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Analysis, Design, and Generalization of Electrochemical ImpedanceSpectroscopy (EIS) Inversion AlgorithmsTo cite this article: Surya Effendy et al 2020 J. Electrochem. Soc. 167 106508

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https://doi.org/10.1149/1945-7111/ab9c82

Analysis, Design, and Generalization of ElectrochemicalImpedance Spectroscopy (EIS) Inversion AlgorithmsSurya Effendy,1 Juhyun Song,1 and Martin Z. Bazant1,2,z

1Massachusetts Institute of Technology, Department of Chemical Engineering, Cambridge, MA 02142, United States ofAmerica2Massachusetts Institute of Technology, Department of Mathematics, 182 Memorial Dr., Cambridge, MA 02142, UnitedStates of America

We introduce a framework for analyzing and designing EIS inversion algorithms. Our framework stems from the observation offour features common to well-defined EIS inversion algorithms, namely (1) the representation of unknown distributions, (2) theminimization of a metric of error to estimate parameters arising from the chosen representation, subject to constraints on (3) thecomplexity control parameters, and (4) a means for choosing optimal control parameter values. These features must be present toovercome the ill-posed nature of EIS inversion problems. We review three established EIS inversion algorithms to illustrate thepervasiveness of these features, and show the utility of the framework by resolving ambiguities concerning three more algorithms.Our framework is then used to design the generalized EIS inversion (gEISi) algorithm, which uses Gaussian basis functionrepresentation, modality control parameter, and cross-validation for choosing the optimal control parameter value. The gEISialgorithm is applicable to the generalized EIS inversion problem, which allows for a wider range of underlying models. We alsoconsidered the construction of credible intervals for distributions arising from the algorithm. The algorithm is able to accuratelyreproduce distributions which have been difficult to obtain using existing algorithms. It is provided gratis on the repository https://github.com/suryaeff/gEISi.git.© 2020 The Electrochemical Society (“ECS”). Published on behalf of ECS by IOP Publishing Limited. [DOI: 10.1149/1945-7111/ab9c82]

Manuscript submitted April 18, 2020; revised manuscript received May 27, 2020. Published June 22, 2020.

Electrochemical impedance spectroscopy (EIS) is a centraltechnique for analyzing electrochemical systems such as corrodingsurfaces,1–4 fuel cells,5–9 batteries,10–14 sensors,15 solar cells,16,17

and biological systems.18,19 The application of a small oscillatingcurrent (or potential) results in a small oscillating potential(or current), which is then studied as a function of the angularfrequency of the oscillation.20–22 The small magnitude of the inputoscillation suppresses nonlinear behaviors in the frequency domain,allowing the system to be studied using relatively simple perturba-tion models.23 The ratio of the oscillating potential to the oscillatingcurrent in frequency domain, i.e., the impedance, often exhibitsfeatures which are easily correlated with the parameters of themodel.

The success of EIS data analysis is contingent on the ability toexpress the system as a quantitative physical model, but it is oftenthe case that the system is not sufficiently well-understood to allowsuch a model to be written. In such cases, the distribution ofrelaxation time (DRT) problem may be solved, wherein the under-lying model is assumed to be a series of relaxation processes with adistribution of characteristic timescales.24–31 The DRT problem is alinear Fredholm integral of the first kind, which can be solved viaFourier, Laplace, or Mellin transform, although questions con-cerning the existence and uniqueness of the resulting solutioninevitably arise.32,33 Similar inverse problems have a long historyin statistical mechanics, where microscopic distributions, such as thedensity of states or the partition function, are directly inverted fromthermodynamic data.34 Resolution of the DRT problem yields thedistribution of characteristic timescales, which can be used toidentify the number, size, and average timescale of physicalprocesses within the system,24,25,27,29 and in some cases, assessthe propriety of the assumed relaxation process.24,27,29 The DRTproblem also allows resolution of overlapping features in theimpedance spectra, whose visual identification is often challenging.The DRT model has found increasing acceptance over the pastdecade, as evidenced by recent works on batteries,25,30,31,35–37 fuelcells,24,27,38–40 and geophysics.26

This acceptance may be partly attributed to the relative maturityof DRT inversion algorithms, i.e., algorithms used to obtain theDRT, which span a range of approaches such as Fourier

transform,9,24 construction of L-curves,26 genetic algorithm,27,28

maximum entropy method,29 various least-squares methods,41,42

various regularization methods,25,30,40,42–44 and Monte Carlomethod.45 Generic inversion toolboxes46 have also been used tosolve the DRT problem.47 This impressive breadth of approachesposes an interesting challenge concerning the classification anddesign of DRT inversion algorithms. As noted by Saccoccio, Han,Chen, and Ciucci, there is a need for “speed and accuracy bench-marks for the various methods”,30 so as to allow one to characterizethe optimality of inversion algorithms, and choose an appropriatealgorithm given the nature of the problem and the availability ofcomputational resource. Concurrently, there is a need for a frame-work which can be used to analyze the optimality of inversionalgorithms, so as to allow the design of an appropriate algorithmgiven the nature of the problem and the availability of computationalresource.

We consider one such framework, founded upon the observationthat there are four features common to all well-posed EIS inversionalgorithms. These features can be broadly stated as (1) therepresentation of unknown distributions, (2) the minimization of ametric of error to estimate the parameters arising from the chosenrepresentation, subject to constraints on (3) the complexity controlparameters, and (4) a means for choosing optimal control parametervalues. For brevity, we will refer to these four features as therepresentation, the interior problem, the complexity control para-meter, and the exterior problem. We illustrate the prevalence of theframework through a review of three established DRT inversionalgorithms, and show the utility of the framework by resolvingambiguities and difficulties concerning three more DRT inversionalgorithms. We then introduce the generalized EIS inversionproblem, which expands upon the concept of DRT to allow for awider range of underlying models, design the corresponding general-ized EIS inversion (gEISi) algorithm, and consider the constructionof credible intervals for the parameters arising from the algorithm.The gEISi algorithm is able to reproduce distributions which exhibita wide range of smoothness, which has been difficult to performusing existing EIS inversion algorithms. We also present validationagainst synthetic data generated from the distributed Randles circuit,which is impossible to analyze using existing EIS inversionalgorithms. The application of the gEISi algorithm to experimentaldata modelled by the distributed Randles circuit will be the subjectof a future companion paper.48zE-mail: [email protected]

Journal of The Electrochemical Society, 2020 167 1065081945-7111/2020/167(10)/106508/21/$40.00 © 2020 The Electrochemical Society (“ECS”). Published on behalf of ECS by IOP Publishing Limited

https://orcid.org/0000-0002-9015-3777

https://github.com/suryaeff/gEISi.git


https://doi.org/10.1149/1945-7111/ab9c82

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Overview of the Concept

We begin with the nomenclature of terms used in the presentwork. The term EIS inversion problem refers to a general class ofproblems in which distributions are inferred from the impedancespectra, assuming some simplified underlying physics. The classicalexample of an EIS inversion problem is the DRT problem, which hasbeen discussed in the introduction :

òw g tg t

wtt=

++¥

¥

¥Z Ri

d R; ,1

10

ˆ ( ( )) ( ) [ ]

The DRT problem uses the Debye model, which assumes anunderlying relaxation process.20 Note that w g t¥Z R; ,ˆ ( ( )) impliesthat the predicted impedance (Z ) has the angular frequency (ω) asindependent variable, and the Ohmic resistance ( ¥R ) as well as theDRT (γ(τ)) as model parameters. The unit of γ(τ) is Ω/s, indicatingthat γ(τ) is not a probability distribution. Data from the literaturesuggests that the domain of γ(τ) can spread over several orders ofmagnitude.49,50 It is thus convenient to perform the substitution

tºv ln( ):

òww

=+

+¥-¥

¥

¥Z R G vG v

i vdv R; ,

1 exp2ˆ ( ( )) ( )

( )[ ]

Here G(v)= τγ(τ) is the DRT on a logarithmic scale. The objectiveof the DRT problem is to estimate G(v) given a vector of measuredimpedance values Z, with Z(ωj) as the jth component of Z.

Florsch, Revil, and Camerlynck proposed a variation to the EISinversion problem, wherein the Debye model is replaced with theHavrilliak-Negami model26:

òww

=+

+a b¥-¥

¥

¥Z R G vG v

i vdv R; ,

1 exp3ˆ ( ( )) ( )

[ ( ( )) ][ ]

Here α and β are constants which are determined prior to solving theEIS inversion problem. In particular, Florsch considered the casewhereby α= 0.5 and β= 1, which corresponds to an infinite seriesof parallel diffusive-capacitive processes. The motivation for usingthe Havrilliak-Negami model is simple, yet significant: if the correctmodel is chosen, the distribution G(v) should exhibit low com-plexity. This supposition is used to conclude that the impedance dataobtained from their archaeological sample can be best explained bythe polarization of an electronically conductive sample.

Yet another variation to the EIS inversion problem is thedistribution of diffusion time (DDT) problem,43 which is appropriatefor cases whereby prior knowledge suggests that the underlyingphysics is better explained by a diffusive process. The model ischosen based on the nature of the diffusive process:

òw w=-¥

¥Y G v G v y v dv; , 4ˆ ( ( )) ( ) ˆ ( ) [ ]

Here Y is the predicted admittance, and y describes the underlyingdiffusion process. Song and Bazant provides a list of expressions fory based on the boundary conditions and the symmetry of the systemunder consideration.43 The integral is performed over the admittancerather than the impedance, so as to reflect the parallel nature ofdiffusion normal to the surface of electrodes. The DDT problem wassolved for the impedance spectra of a silicon-nanowire Li-ion batteryanode,51 yielding a distribution of nanowire radii, which wassubsequently confirmed against scanning electron microscopy imageanalysis.

The works of Florsch, Revil, and Camerlynck26 and Song andBazant43 demonstrate that prior knowledge can be used to constructthe underlying physics of an EIS inversion problem, such that theresulting distribution G(v) corresponds to an observable, as opposedto a mathematical construct whose utility is restricted to theidentification of the number, size, and average timescale of under-lying processes.24,27,29 This motivates the idea of a generalized EIS

inversion problem, in which the form of the underlying physics isnot specified a priori, and the corresponding generalized EISinversion algorithm, which solves the problem independently fromthe underlying physics. We will explore this concept in greater detailin the section “Design of Algorithm” .

The remainder of this section, which discusses the proposedframework, and the section “Analysis” , which reviews selectalgorithms, will focus on the DRT problem, given its prevalencein the literature. However, the concepts outlined therein are alsoapplicable to the more general class of EIS inversion problem.

Framework.—We claim that all well-posed DRT inversionalgorithms possess four common features, namely, representation,interior problem, complexity control parameter, and exterior pro-blem. The representation approximates the unknown distributionwith a discrete or continuous set of basis functions. The mostcommon representation is the sum of triangular basis functions:

åº D=

G v g vg; 5m

M

m m1

( ) ( ) [ ]

D º

--

Î

--

Î

-

--

+

++

v

v v

v vv v v

v v

v vv v v

if ,

if ,

0 otherwise

6m

m

m mm m

m

m mm m

1

11

1

11

⎧

⎨⎪⎪

⎩⎪⎪

( )

[ ]

[ ] [ ]

Here g is the vector of basis function weights with gm as the mthcomponent of g, m is the index of the basis functions, M is the totalnumber of basis functions, and Δm(v) is the basis function. Thisrepresentation is functionally identical to linear interpolation, andpossesses variations according to the treatment of the end-points.41

It is useful to distinguish between fixed-mesh and floating-meshrepresentations. Floating-mesh representations incorporate the meshpoints as parameters in the basis function expansion, while fixed-mesh representations do not. For example, Eq. 5 is a fixed-meshrepresentation, since it is only parameterized by the weights of thebasis function (gm), and not by the location of the mesh points (vm).This distinction is significant, because floating-mesh representationstend to approximate distributions using a smaller number ofparameters. To illustrate this point, consider the true underlyingdistribution shown in Fig. 1a, which is unimodal Cole-Cole centeredat t= 0. We collect 10 noisy impedance measurements per decadebetween ω1= 10−4 and ωJ = 104, and define the accuracy of arepresentation as:

òJ º --¥

¥G v G v G v dv 7true

2[ ( )] [ ( ) ( )] [ ]

Here Gtrue(v) is the true underlying distribution. We invert theunderlying distribution using the RR/RI algorithm introduced in thesection “RR/RI Algorithm” , which employs a fixed-mesh representa-tion, and the gEISi algorithm introduced in the section “GeneralizedEIS Inversion Algorithm” , which employs a floating-mesh represen-tation. The mesh boundaries of the RR/RI algorithm are determinedfrom the upper and lower limits of the measurement angularfrequencies25,26,30,31,41–43:

w= -v ln 8J0 ( ) [ ]

w= -+v ln 9M 1 1( ) [ ]

We then adjust the number of parameters used in the RR/RIalgorithm to match the accuracy of the gEISi algorithm.

The unimodal Cole-Cole distribution is accurately reproduced byboth algorithms, but the RR/RI algorithm requires 86 parameters tomatch the accuracy of the gEISi algorithm, which only requires 6parameters. This order-of-magnitude reduction in the number of

Journal of The Electrochemical Society, 2020 167 106508

parameters does not translate to a reduction in computational time,which rather increases by an order of magnitude. The relativeefficiency of the RR/RI algorithm,30 as well as the majority ofalgorithms employing the fixed-mesh representation,25,26,31,43 arisesfrom the use of near-analytical linearly constrained quadraticprogramming. In contrast, the floating-mesh representation almostinevitably requires stochastic programming to solve,27,28 which canbe computationally expensive. Nevertheless, this reduction in thenumber of parameters will be significant in context of the general-ized EIS inversion problem, which is discussed in the section“Generalized EIS Inversion Problem” .

The representation of the unknown distribution leads to a set ofunknown parameter values, which are estimated via the interiorproblem, subject to a constraint on the complexity control parameter.The interior problem shows the least variation among the fourcommon features, and is typically the minimization of a misfit. Theidea of estimating parameter values subject to a constraint on thecontrol parameter will be demonstrated in a subsequent paragraph.

The third common feature is the complexity control parameter.Implicit in the works of Florsch, Revil, and Camerlynck26 and Songand Bazant,43 which have been discussed in “Overview of theConcept” , is the idea that the application of a proper underlyingmodel tends to result in a distribution with low complexity. As acorollary, if two different underlying models result in two distribu-tions with different complexities, the underlying model yielding thedistribution of lower complexity should be favored. It is important torecognize outright that this is a restatement of Occam’s razor,52–54

i.e., among competing hypotheses, all else being equal, favor thatwhich requires the least assumptions. We will explore the relation-ship between complexity and Occam’s razor in greater depth in thesection “Design of Algorithm” . The existence of complexity is alsorecognized by Hershkovitz, Tomer, Baltianski, and Tsur, who

defines complexity as the number of parameters needed to approx-imate a DRT.27,28

Intuitively, we grasp that a highly fluctuating distribution iscomplex, while a smoothly varying one is not. The idea ofcomplexity is central to the DRT problem, because the ill-posednature of the problem is in effect resolved by limiting the complexityof the distribution. Consider the hierarchical Bayesian algorithm ofCiucci and Chen.25 For an exponential hyperprior, it can be shownthat the interior problem takes the form:

xº - + +*g Z Z Largmin ln 10g

g22

misfit

12

1

penalty

ˆ [( ) ] [ ]

Here L1 is the first derivative operator, ξ is a parameter describingthe width of the hyperprior, and 1· and 2· are the first and secondnorm operators, respectively. This interior problem may be thoughtof as the minimization of model misfit subject to a penalty on thefirst derivative of g. The parameter ξ controls the complexity of theresulting DRT. When ξ becomes large, the penalty becomesinsensitive to changes in the first derivative, yielding a highlyoscillatory distribution. In contrast, when ξ becomes small, due tothe singular nature of the ln[·] operator, L g1 tends to vanisheverywhere, yielding a more uniform distribution.

Hereon, we will define the complexity control parameter as aquantitative constraint on the complicated behavior of a DRT. In thehierarchical Bayesian algorithm, the control parameter is ξ. It isuseful to distinguish between the complexity control parameter andthe complexity, the latter being a description of the complicatedbehavior. In the context of the hierarchical Bayesian algorithm, thecomplexity is the first derivative of the distribution, i.e., L g1

2( ) .The exterior problem selects an appropriate value for the control

parameter. The exterior problem can be quantitative, as is the case formost EIS inversion algorithms, but it can also be qualitative, as wasdone by Boukamp, who uses common sense for the “‘adjustment of the[parameters of the] window function, which should lead to an‘acceptable” DRT”.24 Note that the parameters of the window functionare the control parameters of Boukamp’s DRT inversion algorithm. Incontrast to the the representation, which is largely dominated bytriangular basis functions, or the interior problem, which is typically theminimization of a misfit, the exterior problem exhibits great variation inform. We will discuss these in greater detail in subsequent sections.

The relationship between the four common features described inthis section has been summarized in Fig. 2. EIS inversion algorithmscan be constructed by defining the four common features, which actas standardized modules in a program. We will now demonstrate theprevalence of the four common features in the literature.

Sample Algorithms

In this section, we classify three distinct EIS inversion algorithmsaccording to the four common features. The first algorithm, taken fromSaccoccio, Han, Chen, and Ciucci, consists of ridge regression coupled

Figure 2. The common features of well-posed EIS inversion algorithms.After defining the representation and the complexity control parameter, onecan set up an interior problem which estimates the unknown parameterssubject to a constraint on the control parameter, and an exterior problemwhich determines an appropriate value for the control parameter.

Figure 1. (a) True and inverted distributions corresponding to a unimodalCole-Cole distribution, obtained using the RR/RI algorithm and the gEISialgorithm. (b) The accuracy of the representation as a function of the numberof parameters used in the RR/RI algorithm. The gEISi algorithm returns aDRT containing 6 parameters.


to a real-imaginary cross-validation routine.30,31 The work ofSaccoccio, Han, Chen, and Ciucci is notable for its thoroughinvestigation of the real-imaginary cross-validation routine, whichprovides compelling evidence for the effectiveness of the routine incases whereby the error in the real part of the impedance is independentof the error in the imaginary part of the impedance. Hereon, we willrefer to this algorithm as the ridge regression/real-imaginary cross-validation (RR/RI) algorithm. The second algorithm is the least-squaresvector minimization (LEVM) algorithm of macdonald.41,42 This algo-rithm possesses an implicit exterior problem, which will becomeapparent through a comprehensive analysis of the surrounding litera-ture. The third algorithm, adapted from Hershkovitz, Tomer, Baltianski,and Tsur, provides an example of the four common features in a DRTinversion algorithm which uses a floating-mesh representation.27,28 Indeference to the work of Hershkovitz, Tomer, Baltianski, and Tsur, wewill refer to this algorithm as the impedance spectroscopy analysisusing genetic programming (ISGP) algorithm.

The algorithms considered in this and subsequent sections are notpresented in their original form. To improve ease of understanding,simplifications have been made to parts of these algorithms, andvarious symbols have been changed to align with the present work.Unless stated otherwise, these changes are superficial, and shouldnot alter the behavior of the algorithms substantially.

RR/RI algorithm.—The work of Saccoccio, Han, Chen, andCiucci provides the quintessential example of the four commonfeatures. The DRT problem solved by Saccoccio, Han, Chen, andCiucci is a simplified form of Eq. 2, where the Ohmic resistance hasbeen manually eliminated from the experimental data.

òww

=+-¥

¥Z G v

G v

i vdv;

1 exp11ˆ ( ( )) ( )

( )[ ]

The chosen representation is the sum of triangular basis functions,i.e., Eq. 5, which introduces g as unknown parameter values:

òww

=+-¥

¥Z

G v

i vdvg

g;

;

1 exp12ˆ ( ) ( )

( )[ ]

The interior problem is composed of two optimization subrou-tines:

lº - +g Z Z Largmin im 13g

g22

1 22[ ˆ ] [ ]

lº - +g Z Z Largmin re 14g

g22

1 22[ ˆ ] [ ]

Here λ is the regularization parameter, and re[·] and im[·] are the realand imaginary operators, respectively. These equations are similar toEq. 10, in that they involve the minimization of misfit subject to apenalty on the first derivative of g, but differ in the use of only theimaginary (Eq. 13) and the real (Eq. 14) parts of the data set. Thesuperscripts and denote that the parameters are obtained usingonly the imaginary and real parts of the data set, respectively.

The complexity control parameter is the inverse of the regular-ization parameter (1/λ), and the complexity is the first derivative ofthe DRT. This complexity control parameter makes sense, becauseincreasing λ causes the penalty on the square of the first derivative togrow, resulting in smoother probability distributions.

The exterior problem is the real-imaginary cross-validationroutine, wherein g is used to predict the real part of the data set,and g is used to predict the imaginary part of the data set:

l º - + -l

* Z Z g Z Z gargmin re im 1522

22[ ˆ ( )] [ ˆ ( )] [ ]

The interior problem is embedded within the exterior problem,because g and g are both functions of λ. When excessively lowcontrol parameter values are assumed, the inferred DRT is

over-smoothed, causing the cross-validation error to exhibit a largebias component, and when excessively high control parametervalues are assumed, the inferred DRT is under-smoothed, causingthe cross-validation error to exhibit a large variance component. Thetrade-off between bias and variance is discussed in greater detail bySaccoccio, Han, Chen, and Ciucci.30

LEVM algorithm.—The LEVM algorithm allows for severalrepresentations. The present work will focus on the continuousvariable representation, which has been reported to outperform itsdiscrete and fixed counterparts. This representation takes the form:

åº D=

G v g vg v v; , ; 16m

M

m m1

( ) ( ) [ ]

D º

--

Î

--

Î

-

+ -- +

+

++

v

v v

v vv v v

v v

v vv v vv;

if ,

if ,

0 otherwise

17m

m

m mm m

m

m mm m

1

1 11 1

1

11

⎧

⎨⎪⎪

⎩⎪⎪

( )

[ ]

[ ] [ ]

The location of the mesh points (v) parameterizes the model, makingthis a floating-mesh representation. The interior problem is theminimization of misfit:

º -* *g v Z Z, argmin 18g v,

22ˆ [ ]

The representation and the interior problem of the LEVMalgorithm are well-characterized, but there are no apparent controlparameter and exterior problem. However, we can infer the existenceof the control parameter from the discussion presented byMacdonald on the inversion of water data.41 Macdonald has, ineffect, taken the total number of mesh points (M) as the complexitycontrol parameter, which is sensible, because a more complicatedDRT would require more mesh points to approximate. Conversely,through a priori specification of M, Macdonald constrains howcomplicated the DRT can be. This observation is supported by theanalysis of the output of the interior problem, in which M isincreased to improve the normalized misfita:

º-- -

** *

MJ M

Z Z g vargmin

,

2 119

M

22ˆ ( )

[ ]

As per the case of Eq. 15, the interior problem is embedded withinthe exterior problem, as g* and v* are both functions of M. Thisexterior problem implies that each additional mesh point results indiminishing improvement in misfit, and so the number of meshpoints should be balanced against the degree of freedom left in thedata set (J− 2M− 1). However, in the subsequent discussion on theinversion of n-pentanol and glycerol data, it became evident thatEq. 19 is insufficient. For the glycerol data, with a total of 10 meshpoints, “the [normalized misfit] was about 0.0094, and the estimatedrelative standard deviations of the [g] were all below 0.1 and mostlybelow 0.06”, but when the number of mesh points is increased to 11,“the relative standard deviations of 6 of the 11 [g] were then above0.1, with one above 0.34”, leading Macdonald to reject the solutioncontaining 11 mesh points despite a continued decrease in normal-ized misfit. Macdonald thus implicitly recognizes the statisticalsignificance of g as constraints in the exterior problem:

s g 0.34 20( ) [ ]

Here s[·] is the relative standard deviation operator. Eqs. 19 and 20constitute the full exterior problem.

aEquation 19 is reconstructed from the description of the normalized misfit providedin Appendix B of Ref. 41. This exterior problem assumes that the measurementerror of the real and the imaginary parts of the impedance are completely correlated,thus yielding J independent measurements.


ISGP algorithm.—The ISGP algorithm employs a flexiblerepresentation composed of multiple basis functions27,28:

åº=

G v G G vg g; , ; 21l ll

L

l l1

( { } { }) ( ) [ ]

The lth basis function G v g;l l( ( )) may be Gaussian, Lorentzian,hyperbolic secant, Kirkwood-Fuoss or Cole-Cole, each having threeparameters, or Pearson VII or Havrilliak-Negami, each having fourparameters, or the five-parameter pseudo-Voigt distribution. TheDRT is thus parameterized by the type of the basis function as wellas the parameters of those basis functions. To illustrate the indexing,consider the case whereby L= 2:

= +G v G G G v G vg g g g; , , , ; ; 221 2 1 2 1 1 2 2( ) ( ) ( ) [ ]

p= -

-G v

g

g

v g

gg;

2exp

2231 1

1,1

3,1

2,12

3,12

⎡⎣⎢⎢

⎤⎦⎥⎥( )

( )[ ]

p

p

p=

-

- - -G v

g g

g v g gg;

2

sin 1

cosh cos 1242 2

1,2 3,2

3,2 2,2 3,2

( )([ ] )

( [ ]) ([ ] )[ ]

Thus the DRT is represented by the sum of a Gaussian distribution(G1) with mass g1,1, mean g2,1 and standard deviation g3,1, and aCole-Cole distribution with mass g1,2, mean g2,2 and depression g3,2.In principle, the large number of possible basis functions allows theform of the underlying model to be detected, but the success of suchan approach has not been demonstrated.

The ISGP requires as input two sets of impedance data (Z1 andZ2). The first data set is used to solve the interior problem, which isthe minimization of misfit:

º -* *G g Z Z, argmin 25l lG g,

1 22

l l

{ } { } ˆ [ ]{ } { }

This interior problem is not compatible with gradient-based optimi-zation methods, since it considers the type of basis function ({Gl}) asa decision variable, thus motivating the use of genetic programming.

The ISGP uses the total number of parameters needed torepresent the DRT (M) as the complexity control parameter, whichis in line with the intuition that complicated distributions possessmore features, and thus require more parameters to approximate.Likewise, by constraining M, it becomes possible to limit howcomplicated the DRT can be.

Hershkovitz, Tomer, Baltianski, and Tsur28 reported the fol-lowing exterior problem:

º+

+ -

*MM M

argmin0.8 0.2

1 exp26

M H

12

22

( )[ ]

= - * *GZ Z g, 27l l12

1 22ˆ ({ } { }) [ ]

= - * *GZ Z g, 28l l22

2 22ˆ ({ } { }) [ ]

HereMH is the recommended maximum number of parameters in therepresentation of the DRT, which is specified by the user, 1

2 is themisfit arising from the interior problem, and 2

2 is the validationmisfit. This exterior problem is likely erroneous, because as Mincreases, the objective function in Eq. 26 decreases. This objectivefunction thus favors models with M in excess of MH, whichcontradicts the description of MH as the recommended maximumnumber of parameters. We assume that this is a straightforward caseof mistaking a multiplication for a division, and the exterior problemis supposed to be:

º + ´ + - *M M Margmin 0.8 0.2 1 exp 29M

H12

22[ ] [ ( )] [ ]

In the section “Design of Algorithm” , we will use the ISGPalgorithm with Eq. 29 as the exterior problem.

Analysis

The examples discussed in “Sample Algorithms” illustrate theprevalence of the four common features. In this section, we considerthe utility of the four common features as a tool for analyzing andsupplementing known DRT inversion algorithms. The Fouriertransform algorithm of Schichlein, Müller, Voigts, Krügel, andIvers-Tiffée9 and Boukamp24 lacks parameters which describe thewidth of the window function. An exterior problem which identifiesan optimal width is constructed. The L-curve algorithm of Florsch,Revil, and Camerlynck26 lacks a parameter which describes therotation of the L-curve, which is needed to complete the exteriorproblem. Manipulation of the exterior problem yields equivalentforms which carry statistical meaning, allowing the rotation para-meter to be determined. The maximum entropy algorithm of Hörlinis well-defined,29 but does not seem to work well for certain smoothdistributions, which is unusual for a maximum entropy method. Wewill use the four common features to analyze this unexpectedbehavior.

Fourier transform algorithm.—The Fourier transform algorithmof Schichlein, Müller, Voigts, Krügel, and Ivers-Tiffée9 and thesubsequent modified version published by Boukamp24 are foundedupon the deconvolution theorem, which states that convolutionintegrals are multiplications in the frequency domain:

ò= -

=-¥

¥

h x f y x g y dy

h f g

if and only if

30

( ) ( ) ( )( ) ( ) ( ) [ ]

Here [·] is the Fourier transform, which is defined as:

ò pº --¥

¥ h x h x ixs dsexp 2 31[ ( )] ( ) ( ) [ ]

Note that f, g, h, x and y are dummy variables, and will not be used inthe remainder of this section.

Equation 11 is transformed to a convolution integral byperforming the substitution w= -u ln( ):

ò=+ --¥

¥Z u G v

G v

i v udv;

1 exp32ˆ ( ( )) ( )

( )[ ]

The real and imaginary parts of Z u G v;ˆ ( ( )) are linked by theKramers–Kronig relations. It is thus sufficient to consider theimaginary part of the impedance:

òò

º- = -

º -

-¥

¥

-¥

¥

I Z G v u v dv

G v K u v dv

2 im sech

33

ˆ [ ˆ] ( ) ( )

( ) ( ) [ ]

Here K is the kernel, which describes the underlying model. Thisleads to a deconvolution of the form:

=

FI

K34( ) ( ˆ)

( )[ ]

Similar methods based on the application of convolution theorem onFourier, Laplace, and Mellin transforms have been used extensivelyin the literature on inverse problems in statistical mechanics.34,55–58

In practice, measurement yields a vector of impedance values(Z), which is affected by measurement error. The measured


impedance cannot be directly substituted into Eq. 34. If themeasurement error is a white noise, the Fourier transform ofI=−2 im[Z] yields a finite value at high frequencies:

¥ Ilim constant 35

s( ) [ ]

The Fourier transform of the relaxation kernel vanishes at highfrequencies:

¥ Klim 0 36

s( ) [ ]

As a consequence, if the measured impedance is substituted directlyinto Eq. 34, the Fourier transform of the DRT diverges:

= ¥¥ ¥

FK

Ilim lim 37

s s( ) ( )

( )[ ]

The DRT oscillates wildly, indicating that the problem is ill-posed.The modification proposed by Boukamp concerns the method forsuppressing the high-frequency amplitudes induced by the directsubstitution of I into Eq. 34:

=

F W sK

I38( ) ( ) ( )

( )[ ]

Here W(s) is the window function, which Boukamp defines as:

a b b a b a= + + - +W s s s; ,1

4tanh 1 tanh 1

39

( ) ( [ ( )] )( [ ( )] )

[ ]

The parameters α and β describe the width and decay of the windowfunction, respectively. The window function converges to 0 morerapidly than K( ), thus restoring the property:

= =¥ ¥

F W sK

Ilim lim 0 40

s s( ) ( ) ( )

( )[ ]

We have thus far treated the vector I as if it were a continuousfunction on , and thus subject to Fourier transform. In practice, thedata set can be made continuous on via interpolation andextrapolation, as was done by Boukamp. This procedure is valid,but difficult to explain within the limited scope of the present work.For brevity, the present work employs a closely related formulationbased on the discrete Fourier (DF) transform. The DF transformoperator ([·]) acts on a vector of equispaced data as follows:

å pº -

- ¢ -

¢=¢

i j j

JI I exp

2 1 141j

j

J

j1

⎡⎣⎢

⎤⎦⎥[ ( )] [ ] ( )( ) [ ]

We define the arrangement operator () as:

º¢ Î

¢ Î -¢

-

- -

jJ

jJ

II

I

if 0,2

if2

, 042j

j

j J

1

1 ⎟

⎧⎨⎪⎪

⎩⎪⎪

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎞⎠

[ ( )][ ( )]

[ ( )][ ]

Note that the indexes j and ¢j in Eq. 42 are defined on {1, ... J} and

- ,...J J

2 2⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦{ }, respectively. The arrangement operator allows

direct substitution of I[ ] with I[ ].The Fourier transform algorithm possesses a representation of the

form:

å p

p

º --

+ --

=G v a m

v

v v

b mv

v v

a b; , cos 2 1

sin 2 1 43

m

M

mM

mM

1 1

1

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) [ ]

[ ] [ ]

The parameters a and b are related to the DF transform of thedistribution via:

= a Fre 44[ ( )] [ ]

= b Fim 45[ ( )] [ ]

Here F is the vector of DRT evaluated at v, with G(vm) as the mthcomponent of F. The interior problem is a deconvolution of the form

a b= - - * W sF I s; , 461 1[ ( ) ( ) ( )] [ ]

The symbol is an element-wise division with º K( ) evaluatedat = ¢

-s

v v

j

M 1, where ¢j is the index of I( ), as defined in Eq. 42.

Equation 46 implies that J=M, since the size of F* must be equal toI. Note that for F* to occur in order of increasing t, the data set Ineeds to be input in order of increasing u.

The difficulty faced by Boukamp concerns the selection ofwindow parameters α and 1/β, which we identify as the complexitycontrol parameters of the DRT inversion algorithm. This is sensible,given that increasing α and decreasing β increases the amplitude ofhigh-frequency terms included in F*. The objective, then, is toconstruct an exterior problem to determine appropriate values of αand β.

This can be neatly achieved by utilizing the real part of the dataset in a validation step:

a b º -a b

* * *Z Z F, argmin re 47,

22[ ˆ ( )] [ ]

This exterior problem requires the real part of the measurement errorto be independent of the imaginary part of the measurement error.

We demonstrate the effectiveness of the proposed exteriorproblem using the sample problem of Schichlein, Müller, Voigts,Krügel, and Ivers-Tiffée,9 which consists of three Debye and threeCole-Cole elements connected in series:

åm am

=+ - a

=Z u

R

i uR; , ,

1 exp48

m

m

m1

6

m

ˆ ( )[ ( )]

[ ]

The parameter values ( m aR, , ) of the sample problem are reportedin Table I. The underlying distribution takes the form:

å

å

d m

pa p

a ma p

= -

+-

-- -

=

=

G v R v

R

v2

sin 1

coshcos 1 49

mm m

m

m m

m mm

1

3

4

6

( ) ( )

[( ) ][ ( )]

[( ) ] [ ]

We generate synthetic data for angular frequencies in the domainω= 10−2 to 106 rad s−1 with 10 data points per decade.

The inversion result is reported in Fig. 3. The Fourier transformalgorithm is clearly able to distinguish the six characteristic timescalesof the process, as well as the distribution masses associated with eachcharacteristic timescale. However, the algorithm overestimates thevariances associated with the Debye elements. This is because it is

Table I. List parameter values used for the sample problem. Thefirst three element, i.e., m = 1, ... 3 have αm = 1, and are thus Debyeelements.

m 1 2 3 4 5 6

Rm 1 0.5 1 0.5 1 1μm −11.5 −9.2 −6.9 −4.6 −2.3 0αm 1 1 1 0.85 0.8 0.75


impossible to reproduce the high-frequency component of the Diracdelta distributions while also preventing the interior problem frombecoming ill-posed. Accurate reproduction of the Dirac delta distribu-tions require the Cole-Cole distributions to be highly oscillatory;accurate reproduction of the Cole-Cole distributions require the Diracdelta distributions to be excessively smoothed. This concept is criticalto the design of EIS inversion algorithms, and will be expanded upon inthe section “Design of Algorithm” .

L-curve algorithm.—The L-curve algorithm employs fixed-meshtriangular basis function representation, i.e., Eq. 5. The interiorproblem is the minimization of the regularized misfit:

l

l

º - +

º +

*g Z Z L

g g

argmin

argmin 50

gg

g

22

misfit

1 22

complexity

penalty

ˆ

( ) ( ) [ ]

The complexity control parameter is the inverse of the regularizationparameter (1/λ), and the complexity is the first derivative of thedistribution ( L g1 2

2 ).The conventional definition of the L-curve, which may be attributed

to Hansen,59 is a log-log plot of the complexity against the misfit,parameterized by the regularization parameter. For illustration, we havegenerated the L-curve corresponding to the unimodal Cole-Coledistribution reported in Fig. 1a, which is shown in Fig. 4a. The L-curve algorithm claims that the appropriate value of the regularizationparameter is that which corresponds to the corner of the L-curve, i.e.,somewhere in between 0.1 and 0.0001. This range compares favorablywith the optimal regularization parameter of 7.1× 10−4 obtained usingthe RR/RI algorithm, but suffers from ambiguity in the definition of acorner. This issue is partially resolved by Florsch, Revil, andCamerlynck by defining the corner of an L-curve as the minimumobtained when the L-curve is rotated clockwise, as illustrated in Fig. 4b.The exterior problem takes the form:

Figure 3. (a) Nyquist plot of the impedance data as well as the predictionobtained from the inversion output of the Fourier transform algorithm. Thenoise takes the form = + ´ ´ +Z Z Z N iN0.005 0, 1 0, 1ˆ ∣ ˆ∣ ( ( ) ( )).(b) True and inverted distribution obtained using proposed exterior problem.The black arrows indicate the location and magnitude of the Dirac deltadistributions.

Figure 4. (a) The L-curve corresponding to the unimodal Cole-Cole distribution reported in Fig. 1a. The corner is identifiable but ambiguous. (b) Rotation of theL-curve yields an unambiguous minimum. (c) True and inverted distribution obtained using the proposed exterior problem. The inversion output of the RR/RIalgorithm is included as a reference.


l q qº -l

* * *g gargmin cos ln sin ln 51( ) [ ( )] ( ) [ ( )] [ ]

This exterior problem is incomplete, since the angle of rotation (θ)remains unknown.

The determination of θ is challenging due to the lack of astatistically meaningful interpretation for the rotation of the L-curve.Our objective is to manipulate the exterior problem into a formwhich carries statistical sense, and thus determine an appropriatevalue for θ. Applying the first-order optimality condition to theobjective function of Eq. 50,

l = -*

d

dg 52( ) [ ]

This establishes g* as a function of λ. This relationship must, inparticular, hold true when λ= λ*:

l l= -

* *d

d53( ) [ ]

Applying the first-order optimality condition to the objectivefunction of Eq. 51,

qll

l= -

**

*d

dcot 54

( )( )

( ) [ ]

Combining the previous two expressions,

ql l

l=

* **

cot 55( )( )

[ ]

This establishes a non-obvious relationship between θ and therelative size of the misfit and the penalty. By requiring the misfitto be of the same order as the penalty, as argued by Hansen,60 weobtain the condition θ*= π/4. Application of this angle of rotation tothe L-curve of Fig. 4a yields λ*= 0.040, which corresponds to thegreen curve shown in Fig. 4c. This choice of regularizationparameter results in a DRT which is less oscillatory than that ofthe RR/RI algorithm, but also relatively over-smoothed.

Florsch, Revil, and Camerlynck have defined the L-curve as thelog-log plot of ò-¥

¥G v dv( ) against the misfit, as opposed to the

conventional pairing of complexity and misfit. The analyticaljustification for the efficacy of the L-curve requires the conventionalpairing to be used.60 The use of an unconventional definitionaccounts for the distinct lack of L shape in the curves reported byFlorsch, Revil, and Camerlynck, and the attendant difficulty indetermining λ*. The supplemented exterior problem introduced inthe present work is intended for the conventional pairing. We notethat a well-established exterior problem based on the identificationof the point of maximum curvature exists,59 and is also intended forthe conventional pairing.

Maximum entropy algorithm.—Hörlin applied the maximumentropy algorithm to a modified DRT problem obtained by sub-stituting º -E v G v vexp 2( ) ( ) ( ) and x wºu Z uˆ ( ) ˆ ( ) into Eq. 32:

òx =

´- - - -

-

-¥

¥u E v E v

v u i v u

v udv

;

exp exp

2 cosh56

1

2

1

2( ) ( )ˆ ( ( )) ( )

[ ] [ ]

( )[ ]

This does not change the essential nature of the problem, butmodifies the appearance of the DRT in a useful way. According tothis integral, the Warburg impedance corresponds to a uniformdistribution:

w p= =Z

R

iE v

R57ˆ ⟺ ( ) [ ]

The Debye relaxation process corresponds to a Dirac delta distribu-tion:

w mm

d m=+

= -ZR

iE v R v

1 expexp

258

0

00

⎜ ⎟⎛⎝

⎞⎠ˆ

( )⟺ ( ) ( ) [ ]

Here m tº ln0 0( ) is the log-characteristic timescale of the relaxationprocess. Warburg and relaxation processes are more discernible onE(v) than G(v).

The maximum entropy algorithm employs a fixed-mesh trian-gular basis function representation:

åº D=

E v e v 59m

M

m m1

( ) ( ) [ ]

The interior problem takes the form:

º - * Te e eargmin 60e

( ) ( ) [ ]

sº

-

Je

Z Z e

2612

2

2( )

ˆ ( )[ ]

º - e e e esum ln 62( ) [ ◦ ( )] [ ]

Here is the misfit, T is the temperature, is the entropy, σ2 is themeasurement variance, and sum[·] is the vector summation operator.We use the symbols , T, and to emphasize the connectionbetween the algorithm and thermodynamics. The misfit is theinternal energy of the system, which is composed of quadraticinteractions between the experimental data and the model prediction.The system tends to collapse to the state with the lowest internalenergy, which corresponds to an ill-posed problem, but is preventedfrom doing so by the temperature, which weights the contribution ofentropy. For a fixed distribution mass, the entropy is lowest when thedistribution is a Dirac delta, and highest when the distribution isuniform.

The complexity control parameter is the inverse temperature(1/T) and the complexity is entropy ( ). With increasing tempera-ture, the contribution of entropy increases, thus yielding broaderdistributions which may be thought of as simpler. The exteriorproblem is a constraint on the misfit:

= e 1 63( ) [ ]

This exterior problem is used to calculate the optimal temperature(T*). The evaluation of the misfit requires the measurement varianceto be known, which may be obtained from repeat measurements, orby comparison with the Kramers–Kronig transform of the impe-dance spectra.

We illustrate the difficulty faced by the maximum entropyalgorithm through an example, wherein the the true impedancetakes the form:

ww w

dp

=+

+ = +Zi i

E v v1

1

1 264ˆ ( ) ⟺ ( ) ( ) [ ]

We collect 10 log-equispaced synthetic data per decade over theangular frequency range ω1= 10−4 rad s−1 and ωJ = 104 rad s−1.The resulting probability distribution is inverted over the domainτ1 = 10−6 and τM = 106, as shown in Fig. 5a. The uniformdistribution associated with the Warburg element has dispersedinto a large number of peaks which are difficult to distinguish from


Debye elements with relatively small resistances. As illustrated inFig. 5b, modifying the temperature does not yield a strict improve-ment in the result. At lower temperatures, the Dirac delta distributionbecomes more prominent, but this occurs at the expense ofincreasing the overall oscillation of E(v). At higher temperatures,the algorithm reproduces the uniform part of E(v) with greateraccuracy, but the Dirac delta distribution is suppressed.

The simultaneous reproduction of simple and complex features ina distribution is the central challenge in the design of EIS inversionalgorithms. This problem is not unique to the maximum entropyalgorithm, as evidenced by the inability of the Fourier transformalgorithm to simultaneously reproduce the Dirac delta and the Cole-Cole distributions (see “Fourier Transform Algorithm” ), and hasbeen repeatedly observed in the literature. For example, applicationof the RR/RI algorithm on the fractal element, whose DRT ischaracterized by a smooth domain terminated by a discontinuity,has been reported to output an oscillatory distribution.30 Likewise,the adaptive multi-parameter regularization approach of Žic,Pereverzyev, Subotić and Pereverzyev returns an oscillatory dis-tribution for the fractal element.44 Dion and Lasia are able to invertthe fractal element with minimal oscillation using a genericinversion toolbox.47 However, the interior and exterior problemsused are not reported. To our knowledge, no DRT inversionalgorithm has addressed the simultaneous reproduction of simpleand complex features comprehensively, although the ISGPalgorithm27,28 and the hierarchical Bayesian algorithm25 haveprovided partial resolution to this problem.

The difficulty in simultaneously reproducing the uniform andDirac delta distributions using the maximum entropy algorithm maybe fundamentally attributed to the use of entropy as complexity,which aligns with our intuition, but only in the limit as ¥T ,which yields a uniform distribution. To illustrate this point, considerthe two distributions shown in Fig. 5c, which possess equal entropy.We intuitively perceive the dotted multimodal distribution to bemore complicated than the solid unimodal distribution, in contra-diction to the value of entropy assigned by Eq. 61. As aconsequence, the distribution reported in Fig. 5a may be simplefrom an entropic point-of-view, but complex according to intuition.

Rectification of the maximum entropy algorithm would likelyrequire entropy to be re-defined.

Design of Algorithm

Thus far, complexity has been described in terms of intuition, andhas been associated with the idea of smoothness in distributions.This is true of most of the algorithms listed in “Sample Algorithms”and “Analysis” . The RR/RI algorithm of “RR/RI Algorithm”

considers the first derivative of the distribution as the complexity,and thus smooth distributions with low first derivatives are simple.The Fourier transform algorithm of “Fourier Transform Algorithm”

considers the high-frequency terms of G(v) as the complexity, whosesuppression leads to simple, smooth distributions. In this section, weargue that the intuitive understanding of complexity is incorrect, andthat it should be described in terms of the physics implied by thedistribution, and guided by Occam’s razor.

The works of Florsch, Revil, and Camerlynck26 and Song andBazant43 suggests that when an appropriate underlying model isused, the inverted distribution tends to appear less complicated.Figure 6 illustrates (a) the DRT and (b) the DDT obtained when aWarburg element is inverted using a relaxation and a diffusion-likemodel, respectively. The diffusion-like model returns a less com-plicated distribution relative to the relaxation model, suggesting therelative propriety of the diffusion-like model.

We draw this observation to its logical conclusion, wherein theapplication of an exactly correct underlying model results in thesimplest possible distribution. That is, if the impedance arises due toan integral over the underlying model:

òw w=-¥

¥Z G v z v dv, 65ˆ ( ) ( ) ˆ ( ) [ ]

then the simplest distribution is obtained when w w m=Z z , 0ˆ ( ) ˆ ( ), for

which G(v)= δ(v− μ0). This leads to the counterintuitive conclu-sion that the simplest possible distribution is the Dirac deltadistribution. This conclusion is consistent with Occam’s razor; asillustrated in Fig. 7, for an underlying relaxation model, the Diracdelta distribution corresponds to a single relaxation process. Any

Figure 5. (a) True and inverted distribution obtained using the maximum entropy algorithm. The noise takes the form = + ´Z Z Zˆ ∣ ˆ∣´ +N iN0.005 0, 1 0, 1( ( ) ( )). (b) Effect of temperature on the inverted distribution. (c) We intuitively perceive the dotted multimodal distribution to be

more complicated than the solid unimodal distribution. However, the entropy of the two distributions are equal.


other distribution would imply various combinations of multipletypes of relaxation processes (Fig. 7b) and multiple relaxationprocesses of the same type (Fig. 7c). As a corollary, complexityincreases with modality and variance.

The idea of the Dirac delta distribution as the zero pointof complexity in EIS inversion problems contradicts much of theEIS inversion literature. The Fourier transform algorithm,9,24

L-curve algorithm,26 maximum entropy algorithm,29 the RR/RIalgorithm,30,31 the hierarchical Bayesian algorithm,25 as well asseveral other unmentioned algorithms43,45 all perceive the Diracdelta distribution as possessing maximum complexity. As a con-sequence, these algorithms are not able to identify the Dirac deltadistribution when it occurs in conjunction with smooth features, asillustrated in “Fourier Transform Algorithm” and “MaximumEntropy Algorithm”.

In the remainder of this section, we will use this phenomen-ological picture of complexity to construct an improved EISinversion algorithm, applicable to the generalized EIS inversionproblem, subject to constraints on the allowed computational time.We will also consider the construction of credible intervals for theparameters and distributions arising from the algorithm. We applythe algorithm to a DRT problem exhibiting a wide range ofsmoothness, and compare its performance with the algorithmsdiscussed in “Sample Algorithms”. We also apply the algorithm tothree generalized EIS inversion problems taking the form of adistributed Randles circuit61 possessing nearly overlapping distribu-tion peaks.

Generalized EIS inversion problem.—The generalized EISinversion problem is an abstraction of the DRT problem, in which

the underlying model is specified, but not assumed a priori:

bw=Z Z G v; , 66lˆ ˆ ( { ( )}) [ ]

The model Z is parameterized by b, which takes on definite values,and {Gl(v)}, which is a collection of distributions. Our objective is toconstruct a generalized EIS inversion algorithm which estimates theparameters b and {Gl} given any such model. The generalized EISinversion problem includes models taking the form of a Fredholmintegral of the second kind, which necessitates an iterative solver.

To illustrate the generalized EIS inversion problem, consider theRandles circuit61:

wwt

=+

++

¥ZR

iR 67CT

R

R ZCT

CT

CT D

ˆ ( ) [ ]ˆ

wwt

wt=Z R

i

i

tanh68D D

D

D

ˆ ( ) [ ]

The Randles circuit describes a charge-transfer process withresistance RCT and timescale τCT occurring in series with a diffusionprocess with resistance RD and timescale τD, both located within anelectrical double layer. Due to surface inhomogeneity, τD isdistributed, leading to a diffusion impedance of the form:

òww

w=

-

-¥

¥Z G v G v

i v

i vdv;

exp

tanh exp69D D D

1ˆ ( ( )) ( )( )

( )[ ]

The remainder of the model is recast in the form of the DRTproblem:

òw

w=

++

w

¥

-¥

¥

+

¥

Z R G v G vG v

i vdv R

; , ,

exp70

CT D

CTR

R Z G v;CT

CT D D

ˆ ( ( ) ( ))( )

( )[ ]

ˆ ( ( ))

Equations 69 and 70 constitute a single model of the generalizedform, with ¥R as b, and GCT(v) and GD(v) as {Gl(v)}. Thegeneralized EIS inversion problem corresponding to the Randlescircuit contains an embedded integral, which is incompatible withlinearly constrained quadratic programming. Furthermore, it is aFredholm integral of the second kind.

Idealized EIS inversion algorithm.—The idealized form of thegeneralized EIS inversion algorithm employs a floating-meshGaussian basis function representation:

åm np n

m

n

º

´ --

=G v

R

v

R; , ,2 exp 2

exp2 exp

71

l l l lm

Mm

m

m

m

1

2

l

ll

l

l

l

⎡⎣⎢⎢

⎤⎦⎥⎥

( { })( )

( )( )

[ ]

Here Rml, mml, and nml are the mass, mean, and log-variance of the

mlth basis function of the lth distribution. Typically, the use of a

floating-mesh representation increases the computational time re-quired to solve the problem, since it is incompatible with linearlyconstrained quadratic programming. However, as illustrated by theRandles circuit, the generalized EIS inversion problem will belargely incompatible with linearly constrained quadratic program-ming to begin with, and so this increase in computational time maybe thought of as a sunk cost. In fact, we expect the reduction in thenumber of parameters effected by the use of the floating-meshrepresentation to reduce the overall computational time. The use ofGaussian basis function follows the observation made by Han,Saccoccio, Chen, and Ciucci31 concerning the effectiveness of radialbasis functions.

Figure 6. (a) The DRT and (b) the DDT of a planar Warburg element. TheDRT is obtained using the RR/RI algorithm, while the DDT is obtained usingthe gEISi algorithm introduced in the section "Generalized EIS InversionAlgorithm" set to a planar Warburg model. For completeness, we includethe analytical DRT of the planar Warburg element, i.e., Eq. 57. The solutionfor v ⩾ 4.34 has been excluded to improve visibility. The true impedancecorresponds to a planar Warburg model with d= -G v v ln 0.01( ) ( ( )).


We adapt the interior problem from the RR/RI algorithm ofSaccoccio, Han, Chen, and Ciucci.30 The real part of the idealizedinterior problem takes the form:

b m n n º + +b m n n

R, , , , argmin 72l l lR, , , ,l l l

{ } [ ]{ }

n

n

º-

+-

J

Z Z Z

Z Z Z

re abs

exp

sum im abs

exp73

22

2

[ ˆ ] [ ]( )

[ [ ˆ ] [ ]]( )

[ ]

å å nl lº + n= =

G v exp 74l

L

ll

L

l1 1

1[ ( )] ( ) [ ]

m n

snº

-+ +

J 1 75

2

2

( )( ) [ ]

Here , , and are the misfit, penalty, and hyperprior on themeasurement error, l and λν are the regularization parameters onthe modality and the variance of the distributions, μϵ and s

2 are thehyperprior expectation and variance of the log-variance of measure-ment error, and νϵ is the log-variance of measurement error. Wecalculate the misfit using the real part of the data set and the averageof the imaginary part of the data set to account for models containingpure capacitive terms.26 The imaginary part of the interior problemtakes the symmetric form:

b m n n º + +b m n n

R, , , , argmin 76l l lR, , , ,l l l

{ } [ ]{ }

n

n

º-

+-

J

Z Z Z

Z Z Z

im abs

exp

sum re abs

exp77

22

2

[ ˆ ] [ ]( )

[ [ ˆ ] [ ]]( )

[ ]

The inverse of the regularization parameters, i.e., l1 and 1/λν arethe complexity control parameters. The complexity is the modalityand the variance of the distributions, following the phenomenolo-gical picture of complexity outlined in Fig. 7.

The idealized exterior problem is the real-imaginary cross-validation routine:

b m n

b m n

l l

n n

º

+ + +

nl ln

* * R

R

, argmin , , ,

, , , 78

l l l

l l l

,( { } )

( { } ) ( ) ( ) [ ]

Generalized EIS inversion algorithm.—An application of theidealized generalized EIS inversion algorithm to an experimentaldata set modelled by the distributed Randles circuit results in anunacceptably high computational time. The difficulty lies in thediscrete nature of the [·] operator, which prevents the use ofefficient gradient-based methods, and the inability to obtain initialguesses for l and λν. We obtain a reasonable compromise betweenaccuracy and computational time by using a proxy measure formodality, and eliminating the variance of the distribution from thecomplexity.

The simplified interior problem takes the form:

b m n n º +b m n n

R, , , , argmin 79l l lR, , , ,l l l

{ } [ ]{ }

Figure 7. A phenomenological picture of complexity in EIS inversion problems. (a) The Dirac delta distribution is the simplest possible distribution. (b) Increasein modality and (c) variance correspond to an increase in complexity.


b m n n º +b m n n

R, , , , argmin 80l l lR, , , ,l l l

{ } [ ]{ }

The simplified control parameter is the total number of basisfunctions used:

åº=

M M 81l

L

L1

[ ]

This control parameter appears to be a reasonable approximation ofmodality. In the second validation problem of the section “GeneralValidation” , the Cole-Cole distributions of the charge transfer anddiffusion processes are reasonably approximated by a small numberof Gaussian basis functions, suggesting that M will be of the sameorder as the modality for many problems of practical interest. Theelimination of variance from the complexity results in a slightwidening of the inversion output; this effect is observed in Fig. 6b,wherein a small but nonvanishing variance is observed, which would

Figure 8. The objective of the exterior problem as a function of the complexity control parameter and the inversion output of (a), (b) the RR/RI algorithm, (c),(d) the LEVM algorithm, (e), (f) the ISGP algorithm, and (g), (h) the gEISi algorithm. The objective of the exterior problem and the complexity control parameterare described in (a) “RR/RI Algorithm”, (c) “LEVM Algorithm”, (e) “ISGP Algorithm”, and (g) “Generalized EIS Inversion Algorithm”.


have been otherwise absent for the idealized generalized EISinversion algorithm.

The simplified exterior problem takes the form:

b m n

b m n n n

º

+ + +

*M R

R

argmin , , ,

, , , 82M

l l l

l l l

( { } )

( { } ) ( ) ( ) [ ]

The details on the optimization algorithm used to solve Eqs. 79through 82 are discussed in the Appendix.

Construction of credible intervals.—The details on the construc-tion of credible intervals for b and {Gl(v)} are likewise discussed inthe Appendix. Briefly, we use the Markov chain Monte Carloalgorithm to construct 8 independent sample chains of appropriatelength. Each Monte Carlo sample corresponds to a set of parameterand distribution values drawn from the joint posterior probabilitydensity function of b and {Gl(v)}. We construct the 95% credibleinterval by rejecting the highest and lowest 2.5% of parameter anddistribution values in the combined sample chain.

Comparison with existing algorithms.—We compare the per-formance of the simplified generalized EIS inversion algorithm,hereon referred to as the gEISi algorithm, against the RR/RIalgorithm, the LEVM algorithm, and the ISGP algorithm as wereport them in the present work. The RR/RI algorithm is run withM= 81. For the LEVM algorithm, the standard deviation of g iscalculated using the MATLAB command nlparci. For the ISGPalgorithm, the basis function is fixed to Gaussian, MH is set to 10,and the interior and exterior problems are solved using the non-dominated sorting genetic algorithm-II (NSGA-II) of Deb, Agrawal,Pratap, and Meyarivan.62 We collect synthetic data within theangular frequency range ω1 = 10−2 and ωJ = 102 with 10 datapoints per decade of angular frequency, with the underlying model:

ww w

=+ -

++

Zi i

1

1 exp 2

1

1 exp 283

0.8ˆ ( )

( ) ( ( ))[ ]

The true DRT is the sum of a Dirac delta and a Cole-Coledistribution:

dp

pp

= + +- -

G v vv

21

2

sin 0.2

cosh 0.8 2 cos 0.284( ) ( ) ( )

( [ ]) ( )[ ]

The inversion output of the four algorithms and the correspondingobjective of the exterior problem as a function of the complexity control

parameter are shown in Fig. 8. The exterior problem of the RR/RI,LEVM, ISGP, and gEISi algorithm converges at l = -*log 8.710[ ] ,M*= 4, M*= 6, and M*= 3, respectively. The RR/RI algorithm andthe LEVM algorithm are unable to simultaneously reproduce the sharpand smooth features of the DRT. The failure of the RR/RI algorithmmay be attributed to an inappropriate choice of complexity, while thatof the LEVM algorithm arises due to the tendency for s gm[ ] to divergeas g 0m , suggesting an inappropriate exterior problem. Increasingthe domain of τ results in a marginal improvement in fit but does notchange the qualitative aspects of the solutions. In contrast, by choosinga complexity which approximately aligns with the phenomenologydescribed in “Design of Algorithm” , the ISGP algorithm and the gEISialgorithm are able to reproduce the Dirac delta and the Cole-Coledistribution accurately. The 95% credible interval obtained by the gEISialgorithm likewise bounds the true distribution accurately.

General validation.—We further validate the gEISi algorithmusing three generalized EIS inversion problems taking the form ofEqs. 69 and 70. The distributions of each problem possess nearlyoverlapping peaks, which is reflected in the merging of semicircularfeatures in Figs. 9a, 9d, 9g. Although the concept is not directlyapplicable here, we note that the separation between the distributionpeaks is close to the resolution limit discussed by Florsch, Revil, andCamerlynck,26 which highlights the inherent difficulty of the chosenproblems. The first problem illustrates an idealized situation inwhich the true underlying distributions are unimodal Gaussian, andare thus exactly represented by a small number of basis functions inthe chosen representation. The second problem explores the effect ofbasis function mismatch by using unimodal Cole-Cole distributions,which cannot be exactly approximated by a finite number ofGaussian basis functions. The third problem shows the effect ofincreasing the modality of the true underlying distributions.

We collect 10 noisy impedance measurements per decade ofangular frequency between ω1= 10−2 and ωJ = 106, and invertthe underlying distributions using the gEISi algorithm. The resultsare summarized in Fig. 9. As shown in Figs. 9a, 9d, 9g, theimpedance spectra of all three problems are reproduced satisfactorily.The true underlying distributions of the first (Figs. 9b, 9c) and third(Fig. 9h, 9i) problems are reproduced accurately, with the remainingdifference correctly captured by the 95% credible intervals. However,the true underlying distribution of the second problem, in particularGCT, is not correctly bounded by the corresponding credible interval.This issue is persistent across randomly generated measurement noiseprofiles and stochastic optimization steps. We believe that this issuearises because the mismatch between the single Gaussian basisfunction used to approximate GD is fortuitously captured by the

Table II. True underlying distributions of the generalized EIS inversion problems under consideration and their corresponding parameter values.In all three problems, we set ¥R to 10 and the measurement error to = + ´ ´ +Z Z Z N iN0.005 0, 1 0, 1ˆ ∣ ˆ∣ ( ( ) ( )).

GCT GD

Problem 1-

s pt

s-expR v

2

ln

21

1

12

12

⎡⎣⎢

⎤⎦⎥

( ) -s p

ts

-expR

v1

2

ln

22 2

22

22

⎡⎣⎢

⎤⎦⎥

( )

Problem 2p

a pa t a p

-- - -

R

v2

sin 1

cosh ln cos 11 1

1 1 1

([ ] )( [ ]) ([ ] ) p

a pa t a p

-- - -R v

1

2

sin 1

cosh ln cos 12

1

2 2 2

([ ] )( [ ]) ([ ] )

Problem 3å -

s p

t

s=-

expmR v

12

2

ln

2

m

m

m

m1

1

1

12

12

⎡⎣⎢

⎤⎦⎥

( ) -s p

ts

-expR

v1

2

ln

23 3

32

32

⎡⎣⎢

⎤⎦⎥

( )

Problem 1 Problem 2 Problem 3R1 50 R1 50 R1 25R2 50 R2 50 R2 25σ1 1.5 α1 0.8 R3 50σ2 1.5 α2 0.8 σ1 1.5τ1 0.001 τ1 0.001 σ2 1.5τ2 0.02 τ2 0.02 σ3 1.5

τ1 0.000 05τ2 0.001τ3 0.02


addition of a small Gaussian peak in GCT at t≈−5. This hypothesisis supported by the large credible interval of GD, which suggests thatthe impedance spectra effected by GD can be partially or fullyaccounted for by adjusting the small Gaussian peak in GCT. Thefrequency with which this issue is encountered decreases rapidly withthe peak-to-peak separation of GCT and GD, and becomes nonexistentwhen t tlog10 2 1( ) exceeds 2.

The overall performance of the gEISi algorithm as applied to thedistributed Randles circuit appears to be satisfactory. In all threevalidation problems, the numbers, sizes, and average timescales of

the charge transfer and diffusion processes are correctly estimated,and the true underlying distributions are competently bounded by the95% credible intervals. The gEISi code, implemented in MATLAB,can be downloaded from https://github.com/suryaeff/gEISi.git.

Conclusions

The present work have argued that all well-posed EIS inversionalgorithms possess four common features, namely, the representa-tion, the interior problem, the complexity control parameter, and the

Figure 9. (a), (d), (g) Nyquist plot of the impedance data as well as the prediction obtained from the inversion output of the gEISi algorithm. (b), (e), (h) Trueand inverted distributions corresponding to the charge transfer process. (c), (f), (i) True and inverted distributions corresponding to the diffusion process. Thefirst, second, and third row of figures correspond to the first, second, and third problems listed in Table II, respectively.



exterior problem. These features can be constructed and analyzednearly independently of each other, leading to the concept of aframework for the design of EIS inversion algorithms. The ubiquityof these features has been demonstrated through a review ofestablished DRT inversion algorithms, and are subsequently usedto troubleshoot several ill-defined DRT inversion algorithms, andunderstand the unexpected behavior of others.

Complexity has been linked to model selection, leading to theconclusion that it should be appropriately defined in terms of themodality and the variance of the distribution. This observation, inconjunction with practical considerations on the availability ofcomputational resource, led to the development of the gEISialgorithm, which is applicable to the generalized EIS inversionproblem. The gEISi algorithm employs a floating-mesh Gaussianbasis function representation, coupled to an interior problem whichminimizes a misfit subject to a constraint on the control parameter,which we define as the total number of basis functions. The optimalcontrol parameter value is determined through an exterior problemwhich minimizes the real-imaginary cross-validation error. Thecredible intervals of the parameters and distributions arising fromthe gEISi algorithm are estimated using Markov chain Monte Carloalgorithm. A preliminary comparison between the gEISi algorithmand several established DRT inversion algorithm suggests that it isable to overcome several limitations which has been observed in theliterature. The gEISi algorithm is also validated against severalgeneralized EIS inversion problems taking the form of distributedRandles circuit, which are impossible to analyze using existing DRTinversion algorithms.

It should be made clear, however, that the success of the gEISialgorithm is dependent on the ability to determine the true under-lying model. This allows the determination of the physicochemicalparameters of the system, distributed or otherwise. In the companionpaper,48 we will apply the gEISi algorithm coupled to the distributedRandles circuit to analyze several experimental data sets arisingfrom the impedance of polymer-coated steel surfaces.

Acknowledgments

We gratefully acknowledge support from Dow through theUniversity Partnership Initiative. We would like to thank Hongbo

Zhao, Dimitrios Fraggedakis, and Yu Ren Zhao for their assistancein interpreting the behavior of various algorithms. We would like tothank Professor Francesco Ciucci in clarifying certain aspects of thehierarchical Bayesian algorithm.

Appendix

The resolution of Eqs. 79 through 82 is remarkably difficult,because the model is input by the end-user, and therefore not knowna priori. In such cases, it is prudent to use a combination ofoptimization algorithms; in the present work, three optimizationalgorithms are used, namely, simulated annealing, constrained trustregion, and constrained pattern search. These will be discussedindividually in the subsequent paragraphs.

The construction of credible intervals for b and {Gl(v) necessi-tated the addition of constraints on the interchangeability of basisfunctions. These constraints take the form:

m m + 85m m 1l l[ ]

m m= =+ 86m M m 1l l l 1

[ ]

The first constraint states that the means of the basis functions in thelth distribution must be increasing. In the absence of this constraint,basis functions can interchange, leading to an inflated estimate of thecredible intervals for m nR , ,l l l{ }. The second constraint states thatthe highest mean of the lth distribution must be lower than the lowestmean of the (l+ 1)th distribution. In the absence of this constraint,basis functions can interchange among different distributions,leading to an inflated estimate of the credible intervals for

m nR , ,l l l{ } and {Gl(v)}. The concept of interchangeability amongdifferent basis function is illustrated in Fig. A·1. It is also convenientto set a convention whereby {Gl(v)} is nonnegative:

R 0 87ml [ ]

This arises from the assumption that the end-user knows withcertainty that the form of the model is correct up to a sign. Trial-and-error suggests that this constraint is helpful in reducing thecomputational time needed to solve the exterior problem.

Figure 9. continued


The initial guess for the interior problem at each value of M isgenerated from the previous interior problem by decomposing theprevious optimal fit using a moment-matching equation. The exteriorproblem, i.e., Eq. 82 is solved by sequential enumeration. The gEISicode, implemented in MATLAB, can be downloaded from thesupporting information.

A.1. Simulated annealing.—For brevity, let g be a vector of allparameter values arising from the chosen Gaussian representation,with γg as the gth element of g . The size of g is K+ 3M+ 1, sincethe point parameters b, distributed parameters {Gl(v)}, and the log-variance of the measurement error νϵ give rise to K, 3M, and 1parameter each. Let g* be the maximum likelihood estimate of g ,g p( ) be the proposed step of the simulated annealing, and let g n g,( ) bethe (n, g)th step of the simulated annealing. Finally, let Ns be theeffective number of successful iterations of simulated annealing.

The simulated annealing code requires as inputs the perturbationsize gD and the decorrelation length Nd. The derivation of gD andNd will be discussed in a subsequent section. The pseudocode of thesimulated annealing takes the form:

Here N(0, 1) and U[0, 1] are random numbers generated from thestandard normal and standard uniform distribution, respectively, andp is the acceptance probability of the proposed step. In effect, thispseudocode performs Markov chain Monte Carlo under an increas-ingly relaxed condition as Ns approaches 0. The Monte Carloiterations are done one parameter at a time, with each acceptedstep accounting for a fraction of an effective number of successfuliterations.

A.2. Perturbation size and decorrelation length.—For thesimulated annealing to succeed, the parameter space explored by

the Monte Carlo steps should encompass the maximum likelihoodestimate. In this section, we derive approximate expressions for thedecorrelation length and the perturbation size, defined as the numberand size of successful Monte Carlo steps needed to explore aparameter space encompassing the maximum likelihood estimatewith confidence 1− α, respectively.

Let g g gc º + 2 ( ) ( ) ( ). In a regular Markov chain MonteCarlo algorithm, we have:

g gc c= -

-p min 1, exp

288

p n g2 2 ,⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( ) ( )[ ]

( ) ( )

We want p to be of reasonable size; excessively small p leads toexcessively high computational time, while excessively large p leadsto small, ineffective steps. Setting =p

e

1 , and evaluating g n g,( ) at themaximum likelihood estimate,

g gc c- =* 2 89p2 2( ) ( ) [ ]( )

Let H be the Hessian of χ2 evaluated at the maximum likelihoodestimate. Then around the maximum likelihood estimate, we canapproximate χ2 as:

g g g g g gc c» + - -* * *H1

290p p p2 2 T( ) ( ) ( ) ( ) [ ]( ) ( ) ( )

Since the Monte Carlo simulation is done one parameter at a time,only the diagonal elements of the Hessian matters. We assign theallowed difference in Eq. 89 to the diagonal elements:

g g

g gg

g g g g

ggg

g g g gg g

g

gg g

g=

+ D

=

=

=

´ -+ - -

=<

+ < +

=

= ++ +

D

m

m

+

¢¢

¢

+

+ +

+

*

*

NN N

N

p

p

p p p

N

N

U p

N NK M

Set toSet to 1While is less than ,

Set toFor each element in , perform the update:

0, 1 for the element under consideration, and

otherwise

1 if Eqs. 85 and 86 are satisfied0 otherwise

1 if Eq. 87 is satisfied0 otherwise

min 1,

exp2

if 0, 1

otherwise

If ,

1

3 1Update

EndEnd

End

n g

n g n g

gp g

n gg

gn g

R

R

p p n g n g

n gp

n g

n g n g n g n g

n g

,

s

s d1, ,

,

,

s

d

, ,

1,,

1, 1, , ,

1,

s s

⎪

⎪

⎪

⎧⎨⎩

⎧⎨⎩⎧⎨⎩

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎧⎨⎩

( )

[

( ) ( ) ( ) ( )

[ ]

( ) ( ) ( ) ( )

( )

( ) ( )

( )( )

( )

( ) ( ) ( ) ( )

( )( )

( )

( ) ( ) ( ) ( )

( )


gD »H1

22 91g g g

2, [ ]

By evaluating the diagonal elements of the Hessian at the currentmaximum likelihood estimate, provided that the maximum like-lihood estimate does not fluctuate excessively throughout thesimulated annealing process, a consistent and reasonable perturba-tion size can be obtained.

Next, we consider the decorrelation length. Our strategy is toobtain the size of the parameter space encompassing 1− α of theprobability mass relative to the perturbation size, and calculate thedecorrelation length by approximating successful Monte Carlo stepsas random walk.

Let e º-

jZ Z

Z

re

absj j

j

[ ˆ ])( ) and e º

-j

Z Z

Z

im

absj j

j

[ ˆ ])( ) . We want to obtain some

limiting error e¢ which encompasses 1− α of the probability mass:

ò òe e e e aP = -e

e

e

e

- ¢

¢

- ¢

¢ P d P d 1 92j j j j j( ) ( ) [ ]

For a Gaussian independently and identically distributed error, wehave, as a conservative estimate,

ò p n

e

ne a- = -

e

e

- ¢

¢

d

1

2 exp 2exp

2 exp1 93

jj J

21

2

⎡⎣⎢⎢

⎤⎦⎥⎥( ) ( )

( ) [ ]

Equivalently,

e n a¢ = + -exp 2 norminv1

21 1 94J

12

⎡⎣⎢

⎤⎦⎥( ) ( [ ] ) [ ]

Here norminv[·] is the standard normal inverse operator. Substitutingthe limiting error into Eq. 90, and neglecting the difference in thehyperprior terms,

åg g gen

D D »¢

-

*H1

22

exp 295

j

jmaxT

max

2⎡⎣⎢

⎤⎦⎥( ) ( )

( )( ) [ ]

Here gD max( ) is the size of the parameter space corresponding to thelimiting error. Substituting the expression for limiting error,

g g

g

aD D » + -

- *

JH1

22 norminv

1

21 1

96

JmaxT

max1

2

2⎡⎣⎢

⎤⎦⎥( ) ( ) ( [ ] )

( ) [ ]

We neglect the off-diagonal elements of the Hessian, and split theright-hand side of the equation among the diagonal elements:

gg

aD »

+ - -

+ +

*H

J

K M

1

2

2 norminv 1 1

3 197

g g gmax2

,

1

2

2J

12

⎡⎣ ⎤⎦( )

( [ ] ) ( )

[ ]

It remains to compare the perturbation size to the size of theparameter space corresponding to the limiting error. Taking eachsuccessful Monte Carlo step as a random walk, and approximatingthe misfit g *( ) with the idealized value of 2J,

g

g

a

=D

D

»+ - -

+ +

N

J J

K M

2

2 2 norminv 1 1 2 2

3 198

g

gd

max2

1

2

2J

12

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣ ⎤⎦

( )

( ) ( [ ] ) ( )[ ]

In practice, the Monte Carlo steps do not fully resemble a randomwalk, since they are directed toward g*. To account for this drift, wescale Nd according to the dimensionality of the parameter space:

Figure A·1. The impedance data in (a) is inverted using a model consisting of two distributed relaxation processes, yielding the two distributions shown in (b).The dashed lines of corresponding color is the 95% credible interval of the distributions. (c) In the absence of Eq. 86, the 95% credible interval of thedistributions are indistinguishable and overestimated.


a»

+ - -

+ +N

J J

K M

2 2 norminv 1 1 2 2

3 199d

1

2

2J

12

⎡⎣ ⎤⎦( ) ( [ ] ) ( )[ ]

We also drop the term −2(2J) to allow for a conservative estimate.Eq. 98 has been derived assuming that the full data set has beenused, corresponding to 2J degrees of freedom. For Eqs. 79 and 80,every instance of 2J should be replaced with J+ 1.

A.3. Constrained trust region.—MATLAB’s default constrainedtrust region algorithm fmincon is used to solve Eqs. 79 and 80, settingthe maximum number of function evaluations and the maximum number

of iterations to 100(K+ 3M+ 1)2. The usual constraints on interchan-geability and mass of basis functions apply. Constrained trust region isused to refine the solution obtained using simulated annealing.

A.4. Constrained pattern search.—The execution of con-strained pattern search is identical to the constrained trust region,except done one parameter at a time, and using the default settings offmincon. The constrained pattern search is performed whenever Δγis updated in the simulated annealing pseudocode. Trial-and-errorsuggests that the constrained pattern search is a quick and effectivemethod of accelerating the convergence of the simulated annealing,without sacrificing the stochasticity of the simulated annealing.

A.5. Interior problem.—The simulated annealing, constrainedtrust region, and constrained pattern search are combined to solvethe interior problem. Assume that a rough initial guess of the interiorproblem g 0( ) exists, and refine this initial guess using the optimiza-tion problem:

b m n n º +b m n n

R, , , , argmin 100l l lR, , , ,l l l

{ } [ ]{ }

nº

-

Z Z Zabs

exp1012

2( ˆ ) [ ]( )

[ ]

m n

snº

-+

J2 102

2

2

( )( ) [ ]

The superscript denotes that the parameters are obtained usingboth the imaginary and the real parts of the data set. The interiorproblem code also requires as input the number of basis functionsused to represent each distribution {Ml}, also called the modality orthe case under consideration. The derivation of g 0( ) and {Ml} willbe discussed in a subsequent section. Let Xval be the cross-validation error.

The pseudocode of the interior problem takes the form:

A.6. Sequential enumeration.—The argument of the exteriorproblem (M) is an integer. We thus solve the exterior problem bysequentially increasing M, and then generating new cases and roughinitial guesses for the resulting interior problem. The generation ofnew cases and rough initial guesses will be discussed in subsequentsections. Let *Xval be the current optimal cross-validation error, *Xprevbe the previous optimal cross-validation error, g* be the currentoptimal maximum likelihood estimate, g*prev be the previous optimalmaximum likelihood estimate, *Ml{ } be the current optimal mod-ality, *Ml prev{ } be the previous optimal modality, and {{Ml}} be theset of cases under consideration.

The pseudocode has two parts, corresponding to M= L, i.e., thecase where each distribution is approximated by 1 basis function,and M> L. The pseudocode for M= L takes the form:

g g

= ¼

¼

*

*

*

M

X X

M

Solve the interior problem for 1, , 1

Set to

Set to

Set to 1, , 1

l

l

val val

{ } { }

{ } { }

This pseudocode requires a true initial guess g 0( ) for the interiorproblem, which is provided by the end-user.

g g

g g

g g

g g

g g

g g

g g

g g

= +

= + +

X

X

X X

Set to 0Solve Eq. 100 for the case under consideration, with sub steps:

Perform simulated annealing with initial guess , and set tothe output

Perform constrained trust region with initial guess , and set tothe output

Solve Eq. 79 for the case under consideration, with sub steps:



Solve Eq. 80 for the case under consideration, with sub steps:



val

0

val

val val

‐

‐

( ) ( )‐

( ) ( )

( )


The pseudocode for M> L takes the form:

g g

g g

g g

a- >

<

=

=

* *

* *

* *

* *

*

*

*

*

*

*

X X J

X X

M M

M M

M

X X

X X

M

While 2 2 ,

Set to

Set to

Set to

Generate from

For each element in , perform:

Generate from consistent with the case under

considerationSolve the interior problem for the case under consideration

If ,

Set to the case under considerationEnd

EndEnd

l l

l l

l

l

prev val

prev val

prev

prev

prev

0prev

val val

val val

( )

{ } { }

{{ }} { }{{ }}

{ }

( )

At every iteration of the while loop, the control parameter= åM Ml l increases by 1. The code compares the current optimal

cross-validation error against the previous optimal cross-validationerror, and allows the algorithm to proceed only if the reduction incross-validation error exceeds 2α(2J), where α is the approximateprobability mass left unexplored by the simulated annealing. Notethat 2α arises from a conservative estimate of the confidence of theoptimization problem, while 2J is the idealized value of the misfit.This threshold accounts for incomplete optimization and modelmismatch; since the present work uses only synthetic data, whichdoes not experience model mismatch, α can be set to an arbitrarilylow value. We have opted to set α= 0.01 in “Comparison withExisting Algorithms” , which is effectively vanishing. To provide a

sense of scale, with regards to Fig. 8d, -* *X Xprev val is of the order of100, while 2α(2J) with α= 0.01 is in the order of 1.

More generally, by calibration against the experimental data usedin the companion paper, as well as other experimental data generatedin our laboratory, α is set to a more reasonable value of 0.1. Fordirect comparability with the analysis which will be published in ourcompanion paper,48 in “General Validation” , we set α to 0.1.

A.7. Moment matching.—The pseudocode for the interiorproblem requires the generation of a rough initial guess g 0( ) fromthe previous optimal maximum likelihood estimate g*prev. This isdone by increasing the modality to match the case under considera-tion, and then matching the first two moments of g 0( ) and g*prev. Forclarity, recall that {Ml} is the case under consideration, *Ml prev{ } is theprevious optimal modality, and g is composed of the pointparameters b, the distribution masses {Rl}, the distribution meansml{ }, and the distribution log-variances n l{ } . Let *Ml,prev be the lthelement of *Ml prev{ } , and let *Rm ,prevl

, m*m ,prevl, and n*m ,prevl

be the mlth

distribution mass, mean, and log-variance of the lth distribution,respectively.

The pseudocode of the moment-matching algorithm takes theform:

A.8. Cases under consideration.—The cases under considera-tion {{Ml}} is generated by increasing each element of *Ml prev{ }independently. For example, if =*M 1, 2, 2l prev{ } { }, then

=M 2, 2, 2 , 1, 3, 2 , 1, 2, 3 ;l{{ }} {{ } { } { }} we will then proceedto consider these three cases in the pseudocode of the exteriorproblem. Trial-and-error suggests this heuristic works well over awide variety of EIS inversion problems.

A.9. Construction of credible intervals.—The construction ofcredible intervals for b and {Gl(v)} is done using Markov chainMonte Carlo performed one parameter at a time. The execution ofthe algorithm is identical to simulated annealing, with the relaxationterm removed:

b b

m

n

m

m n

m n m n

m n

w w

=

=

+

=

=

- =

+ + -

+

+ =

-

-

-

- - -

-

- -

*

*

* *

*

* *

* * * * * *

* *

* *

R

Rm

R Rm M

Rm M

m

R R

R Rm M

m M

J

R

Set to

For each element in ,

2if 1

2if 2

2if


exp 2 if 1

exp 2 exp 2if 2

exp 2 if

Set all elements in toabs ln

1

l

m

ml

m ml l

ml l

l

m

m m l

m m m m m m

m ml l

m m l l

lJ

0

0

,prev

1,prev ,prev

1,prev

0

,prev ,prev

1,prev 1,prev 1,prev ,prev ,prev ,prev

1,prev ,prev

1,prev 1,prev

0 1

l

l

l l

l

l

l l

l l l l l l

l l

l l

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

{ }

{ }

[ ]

( [ ]) ( [ ])

[ ]

{ } [ ( )]

( )

( )

( )

( )


g g g g

g g g g

-+ - -

-+ - -

N

Nexp

2

exp2

p p n g n g

p p n g n g

s

d

, ,

replace , ,

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

In principle, the most efficient implementation of the Monte Carloalgorithm consists of a single chain of length ?Nd. However, due tothe various approximations taken in the derivation of Nd, we cannotinterpret it as more than a competent approximation for thedecorrelation length. Trial-and-error suggests that better credibleintervals are obtained by obtaining multiple independent MonteCarlo chains of length Nd. Simulated annealing is used to decorrelatesuccessive runs of Markov chain Monte Carlo algorithm.

It is useful to reiterate some of the definitions used in simulatedannealing. Let g n g,( ) be the (n, g)th step of the algorithm, with n as theindex of Monte Carlo sample and g as the index of parameter underconsideration. After every parameter has been considered once, theindex n increases, and a single Monte Carlo sample is obtained. Wedefine g gºn n,1( ) ( ) to be the nth Monte Carlo sample of the algorithm.Let g n{ }( ) be the collection of samples arising from a single run of theMonte Carlo algorithm, and let g{ } be the collection of samples arisingfrom all runs of the Monte Carlo algorithm. The pseudocode for theconstruction of credible intervals takes the form:

g

ggg g

g gg

gg g

g

= +

*

N

N

N N

Set to 1

Set toWhile 8,

Perform Markov chain Monte Carlo with initial guess , and

collect

Append to

Set to the last element of

Perform simulated annealing with initial guess , and

collect

Set to the last element of1

End

n g

n g

n

n

n g n

n g

n

n g n

MS,

MS,

,

,

,

MS MS

{ }{ } { }

{ }

{ }{ }

( )

( )

( )

( )

( ) ( )

( )

( )

( ) ( )

At this point, we obtain a large collection of samples g{ }. This sampleis post-processed and sorted to give the 95% credible interval:

gb

b ¥

¥

G

G RR

v

v


Calculate andEndFor each element in and , e.g ., ,

Sort in ascending orderReject the highest and lowest 2.5% of the set

End

l

l

{ }{ ( )}

{ ( )}{ }

The first and last elements of each set corresponds to the upper andlower 95% credible interval for the corresponding parameter ordistribution value.

ORCID

Surya Effendy https://orcid.org/0000-0002-9015-3777

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