TRANSP.


Introduction
In future fusion reactors, suprathermal particles (fast ions, FI) will play a pivotal role in the generation of fusion power, as they serve as a crucial source of both energy (heating) and momentum (current drive) [1][2][3].An inadequate confinement of fast ions can lead to a deterioration in reactor performance, and may even result in damage to the first-wall components [4,5].It is imperative to comprehend the mechanisms governing 4 See author list of Zohm et al 2024 Nucl.Fusion https://doi.org/10.1088/1741-4326/ad249d.* Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the transport and loss of suprathermal particles for the realization of a future fusion power plant.One of the primary identified causes for this particle transport and subsequent loss is their interaction with a broad spectrum of magnetic fluctuations.These fluctuations can be intrinsic to the plasma, such as neoclassical tearing modes [6,7], fishbones [8][9][10], or Alfvén eigenmodes [11][12][13][14][15][16][17], or they can result from externally applied perturbations [18][19][20][21].Understanding and controlling these interactions are critical steps towards achieving sustainable fusion energy production.
In this article the first tomographic inversions applied to signals from the imaging neutral particle analyser (INPA) installed at the ASDEX Upgrade tokamak [49][50][51] as well as the analysis of the INPA instrument response and its change with plasma parameters are presented.The INPA diagnostic [52,53] obtains information of the confined FI population analyzing fast neutrals produced in charge exchange (CX) reactions between FI and neutrals injected by a neutral beam injector (NBI).These CX neutrals are ionized in-vessel by an ultra-thin (20 nm) carbon foil and deflected into a scintillator by the local magnetic field.The strike position on the scintillator, observed via a high-resolution CCD camera, allows inferring the energy and radial position of the confined FI (see [49,51] for full details on the measurement principle).Nonetheless, the analysis of INPA signals is not straightforward: the finite size of the collimator limits the resolution of the diagnostic in energy and the finite width of the distribution of NBI neutrals limits the radial resolution.Also, changes in plasma profiles will affect the reionization probability of CX neutrals towards the detector and hence the INPA signal.Even if previous work has been done in the sensitivity of the AUG INPA diagnostic [49,51], the first comprehensive description of the INPA sensitivity is presented in this article.For the weight function formalism, the INPA forward model used for synthetic diagnostic introduced in [51] is reformulated as a matrix equation, which can then be solved via regression techniques to infer the fast-ion distribution.
Section 2 introduces the forward problem, i.e. the calculation of the diagnostic signal given the fast-ion distribution function, section 3 explores the dependence of the INPA weight function to the plasma profiles (nuisance parameters), and section 4 presents the inverse problem and its application to synthetic and experimental signals.

Forward problem: synthetic signal calculation
To calculate the INPA synthetic signal, the FILDSIM code [30] was upgraded to handle the INPA diagnostic.The code can now track the Monte Carlo (MC) markers produced by the FIDASIM code [54], which represent the input CX flux entering the INPA head.These markers are followed until they collide with the collimator or until they impinge on the carbon foil.At this foil, they are ionized and slowed down.Finally, they are deflected towards the scintillator by the tokamak magnetic field.The main changes in the FILDSIM code to accommodate for INPA simulations were the modification of the orbit-following module to handle non-homogeneous electric and magnetic fields, via a Boris leap frog algorithm [55], and the models to simulate the ionization and energy loss in the carbon foil.Furthermore, the capability of including arbitrary geometry from CAD files (via .stlfiles) was added, and a general speed up of the code by two orders of magnitude has been achieved.These upgrades are based on the libraries of the i-HIBPsim code [56].
As already mentioned, this upgraded version of FILDSIM is coupled to the FIDASIM code [54] to produce the INPA synthetic signals and weight functions.The code workflow to calculate the synthetic signal can be seen in figure 1.There are two ways of calculating the synthetic signals: using directly the neutral flux from FIDASIM or using the weight matrix.This latter calculation is based on the fact that the signal in a pixel (i, j), S ij , of the INPA camera, can be written as the following Fredholm integral equation of the first kind: where W ij is the weight (sensitivity) of the diagnostic, F the fast ion distribution function and Ψ stands for the set of variables sampled in the phase space.Assuming toroidal symmetry and averaging over the gyrophase, just four coordinates suffice to span the phase space of fast particles.For example, as TRANSP (NUBEAM) code [57,58]: (R, z, E, λ) 5 ; being major radius, height over the mid-plane, energy and pitch angle, respectively.However, the dependency of the weight function on λ and z is solely coming from geometric origin.Neglecting the dependency on this geometric factor with energy (which comes from the small de-focus of the optics), it is possible to decouple this dependency in an isolate term of the weight matrix: where G represents the diagnostic sensitivity in the (z, λ) space for a given R, and Wij (R, E) the radial and energy dependencies.G satisfies the normalization condition: Despite of the exact analytic form of G, owing the fact that it satisfies this normalization condition, it is direct to write equation (1) as: ) dEdR (4) but the latter is just the fast-ion distribution weighted averaged on (z, λ) over the field of view of the INPA at a point R, ⟨F⟩ INPA .Hence: The numerically calculated G(z, λ|R) and the width of the marginal distributions as a function of R can be seen in figures 2(a)-(c) respectively.Taking a discrete and uniform grid to represent the phase space, equation ( 5) can be written as a matrix problem: where Greek letters span the phase space while Latin indexes the camera pixel space.The same equation can be written in the case that the camera signal is remapped using the strike map, as explained in [51].As the grid is finite, the 2 indexes can be collapsed into one: where # i are the total number of rows of the signal matrix and # β the total number of rows in the fast-ion distribution matrix.Hence, equation ( 6) reads as: where in the last step, the constant grid spacing was included in the weight.The synthetic signals calculated with both methods (direct track of FIDASIM MC markers and matrix product with the INPA weight function) are shown in figure 3, for the plasma profiles shown in figure 4 and the fast-ion distribution of figure 8(e).Except for a minor deviation of the signal level near the magnetic axis (R ∼ 1.75 m), coming from the finite grid size, the agreement of both methods in terms of signal output is excellent.Both are also equivalent in computation time, as the calculation of the weight matrix also requires a full FIDASIM simulation.However, once the weight matrix is computed, synthetic signals for many distribution functions can be computed rapidly, just requiring a matrix multiplication [59].

Zero order dependency
Assuming a mono-energetic NBI, the flux of photons reaching the CCD camera, Φ γ can be written as: where Ω 0 is the solid angle observed by the INPA, r the distance taken from the INPA pinhole, E the energy of the FI, σ CX (v rel ) the cross section, at the relative velocity between the neutral and the FI, F the fast-ion distribution, λ 0 the pitch angle explored at the point r, ⃗ r p the pinhole position, n n the neutral density, A the pinhole area, d 0 the mean free path of CX neutrals in the plasma and P 2 a polynomial which takes into account optical transmission, ionization efficiency in the carbon foil and scintillator yield [51].The weight matrix, in r, E-coordinates would be: From this expression, it is clear that the overall response of the INPA diagnostic sensitivity to ion or electron temperature (T i and T e respectively) is small.As changes in temperatures only affect the signal via changes in the mean free path of the CX neutrals (d 0 ) and given that the energy of the fast CX neutrals is well above the electron and ion temperatures, large changes in T e or T i imply only moderate changes in d 0 .Nonetheless, notice that the experimental INPA signal itself will strongly depend on temperature, as it is proportional to the total number of fast-ions, which roughly scales with the slowing down time (as ∼ T 3/2 e ).On the contrary, changes in density strongly affect the diagnostic weight function.There is a double exponential dependency on density: the penetration of the beam neutrals into the plasma, n n (⃗ r), and the re-ionization of CX neutrals towards the diagnostic, e − |⃗ r| d 0 (⃗ r) with d 0 depending on density.The plasma Z eff (Z eff is defined as: where n i and Z i denote the density and charge of the different ion species, respectively, and n e the electron density.) will also affect significantly to the weight function, as for a fixed electron density, different Z eff implies different ion densities and different beam penetration and re-ionization probability.

Numerical calculation of the INPA weight function
Equation ( 6) can be rewritten as: where the signal is assumed to be remapped in energy and radius at the scintillator, and the weight matrix presented in the introduction was decomposed in three different matrices: • T ij αβ the translation matrix, presented in [51], which relates camera pixel position with points of the phase space.It only depends on the diagnostic characteristics (foil energy loss and ionization rate, scintillator yield and detector geometry) and the magnetic field orientation at the head • O ab ij the optical model, which relates strike points in the scintillator with their position in the camera sensor, this matrix includes the finite focus and distortion of the optics and the overall transmission factor • Ξ ων ab , which is the probability of a cell of the phase space given by ων, to contribute at the emission in a point ab of the scintillator.It is heavily affected by plasma parameters and scales as presented in the section 3.1.The matrix, T ij αβ , can be fully determined by launching Monte Carlo markers as detailed in section 5 of [51].Model of the distortion included in the matrix O ab ij is adjusted to calibration frames, as also described in [51], and the absolute transmission factor is applied following the relation obtained in the first AUG INPA measurements during MHD quiescent phases.The last matrix is obtained from the synthetic signal where a delta fast-ion distribution function is used as input.This latter calculation is equivalent to the calculations done for other FI diagnostics [30,35].
The gross weight function [48,60] of the diagnostic (the sum of all Wαβ shown in equation ( 12)), calculated for the plasma profiles shown in figure 4, can be seen at figure 5(a).
Notice how the weight is heavily biased towards the larger radii, due to the NBI neutral density and towards higher energies, due to the scintillator emission (which scales linearly with energy), transmission through the plasma (the factor d 0 from equation (10), which grows with energy) and efficiency of the carbon foil (which also grows with energy as the foil blocks the pass of low energy particles).At large energies (above 85 keV), it starts to decrease due to the decreasing CX cross sections.The sensitivity of the scintillator pixel centered at R = 1.85 m and E = 60 keV (and with a width in radius of 1.5 cm and 2 keV in energy) can be seen at figure 5(b).As expected, only a small portion of the phase space (10 keV and 6 cm of FWHM in energy and radius, respectively) contributes to the pixel.The size of this weight in energy and radius space reflects the diagnostic resolution.

Dependence on plasma profiles
In order to asses the effect of the plasma profiles in the INPA sensitivity a set of simulations where these profiles were varied was carried out.In each simulation, electron density, temperature and Z eff profiles were varied independently while changes in main ion density, ∆n i , were extracted from quasineutrality.For the ion temperature the following was imposed: ∆T e /T e = ∆T i /T i .The results from the scan can be seen in figure 6. Subplots (a-e) show the overall trend in the instrument response for different values of Z eff while subplots (fk) show the projection of gross weight function when just one of the variables is varied.As explained in section 3.1, changes in plasma temperature have a small effect on the diagnostic response (less than a 5% even for large 25% changes in temperature).On the contrary, changes in the plasma density strongly affect the INPA response.Changes in Z eff have a moderate impact on the overall response of the INPA.This is because the changes in main ion density are related to the changes in Z eff via: where Z imp represent the charge of the main impurity, assumed to be carbon, Z imp = 6.This implies that for moderate values of the effective charge, Z eff ∼ 2, the relative changes in the main ion density would be: ∆ni ni ∼ ∆Z eff 4 .Hence, in the case of ∆Z eff ∼ 2, a large relative variation in Z eff of 25% implies a moderate change in the main ion density of about 10%.

Dependence on fast-ion density
Four different populations of neutral particles can be distinguished in a tokamak: NBI injected neutrals, thermal neutrals, thermal halo neutrals and fast halo neutrals.The NBI neutrals are those directly injected by the NBI.Thermal neutrals are those naturally present in the plasma, for example those released by the reactor wall.The thermal halo neutrals are a cloud of neutrals originated from CX reactions between the NBI neutrals and the thermal ions [61].This cloud of neutrals is shifted towards the direction of plasma rotation.The fast halo neutral are a cloud of neutrals originated from the interaction of the FI with the NBI and thermal halo neutrals.Therefore, the signal can be divided into four contributions: the one coming from the interaction of FI with the NBI neutrals, the one coming from the interaction of FI with the thermal halo neutrals, the one coming from the interaction with the fast halo neutrals and the one coming from the interaction with the thermal neutrals.The thermal halo contribution dominates the AUG INPA signal [51].The density of halo neutrals is related with the plasma main ion density: the larger the  thermal ion density, the larger the halo neutral density.Hence, for a fixed electron density and Z eff , if the concentration of fast-ions grows, the thermal ion density decreases and hence the halo density decreases; thus, reducing the INPA sensitivity.This reduction is not compensated by the increase of the fast halo neutral, as CX reaction cross sections are smaller at higher energies, thus, the fast halo contribution, per neutral, to the sensitivity is smaller than the thermal halo contribution.A scan in fast-ion density was performed, reaching up to 30% of the core electron density.For all cases, a flat fastion profile was assumed.As can be observed in figure 7, the INPA sensitivity decreases up to 40% due to the thermal halo dilution.However, the INPA signal would still be larger with the larger fast-ion concentration, as the increase due to having more fast particles is larger than the decrease due to the smaller sensitivity.
If the concentration of FI in the plasma is large, the INPA inversions should be done as an iterative process: a first inversion should be performed using an estimated concentration of fast-ion to calculate the WF; secondly, the INPA weight function should be recalculated considering the observed amount of FI.The process needs to be repeated until a convergence is reached.Typically, two inversions are enough to obtain changes below 10% on the reconstructed distribution.

Inverse problem: inferring the FI distribution
The inverse problem is the inference of the fast-ion distribution from the measurements using the forward model described in the previous section.The goal is to determine the unknown distribution in equation ( 9) from the measurements.A naive inversion, i.e. using the linear regression, is ill-conditioned.Various regularization techniques can be used to allow inversion as done for CTS [35] or FIDA [38] for example.In this work, two inversion techniques were considered to perform the regularization: 0th-order Tikhonov and Elastic Net.For 0th-order Tikhonov, the cost function to minimize can be written as: where || || 2 , indicates the L 2 norm of the array and α is the regression hyper-parameter.For the case of Elastic Net, it can be written as: where || || 1 indicates the L 1 norm of the array, and l 1 is a second hyper-parameter which balances the relative strength of the 1-norm and 2-norm of the distribution in the regularization.The 0th-order Tikhonov regularization promotes smooth solutions, whereas the Elastic Net promotes sparse solutions.
In both inversions, the fast-ion distribution is obtained by minimizing the cost function: constrained to positive values of F, to avoid non-physical solutions.
Having just one INPA installed at AUG, it is not possible to infer the 4D distribution function as there is no possibility of unravelling the line integration λ and z, presented in section 2, with enough accuracy.Hence, our aim is to reconstruct the average distribution ⟨F⟩ INPA (E, R).

Inversion of synthetic signals
To test the capabilities of each inversion method, 4 fast-ion distribution functions in increasing order of complexity are used: (a) a distribution with a single peak in energy and ρ, (b) a distribution with a single peak in energy and a double peak in ρ, (c) a distribution with a double peak in energy and a single peak in ρ, and a TRANSP slowing down distribution.An overview of these distributions can be seen in figure 8.For each case, the synthetic signal to invert was produced with an independent Monte Carlo simulation from the one used to calculate the instrument function.Different grid sizes were chosen to generate the synthetic signals and inversion, but the problem was kept over-determined (more measurement data points than unknowns), as this will be also chosen for experimental signals.The passive contribution (the signal coming from the CX reactions between the fast-ions and the thermal neutrals) is not considered as it is below 5% for the AUG INPA and only present in low density discharges (line integrated density below 3 • 10 19 m −2 ) [51].
In all inversion examples, noise is added to replicate experimental conditions.To mimic the photon noise in the INPA camera (which is the dominant contribution at the AUG INPA [51]), the following model will be used: where S noisy is the noisy signal to be inverted, S exact is the synthetic signal without any noise applied and η a normally distributed random number with mean zero and variance one.With this definition, k is directly the inverse of the signal-tonoise ratio (SNR).Nonetheless, notice that the synthetic signal coming from the MC simulations will have its intrinsic noise, on the order of a few percent, due to the finite number of markers in the simulations.
Results for the case of 1% of noise added are shown in figures 9 and 10.In figure 9 each row shows the synthetic signal, the ground truth (the input fast-ion distribution, ⟨F⟩ INPA (E, R)) and the inversion results based on the synthetic signals.For the single-energy on-axis FI distribution (subplots a-d), both algorithms reconstruct perfectly the amplitude, center, and width of the distribution, as can be seen in the fits shown in subplots (a) and (e) of figure 10.For the second distribution function, the combination of on-and off-axis, both methods agree perfectly in energy with the ground truth and match the central and outer peaks, but disagree in the highfield side (HFS) peaks.Both of them predict a peak in the HFS of the right amplitude but centred ∼1 cm outwards than the ground truth and a deeper hole in the profile between this and the central peak.The cause of it is under investigation, although it could be associated with the edge of the INPA field of view, which is close to R = 1.55 m.For the case of the third test distribution, the radial location and width of the ground truth is well captured, as well as the energy location and width of the peaks.As depicted in subplot (c) of figure 10, the inversion underestimate the ground truth in the order of 10% in the height of the low energy peaks.This could be due to the fact that the diagnostic sensitivity in that lower energy range is significantly smaller than the sensitivity for large energies.For the TRANSP slowing down distribution, the two inversion methods yield different results.Overall, Elastic Net regularization is further away from the ground truth, both in total number of fast ions and relative peak height, as can be observed in figure 10(d).0th-order Tikhonov regularization match properly the 93 keV peaks and allow the identification of all energy peaks in location and width, although, as in the previous test case, lower energy peaks are slightly underpredicted in amplitude.Regarding the radial distribution: centre, width and amplitude are well captured by the inversion.
The inversions for the case of a 10% of noise are plotted in figures 11 and 12, equivalent to previous figures.As   can be seen, the noise propagates from the signals to the inversions so the 2D distributions present jitter.Nonetheless, the main characteristics of the fast-ion distribution are well captured by the inversion.0th-order Tikhonov regularization still allows to obtain the right number of peaks of the initial distribution and properly locate them in energy, radial position and widths.Moreover, when integrating in one of the dimensions to calculate the marginal fast-ion distributions, the so-called profiles, the jitter partially cancels out and results similar to the previous case with low noise are obtained.Therefore, inversion of INPA signals can yield useful information even under the presence of moderate noise in the signals.

Inversion of experimental signals
As an example of inversion of a real signal, discharge #40 284 at t = 1.17 s was chosen.This time point, corresponding to a scenario of toroidal magnetic field at the axis of −2.5 T, and a plasma current of 800 kA, was selected because it is a representative case of moderate densities (n e (0) ∼ 8 • 10 19 m −3 ), quiescent MHD activity and on-axis NBI injection (both sources 3 and 8 were on at 2.5 MW).The signal was previously analysed in [51].Plasma profiles were already shown in figure 4. The inversion results can be seen in figure 13.An overall agreement with the slowing down distribution predicted by TRANSP can be seen in subplots (c-d), where the marginal distributions from TRANSP and the inversions are compared.The inferred energy profile agrees well with TRANSP, apart from the lowenergy (below 20 keV) and near the NBI#3 injection energy (59 keV).The first can be explained to be due to uncertainties in the background neutral profile introduced in TRANSP to calculate the slowing down distribution.This background profile affect the CX losses of FI, which can reduce significantly the number of FI at low energies, as detailed in [53].The reconstructed FI density is smaller than the TRANSP prediction around 59 keV.This is in line with the observations presented in [51].In figure 18(a) of [51], it can be seen that this overprediction is within errorbars when including uncertainties in the forward model due to plasma profiles.For the radial profile, the TRANSP prediction is more extended radially and has a smaller maxima value than the inversion.This is also in line with previous observations in ASDEX Upgrade, where the experimental signal is found to be less extended in radius than the synthetic signal predicted using TRANSP distribution functions [51].

Conclusions
The FILDSIM code was upgraded to handle INPA simulations.This new code, coupled to FIDASIM, allowed us to calculate in detail the INPA instrument response and its changes due to plasma profiles.The overall INPA diagnostic sensitivity, in energy and radius coordinates, is biased towards large energies and large radius (mainly due to scintillator emission and NBI penetration) and changes exponentially with plasma density.Little changes are observed for plasma temperature, as FI (and fast CX neutrals) have large velocities with respect to the background plasma, so only small differences in CX reactivities take place.This makes the low density and high temperature scenarios to be the more suitable the reconstruction of the FI distribution; because both the INPA sensitivity and FI content in the plasma are larger in this operational range, hence maximizing the INPA signal and increasing the SNR.
The different inversions of the test distributions functions demonstrate the capabilities of the inversion techniques to reconstruct the ground truth even in the presence of a few percent of noise.When aiming to reconstruct just the marginal distributions in energy or radius, up to a moderate level of 10% is tolerable with small harm.In all tested cases, 0thorder Tikhonov inversion performs better than the Elastic Net regularization, as the latter is prone to produce large jitter.Inversion of experimental signal during a MHD quiescent phase shows a good agreement with TRANSP simulations.
In all cases, only the average distribution in the λ and z explored by the INPA could be obtained, due to the lack of a second INPA view.This set a motivation for the installation of such a second view in the near future.Combined inversion techniques, with data from other diagnostic such as FIDA in an orbit tomography formalism will be carried out in the near future.This will open new avenues on the characterization of the FI distribution.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers.The data that support the findings of this study are available upon reasonable request from the authors.

Figure 1 .
Figure 1.Flow diagram for the calculation of synthetic signal with the upgraded FILDSIM code.

Figure 2 .
Figure 2. INPA sensitivity in pitch and z space.(a) Example at R = 1.75 m.(b) Width of the marginal distribution in z, as function of R. (c) Equivalent for the case of the pitch angle.Calculated for shot #40 284 t = 1.17 s.

Figure 3 .
Figure 3.Comparison of the synthetic signals calculated with a full Monte Carlo simulation and using the weight function.(a) 2D synthetic signal.In colour, the Monte Carlo simulation, in grey contours, the signal calculated with the weight function.Dashed green lines indicate the performed cuts along the synthetic signal (shown in subplots b and (c).(b) Energy profiles of the simulations.(c) Radial profile of the simulations.

Figure 5 .
Figure 5. Example of weight function of the INPA diagnostic.(a) Gross weight of the INPA diagnostic.(b) Weight for a pixel centered at R = 1.85 m and E = 60 keV.Calculated for #40 824 and t = 1.17 s.

Figure 6 .
Figure 6.Changes in the weight function due to plasma profile changes.(a)-(e) Relative changes in the total INPA sensitivity as a function of the changes in temperature and density for each change in Z eff .(f)-(k) Projection of the instrument response of the INPA.(f)-(g) are calculated for ∆ne = ∆Z eff = 0, (h)-(i) for ∆Te = ∆ne = 0, and (h)-(i) for ∆Te = ∆Z eff = 0.

Figure 7 .
Figure 7. (a) Changes in the halo density due to the increase of the fast-ion concentration.(b) Changes in the INPA weight function due to the changes in the FI concentration.

Figure 8 .
Figure 8. Used fast-ion distribution functions for the calculation of the synthetic signals.(a) Energy profile of the used distributions.(b) Radial profile, as function of ρ. (d)-(f) Integral of the distributions in z and λ over the field of view of the INPA in z and pitch angle.

Figure 9 .
Figure 9. Inversion of Monte Carlo synthetic signals for a case of 1% of noise added.First column represent the input synthetic signal, second column the ground truth, third one the 0th Tikhonov inversion while the latest one the Elastic Net inversion.

Figure 10 .
Figure 10.Comparison of the profiles from the inversion and the ground truth for a 1% of added noise.Top row, energy profiles, bottom row, radial profiles.

Figure 11 .
Figure 11.Inversion of Monte Carlo synthetic signals for a case of 10% of noise added.First column represent the input synthetic signal, second column the ground truth, third one the 0th Tikhonov inversion while the latest one the Elastic Net inversion.

Figure 12 .
Figure 12.Comparison of the profiles from the inversion and the ground truth for a 10% of added noise.Top row, energy profiles, bottom row, radial profiles.

Figure 13 .
Figure 13.Inversion of experimental data.(a) TRANSP slowing down distribution, for discharge #40 284 at time 1.17 s, normalized to its maximum.(b) Inversion obtained with Tikhonov regularization, normalized to its maximum.(c) Energy profile of the distribution.(d) Radial profile of the distribution.Continuous blue line represent the inversion performed for the measured plasma profiles and the dashed lines the inversion performed for a density varied within the experimental uncertainties.