Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times
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Abstract
Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).
Keywords
EIS DFRT Ill-posed problem RegularizationMathematics Subject Classification
45F05 65R301 Introduction
Up to a certain extent \( g\left( \tau \right) \) provides a circuit model-free representation of essential relaxation times, which are directly connected to the charge transfer process (see, e.g., Song and Bazant 2018). One should bear in mind that not only Voigt circuit, but also other known circuit models, such as a Cole–Cole (Cole and Cole 1941) model, Davidson–Cole (Davidson and Cole 1951) model, Warburg element (Barsoukov and Macdonald 2005), etc., can also be discussed in terms of the Eq. (2).
Note that if operators \( A_{1} ,A_{2} \) are considered to be acting from the space \( L_{2} \left( {0,\infty } \right) \) of real-valued square summable functions on \( \left( {0,\infty } \right) \) then the Eq. (4), due to their finite dimension, are always solvable at least in the sense of least squares. Moreover, least square solutions of (4) can be reduced to the corresponding systems of N linear algebraic equations, such that no additional discretization is required and, as a result, no additional discretization error is introduced. Therefore, the impedance measurements considered as collocation data already hint at a way to approximate the solution of (2).
At the same time, in the EIS literature, one mainly finds two other different approaches for approximate solving of (2). In the first approach, which has been studied in Dion and Lasia (1999), Gavrilyuk et al. (2017) and Renaut et al. (2013), the integral operators \( A_{1} ,A_{2} \) in (4) are additionally discretized by means of quadrature formulas. This approach can also subsume the methods (Boukamp 2015; Boukamp and Rolle 2017; Schichlein et al. 2002) in which the Eq. (2) is reduced to a deconvolution problem by a suitable change of variables, after which a numerical Fourier transform is employed. This procedure is usually conducted by using diverse approximation techniques such as quadrature formula (Boukamp 2015). The second approach, advocated in Saccoccio et al. (2014) and Wan et al. (2015), discretizes the operators \( A_{1} ,A_{2} \) in (4) by projection onto the subspaces of piecewise linear or radial basis functions (RBFs).
In both previously mentioned approaches the level of additional discretization, governed by the number of knots of a quadrature formula or by the number of basis functions, should be properly tuned. Such tuning is especially crucial in the case of noisy impedance measurements when the application of regularization techniques avoids numerical instabilities in solving (4). Then, according to the Regularization theory (see, e.g., Mathe and Pereverzev 2003) the level of additional discretization of \( A_{1} ,A_{2} \) in (4) should be coordinated with the amount of regularization. However, such coordination has not been discussed in the aforementioned literature yet. At the same time, this discretization issue does not even appear in (4) as no additional discretization of the operators \( A_{1} ,A_{2} \) is introduced. Therefore, in the present paper, we study a new approach to obtain \( g\left( \tau \right) \) that avoids any additional discretization of the operators in (4).
Furthermore, it is known that the imaginary and real components \( Z^{\prime \prime } \left( {\omega_{j} } \right), Z^{\prime } \left( {\omega_{j} } \right) \) of the impedance have different importance. A more thorough analysis of Eq. (3) indicates that \( g\left( \tau \right) \) has a greater impact on \( \text{Im} \left( {Z\left( \omega \right)} \right) \) then on \( \text{Re} \left( {Z\left( \omega \right)} \right) \) (see, e.g., Dion and Lasia 1999). In that case it seems reasonable to treat the Eq. (4) with different amount of regularization (i.e., by applying two regularization parameters). At the same time, in the aforesaid literature, regularization of the Eq. (4) is governed by only one regularization parameter that does not allow a desired flexibility in exercising the regularization. There is one exception though, namely the paper (Zhang et al. 2016) that proposes to minimize a multi-parameter version of the Tikhonov regularization functional also over the values of regularization parameters. However, the above minimization problem may have several local minima, and one of them corresponds to zero values of the regularization parameters that leads to unregularized least-squares.
Herein, we propose a new approach that applies a multi-parameter regularization scheme without unnecessary additional discretization. We have also added to this approach an ability to automatically choose regularization parameter values. Note that this kind of endeavor has not been reported in the aforementioned literature. In addition, in order to enable an automatic regularization in the present study, we use the idea (Chen et al. 2015) of an aggregation of regularized solutions corresponding to different values of multiple regularization parameters.
2 Multi-parameter regularization of the collocated impedance equations
From (13), (14) it is clear that for given \( \lambda_{1} ,\lambda_{2} \) the regularized approximate solution \( g_{{\lambda_{1} ,\lambda_{2} }} \) of (4) can be constructed without additional discretization of the integral operators \( A_{1} , A_{2} \).
3 Aggregation of the regularized approximants in weighted norms
While we have described the explicit procedure (13), (14) for approximating the solutions of (2) directly from the impedance measurements without the application of discretization, there is still a question about the choice of the regularization parameters \( \lambda_{1} ,\lambda_{2} \) that determines suitable relative weighting between these measurements. By setting \( \lambda_{1} = 0\,{\text{or}}\,\lambda_{2} = 0 \), one may reduce this question to the case discussed in Gavrilyuk et al. (2017). In another particular case \( \lambda_{1} = \lambda_{2} = \lambda \), one may choose a suitable value of \( \alpha = \lambda^{ - 1} \) by a cross-validation technique, as it was suggested in Wan et al. (2015). In both particular situations, one in fact deals with a single-parameter regularization which is applied in, e.g., Weese (1992) FTIKREG and DRTtools (Wan et al. 2015) software. However, a multi-parameter regularization is much less studied in EIS topic, especially when multiple regularization parameters are employed to construct a common misfit measure as in (5). Herein, we use new findings developed originally for inverse problems of satellite geodesy (Chen et al. 2015) and recently adjusted in the context of EIS (Zic and Pereverzyev 2018).
On the other hand, the components \( F_{m} \) of the vector \( \vec{F} \) in (17) depend on the unknown solution \( g \) of (2), and therefore are inaccessible.
At the same time, in Chen et al. (2015) and Kindermann et al. (2018) we can find an approach to estimate the components of the vector \( \vec{F} \) by using the so-called quasi-optimality criterion in the linear functional strategy. The advantage of this approach is that the values of the scalar products \( F_{m} = \left\langle {g,g^{m} } \right\rangle_{{L_{{2,\nu \left( {W_{min} , W_{max} } \right)}} }} \) of the solution \( g \) can be estimated much more accurately than the solution \( g \) in the norm \( \left\| \cdot \right\|_{{L_{2,\nu } \left( {W_{min} W_{max} } \right)}} \). According to Chen et al. (2015) and Kindermann et al. (2018) we estimate the scalar product \( \left\langle {g,g^{m} } \right\rangle \) by \( \left\langle {f_{\alpha } ,g^{m} } \right\rangle_{{L_{{2,\nu \left( {W_{min} , W_{max} } \right)}} }} \), where \( f_{\alpha } \left( \tau \right) \) is the regularized approximate solutions \( g_{{\lambda_{1} ,\lambda_{2} }} \left( \tau \right) \) given by (13) and constructed for \( \lambda_{1} = 0, \lambda_{2} = \alpha \), with the use of only imaginary part of the impedance data. The reason for this is that the imaginary part may allow better accuracy than the real part of the impedance (see, e.g., Dion and Lasia 1999).
The theoretical analysis of Kindermann et al. (2018) guarantees that the values of \( F_{m} = \left\langle {g,g^{m} } \right\rangle \) are well approximated by the values of \( \tilde{F}_{m} = \left\langle {f_{{\alpha_{sm} }} ,g^{m} } \right\rangle \).
Recall that vector \( \vec{c} \) of the coefficients of the best linear combination of (16) approximating the solution \( g \) of (2) in the norm \( \left\| \cdot \right\|_{{L_{2,\nu } \left( {W_{min} W_{max} } \right)}} \) should solve the matrix vector Eq. (17).
We have described an adaptive procedure that automatically constructs an approximate solution of (2). This procedure should theoretically perform at the level of the best regularized approximant \( g_{{\lambda_{1} ,\lambda_{2} }} \left( \tau \right) \) calculated according to (13), (14) for a given range of \( \lambda_{1} , \lambda_{2} \). The input of the procedure consists of the impedance data \( Z\left( {\omega_{j} } \right), j = 0,1, \ldots ,N - 1 \), the weights \( \gamma_{j} , j = 0,1, \ldots ,N - 1 \) determining the misfits measures, the endpoints \( W_{min} , W_{max} \) of the time window of interest, and the numbers \( \lambda_{0,1} ,\lambda_{0,2} ,\alpha_{0} ,P,Q,S \) defining the range of the regularization parameters under consideration.
4 Experimental
4.1 Synthetic and polluted ZARC2 and FRAC2 impedance data
Parameters used to compute the synthetic ZARC2 and FRAC2 data
Synthetic data | \( R_{\infty } \) (Ω cm^{2}) | R_{1} (Ω cm^{2}) | \( \tau_{0,1} \) (s) | n_{1} | R_{2} (Ω cm^{2}) | \( \tau_{0,2} \) (s) | n_{2} |
---|---|---|---|---|---|---|---|
ZARC2/FRAC2 | 10 | 50 | 0.01 | 0.7 | 50 | 0.001 | 0.7 |
4.2 Analytical DFRT for ZARC2 and FRAC2
4.3 Measured impedance data
For the purpose of this study experimental measurements were performed on solid oxide fuel cells (Fig. 11). The cells were of industrial-size with a chemically active surface of 80 cm^{2}, whereby the operating temperature was set to be 800 °C. The fuel electrode was fed with humidified hydrogen, and the air electrode was supplied with air. For the purpose of this study impedance measurements were performed starting with the open circuit conditions, further loading the cell and decreasing the voltage down to 0.7 V. The EIS measurements were carried out using a galvanostatic technique. The AC amplitude was set to be 4% of the DC values, whereby the voltage was measured. The measurements were performed in a frequency range between 10 kHz and 100 mHz. For more detailed information about the experimental setup, the authors refer to Subotić et al. (2018).
4.4 DFRT software used in this work
The different strategy utilized in available DFRT softwere that apply regularization
Software | Software requirements (license) | Regularization | Regularization parameter choice | Discretization |
---|---|---|---|---|
FTIKREG | None (free^{a}) | Single-parameter | Manual or SC-method (Honerkamp and Weese 1990) | Yes |
DRTtools | Proprietary Matlab (free^{b}) | Single-parameter | Manual | Yes |
DFRT-Py | None (MIT license^{c}) | Mulit-parameter | Automatic | None |
Parameters used to extract DFRT from ZARC2 and FRAC2 impedance data in this work
Software | Regularization parameter | Coefficient^{a} to FWHM | Discretization method | Combined \( \text{Re} \left( {Z\left( \omega \right)} \right) \) and \( \text{Im} \left( {Z\left( \omega \right)} \right) \) |
---|---|---|---|---|
FTIKREG | SC-method | None | Gaussian | Yes |
DRTtools | 10^{−3} | 0.2 | Quadrature | Yes |
DFRT-Py | Automatic | None | None | Yes |
As listed programs (Table 2) apply single- and multi-parameter regularization, it is important to emphasize that DFRT data were constructed from combined \( \text{Re} \left( {Z\left( \omega \right)} \right) \) and \( \text{Im} \left( {Z\left( \omega \right)} \right) \) parts (see Table 3). Interestingly, some researchers prefer the application of only \( \text{Re} \left( {Z\left( \omega \right)} \right) \) part because it is less affected by noise and errors (Ivers-Tiffee and Weber 2017). This approach is a common one, especially when dealing with noisy data. On the other hand, there are papers (Dion and Lasia 1999) that claim that better DFRT results are obtained by the usage of only \( \text{Im} \left( {Z\left( \omega \right)} \right) \). The choice to use only \( \text{Im} \left( {Z\left( \omega \right)} \right) \) can be explained by the fact that DFRT has greater impact on this part of impedance data (see (3)).
5 Results and discussion
5.1 Existing DFRT approaches
According to DFRT literature (Ivers-Tiffee and Weber 2017; Kobayashi and Suzuki 2018), there are numerous approaches to extract the Distribution Function of Relaxation Times (DFRT) data from electrochemical impedance spectroscopy (EIS) data. The majority of reported approaches is based on evolutionary programming (Hershkovitz et al. 2011; Tesler et al. 2010) and Monte Carlo techniques (Tuncer and Macdonald 2006), maximum entropy model (Horlin 1998), Fourier filtering (Boukamp 2015; Schichlein et al. 2002), and regularization techniques (Dion and Lasia 1999; Kobayashi et al. 2016; Kobayashi and Suzuki 2018; Wan et al. 2015; Zic and Pereverzyev 2018). Additionally, the first software to extract DFRT from EIS data is based on Fourier transform technique (Kobayashi and Suzuki 2018). However, in this work we are focused on the regularization techniques that are embedded in FTIKREG, DRTtools and DFRT-Py software (Table 2); and thus, there are several facts that should be discussed.
First, the approaches in FTIKREG and DRTtools are based on discretization methods (Table 2). To be precise, in FTIKREG, all functions and operators are approximated by finite-dimensional vectors and matrices (Weese 1992) whereas, in DRTtools the approximation error is somewhat reduced due to the application of radial basis functions (RBFs) (Wan et al. 2015) as discretization basis. On the other hand, DFRT-Py applies table integrals; and thus, any additional discretization errors are avoided (Zic and Pereverzyev 2018). Second, in DRTtools the regularization parameter should be given a priori, whilst in FTIKREG this parameter can be given manually or it can be obtained by a self-consistent (SC) method (Honerkamp and Weese 1990). However, this method is heavily based on the assumption that the noise is independent standard Gaussian random variable (Honerkamp and Weese 1990), which is frequently not the case when dealing with measured EIS data. Oppositely to DRTtools and FTIKREG, in DFRT-Py, the regularized solutions are aggregated, which allows an automatic regularization (Zic and Pereverzyev 2018). Third, the discretization procedure in DRTtools requires a priori choice of RBF shape parameter (Wan et al. 2015), which indicates that both regularization and the shape parameters have to be properly chosen. Interestingly, this action is avoided when operating with FTIKREG and DFRT-Py, as they do not apply any parameterized basis functions. And finally, DRTtools and FTIKREG apply a single-parameter regularization even when using combined real and imaginary impedance parts, whereas DFRT-Py applies a multi-parameters regularization.
To sum up, the aforementioned software (Table 2) apply different approaches to extract DFRT data from the EIS data; and thus, to properly probe different approaches they were tested by a series of the synthetic and measured data.
5.2 DFRT study of noisy ZARC2 and FRAC2 impedance data
5.3 Effect of missing data points on DFRT study
The idea to study the missing data effect originates from the literature (Boukamp and Rolle 2017; Ivers-Tiffee and Weber 2017) as some authors (Boukamp and Rolle 2017) reported that this effect induces changes in DFRT spectra. On the other hand, one group of authors concluded (Ivers-Tiffee and Weber 2017) that this effect has no impact on DFRT study. However, the conclusions presented in Boukamp and Rolle (2017) and Ivers-Tiffee and Weber (2017) are obtained by using two different software (FTIGREG and DRTtools) that apply single-parameter regularization. Thus, it would be intriguing to see the effect of missing data onto both single- and multi-parameter regularizations techniques.
Furthermore, supplementary calculations with DRTtools indicate that when using only real impedance part (left inset in Fig. 7a), both the depression and the peaks turn out to be more pronounced. At the same time, when using only imaginary impedance part (right inset in Fig. 7a), the depression disappears as two DFRT_{ZARC2} peaks are merged into one. This clearly indicates that the same regularization parameter value (e.g., 10^{−3}) cannot be applied when using single or combined impedance parts in the regularization. To rephrase it, for a proper regularization of combined real and imaginary impedance parts, the multi-parameter regularization (as in DFRT-Py) seems to be unavoidable.
Next, a computation of FRAC2 impedance data shows that both DFRT-Py (Fig. 6b) and DRTtools (Fig. 7b) are not drastically affected by the missing data effect. To be specific, this effect increased oscillation at τ > 0.01 s in Fig. 6b, but there are no additional peaks in Fig. 7b. Moreover, it appears that DRTtools yielded almost identical data in Fig. 7b and in Fig. 7b insets, which suggests that the missing data effect is not observed because RBF cannot properly mimic discontinuities in DFRT_{FRAC2} anyway.
To summarize, it appears that both the missing data effect and the application of additional discretization basis (i.e., RBFs) yield similar DFRT_{ZARC2} and DFRT_{FRAC2} pictures and hinder their distinguishment. At the same time, this kind of problems can be avoided when using DFRT-Py as this software does not apply any unnecessary discretization techniques. However, the full benefit of the application of the regularization without unnecessary discretization will become obvious in the next section when using impedance data corresponding to randomly spaced (in logarithmic scale) frequency values.
5.4 Effect of unequally spaced frequency data on DFRT_{ZARC2} and DFRT_{FRAC2} study
In contrast, DFRT-Py yielded DFRT peaks with no positions offset, but with higher level of oscillation.
5.5 Real experiment data in DFRT_{ZARC2} and DFRT_{FRAC2} study
6 Conclusions
We have tested and analyzed some of the available DFRT programs based on different regularization strategies i.e., FTIKREG and DRTtools apply single-parameter regularization and diverse discretization techniques whereas, DFRT-Py applies the multi-parameter regularization without any additional discretization.
Our tests show that a single-parameter regularization is suitable for moderately corrupted impedance data. On the other hand, a multi-parameter regularization approach is able to handle the cases where the level of data corruption is higher.
In this work, the positions of reconstructed DFRT_{ZARC2} and DFRT_{FRAC2} peaks were always equal to the analytical ones only in the case of DFRT-Py. This clearly supports our belief that a full regularization effect can only be obtained when using multi-parameters regularization and directly applying it to impedance data without any additional discretization.
However, when low quality measured experimental data were analyzed by methods under comparison, the positions of DFRT peaks were not the same. This indicates that both software should be used when dealing with low quality data.
Footnotes
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). M. Žic gratefully acknowledges the stimulation program “Joint Excellence in Science and Humanities” (JESH) of the Austrian Academy of Sciences for providing supporting funds. S. Pereverzyev Jr. gratefully acknowledges the support of the Austrian Science Fund (FWF): Project P 29514-N32.
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