domain: phys spi

text:
Armand Wirgin
email: wirgin@lma.cnrs-mrs.fr

Exact inversion of the quasistatic electric potential field to retrieve the complex permittivity of a dielectric cylinder

Keywords: quasistatic electricity, exact solution of inverse problem, parameter retrieval, real and imaginary parts of permittivity, nuisance parameter uncertainty-induced instabilities

This study concerns the 2D inverse problem of the retrieval, using external field data, of either one of the two physical parameters, constituted by the real and imaginary parts of the permittivity, of a z-independent cylindrical dielectric specimen subjected to an external, z-independent, quasistatic electric field. Six other parameters enter into the inverse problem. They are termed nuisance parameters because: 1) they are not retrieved during the inversion and 2) uncertainty as to their actual values can adversely affect the accuracy of the retrieval of the permittivity. This inverse problem is shown to have an exact, mathematically-explicit, solution, both for continuous and discrete input data, whose properties, with respect to the various nuisance parameter uncertainties, are analyzed, first in a mathematical, and subsequently in a numerical manner for noiseless data. It is found that: a) optimal inversion requires data registered at only a small number of sensors, b) the inverse solution, satisfying pre-existing physical constraints, exists and is unique. Moreover, the inverse solution is shown to be unstable with respect to three nuisance parameter uncertainties, the consequence of which is large retrieval inaccuracy for small nuisance parameter uncertainties, acting either individually or in combination.

A shorter version of this study can be found in: Wirgin A., An exactly-solvable quasistatic electricity inverse problem: retrieval of the complex permittivity of a cylinder taking account of nuisance paramter uncertainty, Prog.In Electromag.Res. B, 62, 1-16 (2015).

for small ε ′′ other nuisance parameters being certain, on the accuracy of the retrieval of ε ′′ 1 7.8 Table of the influence of uncertainty regarding five nuisance parameters, the sixth nuisance parameter being certain, on the accuracy of the retrieval of ε ′

1 Introduction

The retrieval of the complex permittivity (or related physical parameters such as the dielectric constant, index of refraction, absorption coefficient,...) of a homogeneous, isotropic material is a theoretical and experimental electromagnetic inverse (although only relativelyrecently recognized as such) problem of considerable importance. The reason for this is that permittivity is a sensitive indicator of the chemical [START_REF] Bauer | Refractometry[END_REF] and physical identity of natural and man-made materials and of their state (notably in quality control and health monitoring [START_REF] Yilmaz | Detecting vital signs with wearable wireless sensors[END_REF] applications) [START_REF] Hippel | Dielectrics and Waves[END_REF][START_REF] Young | Compilation of the static dielectric constant of inorganic solids[END_REF]. The materials of interest cover a wide range: liquids and colloids (such as industrially-produced organic materials), organic solids (e.g., food [START_REF] Nesvadba | Database of physical properties of agro-food materials[END_REF][START_REF] Soltani | Evaluating banana ripening status from measuring dielectric properties[END_REF], live tissue [START_REF] Yilmaz | Detecting vital signs with wearable wireless sensors[END_REF]), polymers [START_REF] Lee | Non-contacting method of determining DC dielectric constant for a thin insulating polymer layer[END_REF], ceramics, inorganic solids with interesting electronic and optical properties (insulators, conductors, semiconductors, geophysical materials,...) [START_REF] Seitz | The Modern Theory of Solids[END_REF][START_REF] Abelès | Methods for determining optical parameters of thin films[END_REF].

The DC dielectric constant is usually determined by the comparison of the capacity of an empty or air-filled capacitor with that of the capacitor filled with the dielectric material of interest. At low frequencies (10 2 -10 7 Hz), use is made, for this purpose, of a device similar to the Wheatstone bridge. At higher (10-100 MHz) frequencies, the resonant circuit method is employed, whereas in the range 100 MHz-1 GHz the transmission line technique as well as dielectrometry [START_REF] Mamishev | Parameter estimation in dielectrometry measurements[END_REF] are used. At still higher (microwave) frequencies, appeal is made to the transmission line technique and the detection of perturbations (shift of frequency and modification of the Q) of a resonant cavity [START_REF] Yilmaz | Detecting vital signs with wearable wireless sensors[END_REF]. Finally, at optical frequencies, reflectometry [START_REF] Abelès | Methods for determining optical parameters of thin films[END_REF], refractometry [START_REF] Wirgin | Neue Apparate zur Bestimmung des Brechungs und Zerstreuungsvermögens fester und flüssiger Körper[END_REF][START_REF] Abelès | Détermination de l'indice et de l'épaisseur de couches minces[END_REF][START_REF] Bauer | Refractometry[END_REF][START_REF] Abelès | Methods for determining optical parameters of thin films[END_REF], and ellipsometry [START_REF] Vuye | Precision in the ellipsometric determination of the optical constants of very thin films[END_REF] are the dominant techniques for the determination of the complex permittivity.

The aforementioned techniques rely (to match theory to measurement) on the possibility of obtaining homogeneous specimens of prescribed (usually-simple) shape (usually a block, slab, thin film, sphere, cylinder, etc.) and size (e.g., films). In naturally-occurring materials, this is not always possible. For instance, in studies of natural phenomena connected with the scattering of light (interstellar dust, air-borne pollution, powder, granular media [START_REF] Shivola | Mixing models for heterogeneous and granular Media[END_REF] and other divided matter characterizations, material characterization of living bodies (cells, phytoplancton, etc.), the specimens can have complicated shapes (although they are often considered to be spherical or cylindrical [START_REF] Reagan | Light scattering by irregularly shaped particles versus spheres: what are some of the problems presented in remote sensing of atmospheric aerosols?[END_REF][START_REF] Chylek | Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data[END_REF][START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]) and are too small (e.g., films so thin that the matter therein appears to be divided) to be examined by the previously-mentioned techniques [START_REF] Pluchino | Refractive-index measurements of single micron-sized carbon particles[END_REF][START_REF] Chylek | Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data[END_REF][START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF][START_REF] Reagan | Light scattering by irregularly shaped particles versus spheres: what are some of the problems presented in remote sensing of atmospheric aerosols?[END_REF][START_REF] Vuye | Precision in the ellipsometric determination of the optical constants of very thin films[END_REF]. It is thus increasingly recognized that discrepancies between the assumed and actual: size, shape and composition of the specimen (divided versus homogeneous, such as in colloids and metamaterials [START_REF] Shivola | Mixing models for heterogeneous and granular Media[END_REF][START_REF] Smith | Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients[END_REF][START_REF] Chen | Robust method to retrieve the constitutive effective parameters of metamaterials[END_REF][START_REF] Smith | Electromagnetic parameter retrieval from inhomogeneous metamaterials[END_REF][START_REF] Chen | Determining the effective electromagnetic parameters of bianisotropic metamaterials with periodic structures[END_REF]) have to be taken into account in connection with the meaning that is attached to the permittivity determined from the response of the specimens to quasistatic or dynamic (wave-like) electric fields (the latter response incorporates diffraction and/or collective effects, not ordinarily accounted-for in methods relying on reflective or refractive response fields). A second trend of permittivity retrieval inverse problems is the recognition of the necessity of taking into account the uncertainty (of the experimental results [START_REF] Vuye | Precision in the ellipsometric determination of the optical constants of very thin films[END_REF], of certain parameters (and their sensitivity [START_REF] Chen | Robust method to retrieve the constitutive effective parameters of metamaterials[END_REF]) that enter into the retrieval model, of the mathematical ingredients of the retrieval model itself [START_REF] Wirgin | Ill-posedness and accuracy in connection with the recovery of a single parameter from a single measurement[END_REF][START_REF] Sambuelli | Uncertainty propagation using some common mixing rules for the modelling and interpretation of electromagnetic data[END_REF]) in order to evaluate the accuracy of the retrieved parameters, e.g., [START_REF] Vuye | Precision in the ellipsometric determination of the optical constants of very thin films[END_REF][START_REF] Emery | The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters[END_REF][START_REF] Hasar | Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs[END_REF][START_REF] Hasar | Application of a useful uncertainty analysis as a metric tool for assessing the performance of electromagnetic properties retrieval methods of bianisotropic metamaterials[END_REF].

The present investigation is inspired by these two trends.

More specifically, we shall be concerned with quasistatic electrical phenomena [START_REF] Morabito | A fuzzy modeling approach for the solution of an inverse electrostatic problem[END_REF][START_REF] Heubrandtner | The quasi-static approximation for weakly conducting media and applications[END_REF][START_REF] Preis | Time-domain analysis of quasistatic electric fields in media with frequency-dependent permittivity[END_REF][START_REF] Artemev | Inverse electrostatic and elasticity problems for checkered distributions[END_REF] occurring in R n (n = 1, 2, 3) occupied by a heterogeneous, isotropic medium M. Electromagnetic fields are governed, in the space-time framework, by the Maxwell equations ( [START_REF] Stratton | Electromagnetic Theory[END_REF]) ∇ × Ê(x, t) + B,t (x, t) = 0 ,

∇ × Ĥ(x, t) -D,t (x, t) = Ĵ(x, t) ,

wherein Ê and Ĥ are the intensities of the electric and magnetic fields respectively, D the displacement, B the magnetic induction, Ĵ the current density, x the vector from the origin O to a generic point of space, t the time variable, and , t the symbol of partial differentiation with respect to t.

The current density Ĵ is the sum of a conduction current density Ĵc related to the electric field, and an impressed current density Ĵi connected with the impressed charge density ρi by the conservation of charge relation

∇ • Ĵi (x, t) + ρi ,t (x, t) = 0 . (3)
In the quasistatic electrical regime, B,t ≈ 0 so that (1) becomes

∇ × Ê(x, t) = 0 , (4)
whence, by virtue of the identity (for an arbitrary scalar Ŝ) ∇ × ∇ Ŝ = 0, Ê(x, t) = -∇ ψ(x, t) , [START_REF] Artemev | Inverse electrostatic and elasticity problems for checkered distributions[END_REF] in which ψ designates the quasistatic electrical potential. Taking the divergence of (2), and applying the identity (for an arbitrary vector V) ∇ • ∇ × V = 0, gives

-∇ • D,t (x, t) = ∇ • Ĵ(x, t) . (6)
The vector and scalar quantities are expressible as the Fourier integrals

V(x, t) = ℜ ∞ -∞ V(x, ω) exp(-iωt)dω , Ŝ(x, t) = ℜ ∞ -∞ S(x, ω) exp(-iωt)dω , (7)
(wherein ω is the angular frequency), so that (3), ( 5) and ( 6) yield the frequency domain partial differential equations

∇ • J i (x, ω) -iωρ i (x, ω) = 0 . (8)
E(x, ω) = -∇ψ(x, ω) ,

iω∇ • D(x, ω) = ∇ • J(x, ω) = ∇ • J i (x, ω) + J c (x, ω) . (9)
In an isotropic, generally-inhomogeneous, medium, the displacement and the conduction current density are related to the electric field by a dielectric constant ε ′ (x, ω) and loss factor ε ′′ (x, ω) respectively via the constitutive relations:

D(x, ω) = ε ′ (x, ω)E(x, ω) , (11)
J c (x, ω) = ωε ′′ (x, ω)E(x, ω) . ( 12
)
wherein ε ′ and ε ′′ are generally positive (or zero) scalar functions for ω ≥ 0. It ensues that

iω∇ • (ε ′ (x, ω)E(x, ω)) = ∇ • J i (x, ω) + ∇ • (ε ′′ (x, ω)E(x, ω)) , (13)
or

∇ • [(ε ′ (x, ω) + iε ′′ (x, ω)) E(x, ω)] = ∇ • J i (x, ω) iω , (14) whence
∇ • [ε(x, ω)E(x, ω)] = ρ i (x, ω) , (15)
or ∇ • [ε(x, ω)∇ψ(x, ω)] = -ρ i (x, ω) , (16)
wherein ε(x, ω) = ε ′ (x, ω) + iε ′′ (x, ω) , (17)
is the (complex) permittivity such that ε(x, -ω) = ε * (x, ω) ; ω > 0. For For passive materials, ε ′ ≥ 0 and ε ′′ ≥ 0. Usually, ε ′′ << ε ′ at low frequencies. The quasistatic electric field is thus seen to be governed by an inhomogeneous Poisson equation, which, in a homogeneous medium, devoid of impressed sources, becomes the (homogeneous) Laplace equation. Let R n be divided into two domains D 0 and D 1 , separated by the interface I the unit vector normal to which is ν, and let M 0 and M 1 be two homogeneous, isotropic dielectric media (filling D 0 and D 1 respectively) in which the position-independent permittivities are ε 0 and ε 1 respectively. If all space (i.e., D 0 +D 1 ) is occupied solely by M 0 , and to be devoid of impressed charges (i.e., ρ i = 0), but subjected to a uniform electric field E i satisfying

E i = -∇ψ i , ( 18
) then ∇ • (ε 0 ∇ψ i ) = 0 in D 0 ⊂ R n . ( 19
)
The introduction of M 1 into D 1 induces a potential ψ d 0 in D 0 so that the total potential is now

ψ 0 = ψ i + ψ d 0 in D 0 , (20)
whereas the induced and total potentials in D 1 are

ψ 1 = ψ d 1 in D 1 . (21)
Then the problem of the prediction of ψ l ; l = 0, 1, for given E i or ψ i , can be cast in the three-relation form (equivalent to [START_REF] Chen | Determining the effective electromagnetic parameters of bianisotropic metamaterials with periodic structures[END_REF])

∇ • (∇ψ l ) = 0 in D l , l = 0, 1 , (22)
ψ 0 -ψ 1 = 0 on I , (23)
ε 0 ν • ∇ψ 0 -ε 1 ν • ∇ψ 1 = 0 on I , (24)
with uniqueness assured by the condition:

|ψ d l | < ∞ in D l ; l = 0, 1 . (25)
In the preceding lines, the emphasis has been on the forward problem of the prediction of the potential field ψ, assuming that all other ingredients of the configuration and of the solicitation (via ψ i ) are known. Actually, the present investigation is more specifically concerned with the inverse problem (examples of which can be found in [START_REF] Chylek | Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data[END_REF][START_REF] Banks | Estimation Techniques for Distributed Parameter Systems[END_REF][START_REF] Johnson | Parameter estimation by least-squares methods[END_REF][START_REF] Morabito | A fuzzy modeling approach for the solution of an inverse electrostatic problem[END_REF][START_REF] Neittaanmaki | Inverse Problems and Optimal Design in Electricity and Magnetism[END_REF][START_REF] Alessandrini | Examples of instability in inverse boundary-value problems[END_REF][START_REF] Emery | The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters[END_REF][START_REF] Sebaa | Application of the Biot model to ultrasound in bone: inverse problem[END_REF][START_REF] Isakov | Inverse obstacle problems[END_REF][START_REF] Yeh | Characterization of mechanical and geometrical properties of a tube with axial and circumferential guided waves[END_REF][START_REF] Buchanan | Recovery of the parameters of cancellous bone by inversion of effective velocities, and transmission and reflection coefficients[END_REF][START_REF] Young | An application of random projection to parameter estimation in partial differential equations[END_REF][START_REF] Hasar | Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs[END_REF][START_REF] Artemev | Inverse electrostatic and elasticity problems for checkered distributions[END_REF][START_REF] Lefeuve-Mesgouez | Retrieval of the physical properties of an anelastic solid half space from seismic data[END_REF][START_REF] Artemev | Inverse electrostatic and elasticity problems for checkered distributions[END_REF][START_REF] Scotti | Multiparameter identification of a lossy fluid-like object from its transient acoustic response[END_REF][START_REF] Hasar | Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs[END_REF]) of the retrieval of ε (or, more precisely, of ε 1 ) from data relative to ψ (more precisely, ψ 0 ), assuming that all other parameters of the configuration as well as of the solicitation (i.e., the nuisance parameters [START_REF] Emery | The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters[END_REF]) are more or less well-known (i.e., uncertain to some degree).

The chosen physical configuration (in which D 1 is an infinitely-long circular cylinder) will be shown to enable both the forward and inverse problems to be solved in explicit, exact manner so as to make possible a thorough analysis (somewhat in the spirit of [START_REF] Hasar | Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs[END_REF][START_REF] Hasar | Application of a useful uncertainty analysis as a metric tool for assessing the performance of electromagnetic properties retrieval methods of bianisotropic metamaterials[END_REF][START_REF] Emery | The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters[END_REF]) of the influence of nuisance parameter uncertainty on retrieval accuracy. This point merits to be emphasized because it is not often that an other-than-academic inverse problem can be solved exactly (see [START_REF] Wirgin | Ill-posedness and accuracy in connection with the recovery of a single parameter from a single measurement[END_REF] for another example), and it is not commonplace in parameter retrieval problems to be able to evaluate analytically the influence of nuisance prior uncertainty on the accuracy of the retrievals.

At this point, it is necessary to give a definition of uncertainty. Let us first assume that the potential field (i.e., the input data required for the inversion) is the output of a real experiment. In order to conceptualize this experiment, and/or eventually simulate (by a computer code, rather than physically-generate) the data, we need a mathematical model of the physics involved in the experiment. This model, includes, among other things, a set of parameters p = {p 1 , p 2 , ...} to which we must assign values. A way to do this is by measurement. Assume that one of these parameters, say p j is what (we think) is actually measured. It is common to repeat the measurement of p j several times (call these: realizations) while keeping all the other parameters (hopefully) constant. If these various realizations lead to different values of p j then we say that there is some error in the measurement of p j , and the inclination is strong to class these as random errors [START_REF] Emery | The effect of correlations and uncertain parameters on the efficiency of estimating and the precision of estimated parameters[END_REF][START_REF] Coleman | Experimentation, Validation, and Uncertainty Analysis for Engineers[END_REF]. If, on the other hand, the various realizations lead to the same value P j , then the question arises as to whether this value is the true value or something else. If it is something else, then its departure from the true value can be qualified as systematic error [START_REF] Coleman | Experimentation, Validation, and Uncertainty Analysis for Engineers[END_REF]. If systematic error is thought to exist, but cannot be corrected (with the means available to the experimentalist) then a way of taking this error into account is to say that p j lies somewhere within a range of values including P j . If, on the other hand, we don't even go to the trouble of actually measuring p j , then a common way of assigning a value to it is either by guessing, or borrowing the value from a publication. Again, the question arises as to whether this (guessed or borrowed) value P j is the true value. If, as it is reasonable to expect, P j is not the true value of p j (all the more so than we don't know (the true) p j ), then we can take into account this error by saying that p j lies somewhere within a range of values including P j . In what follows, we qualify a parameter as being uncertain by the fact that its assigned value, resulting from experiment, guessing or borrowing from a published result, is incorrect in a sense akin to systematic measurement error.

Description of the physical configuration

A circular cylinder, occupied by the homogeneous, isotropic medium M 1 (in which the permittivity is ε 1 ) is introduced into another homogeneous, isotropic medium M 0 (in which the permittivity is ε 0 ) of infinite extent and is submitted to an electric field E i whose direction is assumed to be constant at all points of space. The z axis (of the cartesian coordinate system Oxyz forms the axis of the cylinder and the circular disk Ω 1 , with center at O, constitutes the support of the cylinder in the xy plane. The unbounded region exterior to Ω 1 (in the xy plane) is Ω 0 .

The incident electric field vector E i is assumed to lie in the xy plane and to be independent of z. The circular boundary of Ω 1 is Γ, the outward unit vector normal to which is ν. Consequently, the incident and induced fields are independent of z, i.e., the problem is two-dimensional, with z the ignorable coordinate (fig. 1).

The effect of the primary field E i = -∇ψ i on the cylinder is to induce a secondary field E d = -∇ψ d . In the so-called forward problem, the task is to predict this secondary field, whereas in the inverse problem, the associated potential ψ d (combined with ψ i ) constitutes the data, which, by means of an inversion scheme, is analyzed to enable the retrieval of the constitutive parameter ε 1 of M 1 .

The units of E, ψ, ε, a, θ i , b and θ are: volt/m, volt, f arad/m, m, • or rad, m, • or rad respectively.

Let r, θ designate the polar coordinates of a point P in the x-y plane and let x designate the vector joining O to P . The parametric equation of Γ is r = a ; ∀θ ∈ [0, 2π[, with a (m) the radius of the circular disk Ω 1 . The (incident) angle between E i and the x axis is θ i ( • or rad).

The total (primary plus secondary) potential field is assumed to be sensed at various points on a circle (concentric with Γ) of radius b > a. The polar angle ( • or rad) at which a generic point-like sensor is located is θ (with respect to the positive x axis).

In the first part of this document, the objective will be: given e i (amplitude of ψ i ), θ i , a and ε l ; l = 0, 1, find the total potential fields at one or more positions (starting with θ = θ b ) on the circle r = b. It is assumed that the location of the axis of the cylinder is known (and coincides with the z-axis) and that the number, and angular, positions of the point-like (in In the second, main, part of this document, the objective will be: given the total potential fields registered at one or more sensors located at angular positions starting with θ = θ b on the circle r = B (analogous to, but different from, b due to uncertainty of this parameter), as well as the set of parameters E i , Θ i , A and E 0 (analogous to e i , θ i , a and ε 0 , but integrating uncertainties), find E ′ 1 (analogous to ε 1 ). Actually, the exact solution (to simulate measured data concerning the total field) obtained in the first part of this document will be employed as the data (the corresponding model is termed the data simulation model) to solve the inverse problem in the second part of the document. In addition, we appeal to a parameter retrieval model, also based on the aforementioned physical configuration, to recover the constitutive parameter of the cylinder. Some of the fixed (during the inversion) parameters (the so-called nuisance parameters) of the retrieval models will be assumed to be not precisely known. The effect of various amounts of nuisance parameter uncertainty on the accuracy of the retrievals will be studied in depth in the sequel.

3 The forward problem

Boundary-value problem

The generic potential ψ(x) is related to the generic vectorial electric field E(x) by E(x) = -[ψ ,r (x)i rr -1 ψ ,θ (x)i θ ], with i r , i θ the unit vectors corresponding to r, θ. Furthermore,

ν = i r , so that ν • ∇ψ(x) = ψ ,r (x).
The assumed primary electric potential which satisfies

1 r ∂ ∂r r ∂ψ i (r, θ) ∂r + 1 r 2 ∂ 2 ψ i (r, θ) ∂θ 2 = 0 ; ∀(r, θ) ∈ Ω 0 , (26)
is ψ i (r, θ) = -e i r cos(θ -θ i ) , (27)
with e i a constant amplitude term. Applied to our cylindrical geometry problem, the governing equations of sect. 1 become

1 r ∂ ∂r r ∂ψ d l (r, θ) ∂r + 1 r 2 ∂ 2 ψ d l (r, θ) ∂θ 2 = 0 ; ∀(r, θ) ∈ Ω l ; l = 0, 1 , (28)
ψ 0 (a, θ) -ψ 1 (a, θ) = 0 ; ∀θ ∈ [0, 2π[, (29)
ε 0 ψ 0,r (a, θ) -ε 1 ψ 1,r (a, θ) = 0 ; ∀θ ∈ [0, 2π[ . ( 30
)
|ψ d l (r, θ)| < ∞ ; ∀r, θ ∈ Ω l ; l = 0, 1 , (31)

Field representations and application of the boundary conditions

By separation of variables, we obtain the following electric potential field representations satisfying ( 27)- [START_REF] Neittaanmaki | Inverse Problems and Optimal Design in Electricity and Magnetism[END_REF]; note that a term such as ln r does not satisfy [START_REF] Neittaanmaki | Inverse Problems and Optimal Design in Electricity and Magnetism[END_REF] either in Ω 0 (notably at r → ∞) or Ω 1 (notably at r = 0):

ψ d 0 (x) = ∞ n=0 r -n [A n cos(nθ) + A ′ n sin(nθ)] ; ∀x ∈ Ω 0 , (32)
ψ d 1 (x) = ∞ n=0 r n [B n cos(nθ) + B ′ n sin(nθ)] ; ∀x ∈ Ω 1 , (33)
with the understanding that A ′ 0 = B ′ 0 = 0. This suggests writing ψ i as:

ψ i 1 (x) = ∞ n=0 r n [C n cos(nθ) + C ′ n sin(nθ)] , (34)
wherein

C n = -e i cos(nθ i )δ n1 , C ′ n = -e i sin(nθ)δ n1 , (35)
and δ nm is the Kronecker delta symbol. Eqs. ( 29)-( 30) lead to:

A m = e i a 2 ε 1 -ε 0 ε 1 + ε 0 cos(θ i )δ m1 , A ′ m = e i a 2 ε 1 -ε 0 ε 1 + ε 0 sin(θ i )δ m1 , (36)
B m = -e i 2ε 0 ε 1 + ε 0 cos(θ i )δ m1 , B ′ m = e i 2ε 0 ε 1 + ε 0 sin(θ i )δ m1 , (37)
whence, by virtue of ( 32)- [START_REF] Nesvadba | Database of physical properties of agro-food materials[END_REF], and taking, for convenience, A 0 = B 0 = 0:

ψ d 0 (x) = e i a 2 r ε 1 -ε 0 ε 1 + ε 0 cos(θ -θ i ) ⇒ ψ 0 (x) = e i -r + a 2 r ε 1 -ε 0 ε 1 + ε 0 cos(θ -θ i ) , (38)
ψ 1 (x) = ψ d 1 (x) = -e i r 2ε 0 ε 1 + ε 0 cos(θ -θ i ) . (39)
Note that these expressions are identical to those in [START_REF] Morse | Methods of Theoretical Phyics[END_REF], p. 1185 when θ i = 0.

4 Exact solution of the inverse problem

General considerations

The inversion consists in obtaining estimates E ′ 1 , E ′′ 1 of the sought-for constitutive parameters ε ′ 1 , ε ′′ 1 by minimizing a cost functional which expresses the discrepancy between the data simulation model (ψ 0 ) of the electric potential and the parameter retrieval model (Ψ 0 ) of the electric potential on the circle of radius b (which becomes B if it is uncertain) in the angular interval of observation [0, 2π[. This cost functional is:

K(E ′ 1 , E ′′ 1 ) = 2π 0 ψ 0 (b, θ|e i , θ i , a, ε 0 , ε ′ 1 , ε ′′ 1 ) -Ψ 0 (B, θ|E i , Θ i , A, E 0 , E ′ 1 , E ′′ 1 ) 2 dθ 2π 0 ψ 0 (b, θ|e i , θ i , a, ε 0 , ε 1 ) 2 dθ . (40)
K is actually replaced, in the numerical context, and to account for the discrete nature of the physical sensing process, by another cost functional obtained by adopting a simple quadrature rule for the integrals:

K ≈ K (N ) (E ′ 1 , E ′′ 1 ) = δ θ N n=1 ψ 0 (b, θ n |e i , θ i , a, ε 0 , ε ′ 1 , ε ′′ 1 ) -Ψ 0 (B, θ n |E i , Θ i , A, E 0 , E ′ 1 , E ′′ 1 ) 2 δ θ N n=1 ψ 0 (b, θ n |e i , θ i , a, ε 0 , ε ′ 1 , ε ′′ 1 ) 2 , (41)
wherein δ θ = 2π/N, and θ n = θ b + δ θ 2 + (n -1)δ θ are the actual angles (for n = 1, 2, ..., N ; θ b a chosen starting angle and θ e a chosen ending angle) at which the electric potential is sensed.

Note that the set of parameters (lower case letters and symbols) is different in the simulated data model from the corresponding set (upper case letters and symbols) in the parameter retrieval model; this expresses the fact that the subset of nuisance parameters (all the parameters except ε 1 ) may be not well-known to us before, and during, the inversion.

We have

ψ 0 (b, θ) = e i -b + a 2 b ε ′ 1 + iε ′′ 1 -ε 0 ε ′ 1 + iε ′′ 1 + ε 0 cos(θ -θ i ) = f(e i , a, b, ε 0 , ε ′ 1 , ε ′′ 1 ) cos(θ -θ i ) , (42)
so that

Ψ 0 (B, θ) = E i -B + A 2 B E ′ 1 + iE ′′ 1 -E 0 E ′ 1 + iE ′′ 1 + E 0 cos(θ -Θ i ) = F(E i , A, B, E 0 , E ′ 1 , E ′′ 1 ) cos(θ -Θ i ) . (43
) It ensues that:

K(E 1 ) = 2π 0 f cos(θ -θ i ) -F cos(θ -Θ i ) 2 dθ 2π 0 f cos(θ -θ i ) 2 dθ . ( 44
)
and

K (N ) (E 1 ) = δ θ N n=1 f cos(θ n -θ i ) -F cos(θ n -Θ i ) 2 dθ δ θ N 0 f cos(θ n -θ i ) 2 dθ . (45)
4.2 Finding a constitutive parameter of the cylinder by minimizing K

Let G be any one of the capital-letter parameters and κ = cos(Θ iθ i ). From [START_REF] Soltani | Evaluating banana ripening status from measuring dielectric properties[END_REF] we obtain

K(G) = f 2 -2κℜ(f * F) + F 2 f 2 , (46)
wherein the symbol * designates the complex conjugate operator. An extremum of the cost functional with respect to the variable G (henceforth, E ′ or E ′′ ) is found for ∂K(G) ∂G = 0 or

ℜ ∂F * ∂G ℜ(F -κf) -ℑ ∂F * ∂G ℑ(F -κf) = 0 . ( 47
)
It is easily found that:

ℜ(F -κf) = E i -B + A 2 B E ′ 2 1 + E ′′ 2 1 -E 2 0 E 1 + E 0 2 -e i κ -b + a 2 b ε ′ 2 1 + ε ′′ 2 1 -ε 2 0 ε 1 + ε 0 2 , (48)
ℑ(F -fκ) = E i A 2 B 2E ′′ 1 E 0 E 1 + E 0 2 -e i κ a 2 b 2ε ′′ 1 ε 0 ε 1 + ε 0 2 , ( 49
)
∂F * ∂E ′ 1 = E i * A 2 2E * 0 B (E 1 + E 0 ) 2 2 E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 + 2i E ′ 1 E ′′ 1 + E 0 E ′′ 1 := F E i * A 2 2E * 0 B (E 1 + E 0 ) 2 2 , ( 50
) ∂F * ∂E ′′ 1 = -i ∂F * ∂E ′ := -iF E i * A 2 2E * 0 B (E 1 + E 0 ) 2 2 . ( 51
)
Henceforth, we shall assume that E i and E 0 are positive real, so that dividing (47) by

E i * A 2 2E * 0 B (E 1 +E 0 ) 2 2 = 0 yields either ℜ(F )ℜ(F -κf) -ℑ(F )ℑ(F -κf) = 0 , (52)
(for the determination of

E ′ 1 ) or ℜ(-iF )ℜ(F -κf) -ℑ(-iF )ℑ(F -κf) = 0 , (53)
(for the determination of E ′′ 1 ) wherein

F = E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 + 2i E ′ 1 E ′′ 1 + E 0 E ′′ 1 , (54)
4.

3 Finding E ′ 1 of the cylinder by minimizing K Eq.( 54) tells us that

ℜ(F ) = E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 , (55)
ℑ(F ) = 2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 , (56)
so that (52) becomes

E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 ℜ(F -κf) -2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 ℑ(F -κf) = 0 . ( 57
)
On the other hand, we can write [START_REF] Hippel | Dielectrics and Waves[END_REF] and [START_REF] Vuye | Precision in the ellipsometric determination of the optical constants of very thin films[END_REF] as:

ℜ(F -κf) = F -B 2 + A 2 E ′ 2 1 + E ′′ 2 1 -E 2 0 E 1 + E 0 2 -f g , (58)
ℑ(F -fκ) = F A 2 2E ′′ 1 E 0 E 1 + E 0 2 -f h , (59)
wherein

F := E i B , f := e i κ b , g := -b 2 + a 2 ε ′ 2 1 + ε ′′ 2 1 -ε 2 0 ε 1 + ε 0 2 , h := 2a 2 ε ′′ 1 ε 0 ε 1 + ε 0 2 , ( 60
)
so that ( 52) becomes

E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 -F B 2 + F A 2 E ′ 2 1 + E ′′ 2 1 -E 2 0 E 1 + E 0 2 -f g - 2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 F A 2 2E ′′ 1 E 0 E 1 + E 0 2 -f h = 0 . (61)
The product of this equation with

E 1 + E 0 2 = E ′ 2 1 + E ′′ 2 1 + 2E ′ 1 E 0 + E 2 0 = 0 yields E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 -(F B 2 + f g)(E ′ 2 1 + E ′′ 2 1 + 2E ′ 1 E 0 + E 2 0 ) + F A 2 (E ′ 2 1 + E ′′ 2 1 -E 2 0 ) - 2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 F A 2 2E ′′ 1 E 0 -f h(E ′ 2 1 + E ′′ 2 1 + 2E ′ 1 E 0 + E 2 0 ) = 0 , (62)
which can be cast into the form of the quartic equation [4]

C 4 E ′ 4 1 + C 3 E ′ 3 1 + C 2 E ′ 2 1 + C 1 E ′ 1 + C 0 = 0 , (63)
whose coefficients are:

C 4 = F A 2 -(F B 2 + f g) , (64)
C 3 = 2E 0 F A 2 -4E 0 (F B 2 + f g) + 2E ′′ 1 f h , (65)
C 2 = -2E 2 0 (F B 2 + f g) + 6E 0 E ′′ 1 f h , (66)
C 1 = -2E 0 k + F A 2 -2E 0 (k + + k -)(F B 2 + f g) + (2k + + 4E 2 0 )E ′′ 1 f h , (67)
C 0 = -k 2 + F A 2 -k + k -(F B 2 + f g) + 2k + E 0 E ′′ 1 f h , (68)
with

k ± := E 2 0 ± E ′′ 2 1 . (69)
It ensues from these formulae that:

C 1 = 2E 0 C 0 k + + k + (C 3 -2E 0 C 4 ) , (70)
C 2 = C 0 k + + 2E 0 (C 3 -2E 0 C 4 ) + k + C 4 , (71)
so that the quartic equation can re-written as

C 4 E ′ 4 1 +C 3 E ′ 3 1 + C 0 k + +2E 0 (C 3 -2E 0 C 4 )+k + C 4 E ′ 2 1 + 2E 0 C 0 k + +k + (C 3 -2E 0 C 4 ) E ′ 1 +C 0 = 0 , (72) or (E
′ 2 1 + 2E 0 E ′ 1 + k + ) C 4 E ′ 1 (E ′ 1 -2E 0 ) + C 3 E ′ 1 + C 0 k + = 0 . (73)
Thus, the solutions of the quartic equation can be obtained from the solutions of the two quadratic equations:

E ′ 2 1 + 2E 0 E ′ 1 + k + = 0 . ( 74
)
C 4 E ′ 1 (E ′ 1 -2E 0 ) + C 3 E ′ 1 + C 0 k + = 0 . ( 75
)
The two roots of (74) are:

E ′ (1) 1 = -E 0 -iE ′′ 1 E ′ (2) 1 = -E 0 + iE ′′ 1 , (76)
whereas the two roots of (75) are:

E ′ (-) 1 = -k + (C 3 -2E 0 C 4 )- √ k 2 + (C 3 -2E 0 C 4 ) 2 -4k + C 4 C 0 2k + C 4 E ′ (+) 1 = -k + (C 3 -2E 0 C 4 )+ √ k 2 + (C 3 -2E 0 C 4 ) 2 -4k + C 4 C 0 2k + C 4 . (77)
Eqs. ( 76)-(77) represent the exact solutions of the inverse problem of the identification of the sole parameter E ′ 1 . Recall that it was assumed that E 0 , E ′ 1 and E ′′ 1 (of the same nature as

ε 0 , ε ′ 1 , ε ′′ 1 respectively) are positive real. Thus, E ′ (1) 1 and E ′ (2) 1
are not admissible solutions. Whether E ′ (±) 1 are admissible or not can be decided empirically, or by other means, using this same criterium (a solution is admissible if it is positive real).

Perturbation solutions of E

′ (±) 1 for small ε ′′ 1
Let us return to (75) which can be written as

2 j=0 U j u j = 0 , (78)
wherein:

U 0 = C 0 k + , U 1 = C 3 -2E 0 C 4 , U 2 = C 4 , (79)
and

u 0 = 1 , u 1 = E ′ 1 , u 2 = E ′ 2 1 = u 2 1 . (80)
We expand U j and u j in series of powers of the supposedly-small quantity

δ := ε ′′ 1 , (81)
that is,

U j = ∞ k=0 U (k) j δ k , u j = ∞ l=0 u (l) j δ l , (82)
with the understanding u

(l) 0 = δ l0 , (83)
(in which δ l0 is the Kronecker delta symbol) and

u (l) 2 = ∞ l 1 =0 u (l 1 ) 1 δ l 1 ∞ l 2 =0 u (l 2 ) 1 δ l 2 = ∞ l=0 l l 1 =0 u (l 1 ) 1 u (l-l 1 ) 1 . ( 84
) whence u (0) 2 = u (0)2 1 , u (1)
2 = 2u (0) 1 u (1) 1 , u (2)
2 = 2u (0) 1 u (2) 1 + u (1)2 1 , (85)
and so on. The introduction of (82) into (78) gives rise to

∞ m=0 2 j=0 m l=0 U (m-l) j u (l) j δ m = 0 , (86)
whence

2 j=0 U (0) j u (0) j = 0 , (87) 2 j=0
[U (0) j u

(1)

j + U (1) j u (0) j ] = 0 , ( 88
) 2 j=0 [U (0) j u (2) j + U (1) j u
(1)

j + +U (2) j u (0) j ] = 0 , (89)
and so on. More explicitly, we have:

U (0) 0 u (0) 0 + U (0) 1 u (0) 1 + U (0) 2 u (0) 2 = 0 , (90)
U (0) 0 u (1) 0 + U (1) 0 u (0) 0 + U (0) 1 u (1)
1 + U

(0) 1 + U (0) 2 u (1) 2 + U (1) 2 u (0) 2 = 0 , (1) 1 u
U (0) 0 u (2) 0 + U (91)
(1)

0 + U (2) 0 u (0) 0 + U (0) 1 u
(2)

1 + U (1) 1 u
(1)

1 + U (2) 1 u (0) 1 + U (0) 2 u (2) 2 + U (1) 2 u
(1)

2 + U (2) 2 u (0) 2 = 0 , (92)
and so on.

Taking into account (83) and ( 85) enables ( 90)-(92) to be written as:

U (0) 0 + U (0) 1 u (0) 1 + U (0) 2 u (0)2 1 = 0 , (93)
U (1) 0 + U (0) 1 u (1) 1 + U (1) 1 u (0) 1 + U (0) 2 2u (0) 1 u
(1)

1 + U (1) 2 u (0)2 1 = 0 , (94)
U (2) 0 +U (0) 1 u (2) 1 +U (1) 1 u (1) 1 +U
(2)

1 u (0) 1 +U (0) 2 (2u (0) 1 u
(2)

1 +u (1)2 1 )+U (1) 2 2u
(0)

1 u (1) 1 +U (2) 2 u (1)2 1 = 0 . ( 95
)
The first of these last three relations is a quadratic equation for the unknown u 1 . Thus, the perturbation method consititues an iterative scheme for the obtention of the u (l) 1 ; l = 0, 1, 2, .... We now address the problem of the U (k) j . To do this, we must first recall (79) and the expressions for C 4 , C 3 and C 0 :

C 4 = F A 2 -(F B 2 + f g) , (96)
C 3 = 2E 0 F A 2 -4E 0 (F B 2 + f g) + 2E ′′ 1 f h , (97)
C 0 = -k 2 + F A 2 -k + k -(F B 2 + f g) + 2k + E 0 E ′′ 1 f h , (98)
so that:

U 0 = C 0 k + = -k + F A 2 -k -(F B 2 + f g) + 2E 0 E ′′ 1 f h , (99)
U 1 = C 3 -2E 0 C 4 = -2E 0 (F B 2 + f g) + 2E ′′ 1 f h , (100)
U 2 = C 4 = F A 2 -(F B 2 + f g) . (101)
Recall that:

g = -b 2 + a 2 ε ′ 2 1 + ε ′′ 2 1 -ε ′ 2 0 ε 1 + ε 0 2 , h = 2a 2 ε 0 ε ′′ 1 ε 1 + ε 0 2 , ( 102
)
which can be written as:

g = [-b 2 (ε ′ 2 1 + ε 2 0 ) + a 2 (ε ′ 2 1 -ε 2 0 )] + (-b 2 + a 2 )ε ′′ 2 1 (ε ′ 1 + ε 0 ) 2 + ε ′′ 2 1 , h = 2a 2 ε 0 (ε ′ 1 + ε 0 ) 2 + ε ′′ 2 1 ε ′′ 1 . (103)
or (with the new condensed notations and δ := ε ′′ 1 )

g = A + Bδ 2 C + δ 2 , h = 2a 2 ε 0 δ C + δ 2 (104)
Under the (small-δ) hypothesis (implicit in the perturbation method)

0 ≤ δ << 1 , (105)
we find

g = A C + 1 C B - A C δ 2 + O(δ 4 ) := g (0) + g (2) δ 2 + O(δ 4 ) , (106)
h = 2a 2 ε 0 C δ - 2a 2 ε 0 C 2 δ 3 + O(δ 5 ) := h (1) δ + h (3) δ 3 + O(δ 5 ) , (107)
Consequently:

U 0 = [-k + F A 2 -k -F B 2 ] -k -f (g (0) + g (2) δ 2 ) + 2E 0 E ′′ 1 f (h (1) δ + h (3) δ 3 ) + O(δ 4 ) = [-k + F A 2 -k -F B 2 -k -f g (0) ] + [2E 0 E ′′ 1 f h (1) ]δ + [-k -f (g (0) ]δ 2 + O(δ 3 ) , (108)
whence

U (0) 0 = [-k + F A 2 -k -F B 2 -k -f g (0) ] , U (1)
0 = [2E 0 E ′′ 1 f h (1) ] , U (2) 0 = [-k -f g (2) ] (109) U 1 = [-2E 0 F B 2 ] + [-2E 0 f g (0) + g (2) δ 2 )] + [2E ′′ 1 f (h (1) δ + h (3) δ 3 )] + O(δ 4 ) = [-2E 0 F B 2 -2E 0 f g (0) ] + [2E ′′ 1 f h (1) ]δ + [-2E 0 f g (2) ]δ 2 + O(δ 3 ) , (110)
whence

U (0) 1 = [-2E 0 F B 2 -2E 0 f g (0) ] , U (1)
1 = [2E ′′ 1 f h (1) ] , U (2) 1 = [-2E 0 f g (2) ] , (111)
U 2 = [F A 2 -F B 2 ]-f (g (0) +g (2) δ 2 )+O(δ 4 ) = [F A 2 -F B 2 -f g (0) ]+[-f g (2) ]δ 2 +O(δ 4 ) , (112) whence U (0) 2 = [F A 2 -F B 2 -f g (0) ] , U (1)
2 = 0 , U (2)
2 = [-f g (2) ] .

(113)

The solution of (93) is

u (0) 1 = -U (0) 1 ± U (0)2 1 -4U (0) 2 U (0) 0 2U (0) 2 , (114)
or

u (0) 1 = E 0 (F B 2 + f g (0) ) ± E 2 0 (F A 2 ) 2 + E ′′ 2 1 (F A 2 -F B 2 -f g (0) ) 2 F A 2 -F B 2 -f g (0) , (115)
Since we assumed ε ′′ 1 to be small, it is not illogical to assume E ′′ 1 also to be small, i.e., 0

< E ′′ 1 << 1 . (116)
With the change of variables

α = F A 2 , β = F B 2 + f g (0) , (117)
and recalling that

k ± = E 2 0 ± E ′′ 2
1 , it is easily shown that

U (0)2 1 -4U (0) 2 U (0) 0 = 2 E 2 0 α 2 + E ′′ 2 1 (α -β) 2 = 2 E 0 α + (α -β) 2 2E 0 α E ′′ 2 1 + O(E ′′ 4
1 ) , (118) so that

u (0) 1 = u (0)± 1 := E 0 (β ± α) α -β ± E ′′ 2 1 2(α -β)E 0 α + O(E ′′ 4 1 ) , (119)
or

u (0)- 1 = -E 0 - E ′′ 2 1 2(α -β)E 0 α + O(E ′′ 4 1 ) , (120)
u (0)+ 1 = E 0 (α + β) α -β + E ′′ 2 1 2(α -β)E 0 α + O(E ′′ 4 1 ) , (121)
The negative nature of u (0)-1 makes this solution inadmissible, so that only u

(0)+ 1
is admissible. This finding translates to the choice of sign in (115), so that the unique, exact solution for u

(0) 1 is u (0) 1 = E 0 (F B 2 + f g (0) ) + E 2 0 (F A 2 ) 2 + E ′′ 2 1 (F A 2 -F B 2 -f g (0) ) 2 F A 2 -F B 2 -f g (0) , (122)
or, after integrating the small nature of

E ′′ 1 u (0) 1 = E 0 F A 2 + F B 2 + f g (0) F A 2 -F B 2 -f g (0) + O(E ′′ 2 1 ) . ( 123
)
The next step in the perturbation procedure is to obtain u

(1)

1 from u (0)
1 . This is done via (94) from which we obtain

u (1) 1 = - U (1) 0 + U (1) 1 u (0) 1 + U (1) 2 u (0)2 1 U (0) 1 + U (0) 2 2u (0) 1 , (124)
But, instead of writing the explicit expression for u

(1)

1 , we recall that here u 1 stands for E ′ 1 , and to first order in ε ′′

u 1 = E ′ 1 ≈ u (0) 1 + u (1) 1 ε ′′ 1 , (125)
or, neglecting terms of order E ′′ 1 and ε ′′ 1 (due to the assumed smallness of E ′′ 1 and ε ′′ 1 ), the approximate solution for

E ′ 1 is E ′ 1 ≈ E 0 F A 2 + F B 2 + f g (0) F A 2 -F B 2 -f g (0) , (126)
with the understanding that, on account of the what the perturbation analysis showed concerning the admissible solution, the exact solution for E ′ 1 is The result embodied in (127) calls for the following comments:

E ′ 1 = -k + (C 3 -2E 0 C 4 ) + k 2 + (C 3 -2E 0 C 4 ) 2 -4k + C 4 C 0 2k + C 4 . ( 127
1-It shows that the solution to the inverse problem of the identification of the sole parameter E ′ 1 exists, even in the presence of nuisance parameter uncertainties; 2-it shows that it is possible to obtain the mathematically-explicit and exact solution to the given inverse problem, even in the presence of nuisance parameter uncertainties; 3-it shows that this solution is unique for a given set of parameters

ε ′ 1 , ε ′′ 1 , ε 0 , a, e i , θ i , b, E ′′ 1 , E 0 , A, E i , Θ i , B
, subject to the assumed physical constraint (i.e., the real part of the permittivity should be positive real); 4-it shows that the accuracy of the retrieval of the real part of the permittivity (E ′ 1 ) is conditioned, albeit in a complex manner, by the uncertainty of the nuisance parameters E 0 , A, E i , Θ i , B;

The result embodied in (126) 5-shows that the accuracy of the retrieval of E ′ 1 is weakly-conditioned by ε ′′ 1 and E ′′ 1 since the dependence of E ′ 1 on these parameters is at least of order E ′′ 1 or ε ′′ 1 (and therefore small due to ε ′′ 1 and E ′′ 1 having been assumed to be small).

4.3.3

Preliminaries concerning the dependence of the retrieval error of E ′ 1 on the nuisance parameter uncertainties Due to the fifth comment in sect. 4.3.2 it is no longer necessary to delve on the issue of the dependence of the retrieval error of E ′ 1 on the (assumed-small) nuisance parameter E ′′ 1 . Thus, from now on, we deal with (126)

E ′ 1 = E 0 F A 2 + F B 2 + f g (0) F A 2 -F B 2 -f g (0) := E 0 ε ′ 1 X + + ε 0 Y + ε ′ 1 Y -+ ε 0 X - . ( 128
)
wherein

X ± = (F A 2 + f a 2 ) ± (F B 2 -f b 2 ) , Y ± = (F A 2 -f a 2 ) ± (F B 2 -f b 2 ) . ( 129
)
To unravel the complexity alluded to in the fourth comment in sect. 4.3.2, first suppose that the nuisance parameters A, B, E i , Θ i are known precisely, i.e., E i = e i , A = a, B = b, Θ i = θ i , whence X + = X -and Y ± = 0, so that

E ′ 1 = Ě′ 1 = E 0 ε ′ 1 ε 0 , (130)
which shows that the the relative error of the retrieved parameter, i.e.,

ε ε ′ 1 = E ′ 1 -ε 1 ε 1 = E ′ 1 ε 1 -1 , (131)
depends linearly on the ratio E 0 ε 0 , or, equivalently, the relative error of the retrieved constitutive parameter of the cylinder equals the relative uncertainty of the nuisance parameter concerning the constitutive parameter of the host

δ ε 0 = E 0 -ε 0 ε 0 = E 0 ε 0 -1 , (132)
which fact translates to

ε ε ′ 1 = δ ε 0 . ( 133
)
Returning to (128), which can now be written as

E ′ 1 = Ě′ 1 + E 0 ε 0 ε 0 ε ′ 1 (X + -X -) + (ε ′ 2 1 Y + -ε 2 0 Y -) ε ′ 1 Y -+ ε 0 X - , (134)
we find that

E ′ 1 ≈ Ě′ 1 , (135)
for small uncertainties of the nuisance parameters E i , A, B, Θ i .

Properties of E ′

1 as a function of B We can write

E ′ 1 (B) = E 0 E i b(B 2 + A 2 ) -e i BM -E i b(B 2 -A 2 ) -e i BM = -E 0 B 2 -e i M E i b B + A 2 B 2 -e i M E i b B -A 2 := -E 0 N D . ( 136
)
wherein

M := -κg (0) = κ (ε ′ 1 + ε 0 )b 2 -(ε ′ 1 -ε 0 )a 2 ε ′ 1 + ε 0 . ( 137
)
Since we assume a relatively-small uncertainty on the nuisance parameter Θ i , it follows that

|Θ i -θ i | < π/2, whence κ > 0. Also, since we assumed that ε 0 > 0, ε ′ 1 > 0, b 2 > a 2 , it follows that M > 0 . (138)
Finally, recall that we assume E 0 > 0. Now, assume that δ > 0 and consider

E ′ 1 (B 0 ± δ). It is easy to show that E ′ 1 (B 0 + δ) = E ′ 1 (B 0 -δ) , (139) entails
B 0 = e i M 2E i b > 0 , (140)
so that

E ′ 1 (B) = -E 0 B 2 -2B 0 B + A 2 B 2 -2B 0 B -A 2 . ( 141
) whence E ′ 1 (B 0 ) = E 0 . ( 142
)
Differentiating gives

dE ′ 1 (B) dB = 4E 0 (B -B 0 )A 2 B 2 -2B 0 B -A 2 , ( 143
) whence dE ′ 1 (B 0 ) dB = 0 . ( 144
)
The equation D = 0 has two solutions

B ± = e i M E i b ± e i M E i b 2 + 4A 2 2 = B 0 ± B 2 0 + A 2 . ( 145
)
at which E ′ 1 blows up. It is generally admitted that nonlinear inverse problems are ill posed ( [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equations[END_REF], [START_REF] Banks | Estimation Techniques for Distributed Parameter Systems[END_REF], [START_REF] Isakov | Inverse obstacle problems[END_REF]) which means that the solution either does not exist, or is not unique (our particular inverse solution was shown to be unique when certain physical constraints are satisfied) and is unstable in the case of existence. Instability is usually defined ( [START_REF] Banks | Modelling and estimating uncertainty in parameter estimation[END_REF]) by retrievals not being continuouslydependent on variations of the data. We, on the other hand, find that the function E ′ 1 (B) diverges at B -and B + , which suggests a new form of instability, first observed, but not explained, in [START_REF] Lefeuve-Mesgouez | Retrieval of the physical properties of an anelastic solid half space from seismic data[END_REF].

Borrowing a notion first expressed in [START_REF] Scotti | Multiparameter identification of a lossy fluid-like object from its transient acoustic response[END_REF], we define retrieval instability induced by nuisance parameter uncertainty as that which occurs when a very small variation of a nuisance parameter B (and perhaps other nuisance parameters) produces a very large variation of a retrieved parameter (at present, E ′ 1 ). Now consider E ′ 1 (B + ± δ), with 0 < δ << 1 defined as previously. We find

E ′ 1 (B + ± δ) = -E 0 A 2 ± δ B 2 0 + A 2 ±δ B 2 0 + A 2 ≈ ∓E 0 A 2 δ B 2 0 + A 2 , ( 146
)
which shows that for small positive δ,

E ′ 1 (B + -δ) > 0 , E ′ 1 (B + + δ) < 0. . (147)
Similarly, for ε >> 1,

E ′ 1 (B + ± ε) ∼ -E 0 ; ε → ∞ , (148)
which shows that E ′ 1 (B) tends to the negative value -E 0 for B → ±∞. This result, which at first appears to be surprising since it was assumed at the outset that E ′ 1 > 0, actually means that no physically-meaningful result can be obtained for large uncertainty of |B|. In other words, the solutions obtained for B > B + and B < B + must be rejected.

The analytical properties (symmetries, behavior in the neighborhoods of B 0 , B -and B + ) of the function E ′ 1 (B), deduced from the preceding formulae, are exhibited in fig. ??. It should be noted, in this figure, that it is possible to retrieve a physically-meaningful E ′ 1 (i.e., that is positive real) only for B + > B ≥ 0.

Properties of E ′

1 as a function of A We can write

E ′ 1 (A) = E 0 A 2 -e i BM E i b + B 2 A 2 -e i BM E i b -B 2 = E 0 A 2 -C 2 + B 2 A 2 -C 2 -B 2 := E 0 N D , (149)
wherein

C 2 := e i BM E i b (150)
It is immediately apparent that E ′ 1 (A) is symmetrical with respect to A = 0, i.e.,

E ′ 1 (-A) = E ′ 1 (A) . (151)
The equation D = 0 has two solutions

A ± = ± √ B 2 -C 2 . ( 152
)
at which E ′ 1 blows up. We can write

E ′ 1 (A) = E 0 A 2 + A 2 + A 2 -A 2 + , (153)
whence the result

E ′ 1 (0) = -E 0 . (154)
It also follows (for 0 < δ << 1) that

E ′ 1 (A + ± δ) = E 0 2A 2 + ± 2A + δ + δ 2 ±2A + δ + δ 2 , ( 155
) whence E ′ 1 (A + ± δ) ∼ ±E 0 A + δ ; δ → 0 , (156)
which means that

E ′ 1 > 0 for A A + , E ′ 1 < 0 for A A + and lim A→A + -δ = -∞. Next, consider dE ′ 1 (A)/dA: dE ′ 1 (A) dA = -4E 0 A 2 + A (A 2 -A 2 + ) 2 , (157)
from which it follows that dE ′ 1 (A)/dA = 0 for A = 0, dE ′ 1 (A)/dA < 0 for A > 0 and dE ′ 1 (A)/dA > 0 for A < 0. The analytical properties (symmetries, behavior in the neighborhoods of A = 0, A -and A + ) of the function E ′ 1 (A), deduced from the preceding formulae, are exhibited in fig. 3 in which one should note that all negative solutions for E ′ 1 are forbidden. It should be noted, in this figure, that it is possible to retrieve a physically-meaningful E ′ 1 (i.e., that is positive real) only for ∞ > A > A + .

E ′ 1 (E i ) = -E 0 B 2 + A 2 B 2 -A 2 E i -E i 0 E i -E i + , (158)
wherein:

E i 0 = e i BM b(B 2 + A 2 ) , E i + = e i BM b(B 2 -A 2 ) > E i 0 . (159)
It ensues that:

E ′ 1 (E i 0 ) = 0 , (160)
Moreover, E ′ 1 (E i ) blows up at E i = E i + , and from the fact that

E ′ 1 (E i ) ∼ -E 0 B 2 + A 2 B 2 -A 2 := E ′ ∞ 1 ; E i → ±∞ , (161)
we find that E ′ ∞ 1 < 0 (because, by hypothesis, b > a). Now we seek the behavior of E ′ 1 (E i ) in the neighborhood of E i 0 . Assume that δ > 0, so that

E ′ 1 (E i 0 ± δ) = ∓E 0 B 2 + A 2 B 2 -A 2 δ E i 0 -E i + ± δ , (162)
whence the asymptotic behavior, for small positive δ

E ′ 1 (E i 0 ± δ) = ±E 0 B 2 + A 2 B 2 -A 2 δ E i + -E i 0 ; δ → 0 , (163)
and the result that, for small positive δ,

E ′ 1 (E i ) > 0 at points E i = E i 0 + δ, and E ′ 1 (E i ) < 0 at points E i 0 = E i 0 -δ. Similarly, E ′ 1 (E i + ± δ) = ∓E 0 B 2 + A 2 B 2 -A 2 E i + -E i 0 ± δ δ ∼ ∓E 0 B 2 + A 2 B 2 -A 2 E i + -E i 0 δ ; δ → 0 , (164)
whence the result that, for small positive δ, E ′ 1 (E i ) < 0 at points E i = E i + +δ, and

E ′ 1 (E i ) > 0 at points E i 0 = E i + -δ. Finally, dE ′ 1 (E i ) dE i = E 0 B 2 + A 2 B 2 -A 2 E i + -E i 0 (E i -E i + ) 2 , (165)
which means that

dE ′ 1 (E i ) dE i
> 0 for all finite E i and

dE ′ 1 (E i ) dE i
→ 0 for E i → ±∞. The analytical properties (behavior in the neighborhoods of E i 0 , E i + of the function E ′ 1 (E i ), deduced from the preceding formulae, are exhibited in fig. 4 in which one should note that all negative solutions for E ′ 1 are forbidden. It should be noted, in this figure, that it is possible to retrieve a physically-meaningful E ′ 1 (i.e., that is positive real) only for E i + > B > E i 0 .

Comments on the analytical properties of E

′ 1 (A), E ′ 1 (E i ), E ′ 1 (B)
In sect 4.3.2, it was stated that the explicit formula for E ′ 1 demonstrates the existence of a solution of the inverse problem, even in the presence of nuisance parameter uncertainties. We now see that this statement must be interpreted in the following sense: a physicallyadmissible solution (i.e., positive and not infinite) for E ′ 1 exists only for a range of nuisance parameter uncertainties (i.e., those for which E ′ 1 is positive and not infinite). Furthermore, it was stated in sect 4.3.2 that the explicit formula for E ′ 1 constitutes a unique solution for a given set of parameters

ε ′ 1 , ε ′′ 1 , ε 0 , a, e i , θ i , b, θ b , E ′′ 1 , E 0 , A, E i , Θ i , B.
However, this does not mean that a physically-admissible E ′ 1 cannot arise from more than one sets of nuisance parameters, as is illustrated in fig. 2 for B in the neighborhood of B 0 (i.e., the two values B 0 ± δ give rise to the same E ′ 1 ).

4.4 Finding the real part of the permittivity of the cylinder by minimizing

K (N )
The method of obtaining the real part of the permittivity of the cylinder outlined in sect. 4.2 is somewhat unrealistic in that it supposes that the data is registered at a continuum of points on the sensing circle r = b. In reality, the data is registered at discrete locations on this circle and the number N of these locations is finite. Of course, N will have an influence on the accuracy of the retrieval and it is this influence that we shall now examine. From ( 45) we obtain

K (N ) (E ′ 1 ) = 1 f 2 δ θ N n=1 cos 2 (α + (n -1)δ θ ) × δ θ N n=1 f 2 cos 2 (α + (n -1)δ θ ) -2ℜ(f * F) cos(α + (n -1)δ θ ) cos(β + (n -1)δ θ )+ F 2 cos 2 (β + (n -1)δ θ ) , (166)
wherein α = θ b + δ θ 2 -θ i and β = θ b + δ θ 2 -Θ i , δ θ = θ e -θ b N , θ n = θ b + δ θ 2 + (n -1)δ θ .
Note that since δ θ depends on N, α and β also depend on N. Also note that, contrary to what is assumed in K, i.e., θ eθ e = 2π, here we admit arbitrary θ b and θ b , with the only restriction that θ e > θ b . Consider

σ (N ) (α, β) = δ θ N n=1 cos α + (n -1)δ θ cos β + (n -1)δ θ , (167)
It is straightforward to show that

σ (N ) (α, β) = δ θ 2 N cos(α -β) + cos α + β + (N -1)δ θ sin(Nδ θ ) sin(δ θ ) . ( 168
)
and, on account of (166)

K (N ) (E ′ 1 ) = |f 2 σ (N ) (α, α) -2ℜ(f * F)σ (N ) (α, β) + F 2 σ (N ) (β, β) f 2 σ (N ) (α, α) . ( 169
)
For θ eθ b = 2π, the fact that lim

N →∞ σ (N ) (α, β) = π cos(α -β) = π cos(θ i -Θ i ) = πκ , (170)
gives rise to the expected result lim

N →∞ K (N ) (E ′ 1 ) = K(E ′ 1 ) , (171)
Recall that our goal was to obtain an estimation of E ′ 1 by minimizing

K (N )) (E ′ 1 )
. The procedure is the same as in sect. 4.2 and yields the exact (mathematical) solution

E ′ (N ) 1 = -k + (C (N ) 3 -2E 0 C (N ) 4 ) + k 2 + (C (N ) 3 -2E 0 C (N ) 4 ) 2 -4k + C (N ) 4 C (N ) 0 2k + C (N ) 4 , (172)
and the approximate solution

E ′ (N ) 1 ≈ E 0 F A 2 + F B 2 + f (N ) g (0) F A 2 -F B 2 -f (N ) g (0) , (173)
wherein:

C 4 = F A 2 -(F B 2 + f (N ) g) , (174)
C 3 = 2E 0 F A 2 -4E 0 (F B 2 + f (N ) g) + 2E ′′ 1 f (N ) h , (175)
C 0 = -k 2 + F A 2 -k + k -(F B 2 + f (N ) g) + 2k + E 0 E ′′ 1 f (N ) h , (176)
f (N ) = e i κ (N ) b , (177)
κ (N ) = σ (N ) (α, β) σ (N ) (β, β) . ( 178
)
Eqs. ( 172)-( 173) are the (mathematically) exact and approximate solutions respectively to the inverse problem for discrete data in the interval [θ b + δ θ 2 , θ eδ θ 2 ]. Eqs. ( 174)-( 178)) show that the accuracy of the retrieval of ε ′ 1 is conditioned, not only by the uncertainty of the nuisance parameters E ′′ 1 , E 0 , A, E i , B, Θ i , but also by the number N of data samples.

It is of some interest to see how the choice of N influences the retrieval of ε ′ 1 , either via the explicit formula (172) or via minimization of the cost functional K (N ) in (173). To do this, we consider solely the case θ e -θ b = 2π, keeping in mind the reference solution obtained by minimization of the cost functional K. From (168) it ensues that

κ (N ) (α, β) = cos α cos β ; N = 1, 2 . ( 179
)
κ (N ) (α, β) = cos(α -β) ; N ≥ 3 . ( 180
) wherein α = π N -θ i , β = π N -Θ i . (181)
Consequently,

κ (1) = cos θ i cos Θ i , κ (2) = sin θ i sin Θ i , κ (N ≥3) = cos(θ i -Θ i ) = κ . ( 182
)
This result tells us that, in the case θ eθ b = 2π, the retrieval depends on N for small N (= 1, 2), but no longer depends on N for N ≥ 3, which suggests that the optimal number N of sensors is N = 3 separated (in terms of angle θ) by 2π/3, on the sensing circle of radius b ≥ a. Note also that owing to the result κ (N ≥3) = κ, it ensues that

K (N ≥3) = K , (183)
which means that the employment of the term 'optimal', is all the more justified that the N = 3 inversion gives rise to the reference retrieval (obtained by minimization of K).

Finding E ′′ of the cylinder by minimizing K

Eq.( 54) tells us that

ℜ(-iF ) = ℑ(F ) = 2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 , (184)
ℑ(-iF ) = -ℜ(F ) = -E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 , (185)
so that [START_REF] Young | An application of random projection to parameter estimation in partial differential equations[END_REF] becomes

2 E ′ 1 E ′′ 1 + E 0 E ′′ 1 ℜ(F -κf) + E ′ 2 1 -E ′′ 2 1 + 2E 0 E ′ 1 + E 2 0 ℑ(F -κf) = 0 , (186)
or, more explicitly:

F A 2 4E 0 E ′′ 1 E + E 1 + E 0 4 F -B 2 + A 2 E ′′ 2 1 + E + E - E 1 + E 0 2 -f g + F A 2 2E 0 (E 2 + -E ′′ 2
1 )

E 1 + E 0 4 F A 2 2E 0 E ′′ 1 E 1 + E 0 2 -f h = 0 , (187)
wherein

E ± := E ′ 1 ± E 0 , ε ± := ε ′ 1 ± ε 0 . (188)
After division by F A 2 2E 0 E 1 +E 0 6 = 0, (187) becomes

(2E ′′ 1 E + ) -[F B 2 + f g] E 1 + E 0 2 + F A 2 (E ′′ 2 1 + E + E -) + E 2 + -E ′ 2 1 F A 2 2E 0 E ′′ 1 -f h E 1 + E 0 2 = 0 , (189)
This relation can be cast into the form of the fourth-order (in terms of E ′′ ) (quartic) algebraic equation

O 4 E ′′ 4 1 + O 3 E ′′ 3 1 + O 2 E ′′ 2 1 + O 1 E ′′ 1 + O 0 = 0 , (190)
wherein

O 4 = f h , O 3 = 2 [F A 2 (E + -E 0 ) -(F B 2 + f g)E + ] , O 2 = 0 , O 1 = 2E 2 + {[F A 2 E --(F B 2 + f g)E + ] + F A 2 E 0 } , O 0 = -f hE 4 + . (191)
It ensues from these formulae that:

O 1 = E 2 + O 3 , O 0 = -O 4 E 4 + , (192)
so that the roots of the quartic equation can be found from those of the two quadratic equations

E ′′ 2 1 + E 2 + = 0 , (193)
O 4 E ′′ 2 1 -E 2 + + O 3 E ′′ 1 = 0 , (194)
The two roots of the first quadratic equation are:

E ′′ (1) 1 = -iE + , E ′′ (2) 1 = iE + , (195)
and the two roots of the second quadratic equation are: are not admissible solutions. Whether E ′′ (±) 1 are admissible or not can be decided empirically, or by other means, using this same criterium (a solution is admissible if it is positive real).

E ′′ (-) 1 = -O 3 - √ O 2 3 +4O 2 4 E 2 + 2O 4 , E ′′ (+) 1 = -O 3 + √ O 2 3 +4O 2 4 E 2 + 2O 4 . ( 196

Comments on the exact and approximate solutions for E ′′ 1

The result embodied in (223) calls for the following comments:

1-It shows that the solution to the inverse problem of the identification of the sole parameter E ′′ 1 exists, even in the presence of nuisance parameter uncertainties; 2-it shows that it is possible to obtain the mathematically-explicit and exact solution to the given inverse problem, even in the presence of nuisance parameter uncertainties; 3-it shows that this solution is unique for a given set of parameters;

ε ′ 1 , ε ′′ 1 , ε 0 , a, e i , θ i , b, E ′′ 1 , E 0 , A, E i , Θ i , B
, provided a physical constraint (i.e., that E ′′ 1 be positive real) is satisfied; 4-it shows that the accuracy of the retrieval of the imaginary part of the permittivity (E ′′ 1 ) is conditioned, albeit in a complex manner, by the uncertainty of the nuisance parameters

E ′ 1 , E 0 , A, E i , Θ i , B;
The result embodied in (222)

5-shows that the accuracy of the retrieval of E ′′ 1 is strongly-conditioned by ε ′′ 1 and E ′ 1 since the dependence of E ′′ 1 on ε ′′ 1 is linear and that on E ′ 1 is quadratic, 6-shows nevertheless that E ′′ 1 is usually small due to ε ′′ 1 having been assumed to be small, 7-shows that, due to the possibility of vanishing denominator in (222)), the retrieval of E ′′ 1 , like that of E ′ 1 , can be unstable with respect to certain nuisance parameters.

Finding the imaginary part of the permittivity (E ′′ ) of the cylinder by minimizing K (N )

The method of obtaining the imaginary part of the permittivity of the cylinder outlined in sect. 4.5 is somewhat unrealistic in that it supposes that the data is registered at a continuum of points on the sensing circle r = b. In reality, the data is registered at discrete locations on this circle and the number N of these locations is finite. Of course, N will have an influence on the accuracy of the retrieval and it is this influence that we shall now examine. From (45) we obtain

K (N ) (E ′′ 1 ) = 1 f 2 δ θ N n=1 cos 2 (α + (n -1)δ θ ) × δ θ N n=1 f 2 cos 2 (α + (n -1)δ θ ) -2ℜ(f * F) cos(α + (n -1)δ θ ) cos(β + (n -1)δ θ )+ F 2 cos 2 (β + (n -1)δ θ ) , (224)
wherein all quantities are as in sect. 4.4, and find, as in sect. 4.4,

K (N ) (E ′′ 1 ) = |f 2 σ (N ) (α, α) -2ℜ(f * F)σ (N ) (α, β) + F 2 σ (N ) (β, β) f 2 σ (N ) (α, α) . ( 225
)
For θ eθ b = 2π, the fact that lim

N →∞ σ (N ) (α, β) = π cos(α -β) = π cos(θ i -Θ i ) = πκ , (226)
gives rise to the expected result lim

N →∞ K (N ) (E ′′ 1 ) = K(E ′′ 1 ) , (227)
Recall that our goal was to obtain an estimation of E ′′ 1 by minimizing K (N )) (E ′′ 1 ). The procedure is the same as in sect. 4.5 and yields the exact solution

E ′′ (N ) 1 = -O (N ) 3 + O (N )2 3 + 4O (N )2 4 E 2 + 2O (N ) 4 . ( 228
)
and the approximate solution

E ′′ (N ) 1 ≈ E 2 + f (N ) h (1) F A 2 -F B 2 -f (N ) g (0) ε ′′ , (229) wherein:
O (N ) 4 = f (N ) h , (230)
O (N ) 3 = 2E 0 F A 2 -4E 0 (F B 2 + f (N ) g) , (231)
f (N ) = e i κ (N ) b , (232)
κ (N ) = σ (N ) (α, β) σ (N ) (β, β) . (233)
Eqs. ( 228) and ( 228) are the (mathematically) exact and approximate solutions to the inverse problem for discrete data in the interval [θ b + δ θ 2 , θ eδ θ 2 ]. These relations, as well as ( 230)-( 233), show that the accuracy of the retrieval of E ′′ 1 is conditioned, not only by the uncertainty of the nuisance parameters E ′ 1 , E 0 , A, E i , B, Θ i , but also by the number N of data samples.

As in sect. 4.4, we can show that, in the case θ eθ b = 2π, the retrieval depends on N for small N (= 1, 2), but no longer depends on N for N ≥ 3, which suggests that the optimal number N of sensors is N = 3 separated (in terms of angle θ) by 2π/3, on the sensing circle of radius b ≥ a. Moreover, the term 'optimal', is all the more justified that the N = 3 inversion gives rise to the reference retrieval (obtained by minimization of K).

Inversion by solving numerically the quartic equations

We saw in sects. 4.3, 4.5, 4.4, 4.6, that the next-to-final step of the inverse problem boiled down to the resolution of either the quartic equations

C (N ) 4 E ′ 4 1 + C (N ) 3 E ′ 3 1 + C (N ) 2 E ′ 2 1 + C (N ) 1 E ′ 1 + C (N ) 0 = 0 , (234)
or the quartic equation

O (N ) 4 E ′′ 4 1 + O (N ) 3 E ′′ 3 1 + O (N ) 2 E ′′ 2 1 + O (N ) 1 E ′′ 1 + O (N ) 0 = 0 , (235)
with N = ∞ or N ≥ 3.

Consider the general problem of finding the n (an integer) complex roots (z 1 , z 2 , ...z n ) of the n-th order polynomical equation

n j=1 a n z n = 0 . ( 236
)
By employing the Viète formulae for the product of the roots one is led to the formation of the companion matrix

          -a 1 a 0 -a 2
          . ( 237
)
so that finding the eigenvalues λ j = 1 z j ; j = 1, 2, ..., n of this matrix is equivalent to finding the solutions λ j = 1 z j ; j = 1, 2, ..., n to the equation

det           -a 1 a 0 -λ -a 2 a 0 . . . . -an a 0 1 -λ 0 . . 0 0 0 1 0 . . . . . . . . . . . . . . . . . . . . . . 1 0 . 0 0 . . . 1 -λ           = 0 , (238)
so that the determination of the roots of the polynomial equation reduces to the determination of the eigenvalues of the companion matrix. This can be done by any standard (such as the QR) technique [START_REF] Trefethen | Numerical Linear Algebra[END_REF].

manner. On the contrary, our inverse problem was shown to possess mathematical, explicit, exact solutions. Thus by confronting the two types (i.e., numerical and mathematical) of solutions enables us to test the quality of the numerical solutions.

The numerical minimization of the cost functional K (N ) can be carried out by a large variety of schemes; our choice was that of the Nelder-Mead Simplex algorithm [START_REF] Nelder | A Simplex method for function minimization[END_REF] implemented in MATLAB by the function fminsearch. This algorithm is a geometrical, rather than gradient-based, optimization manner of finding the parameter (or parameters if more than one parameters are to be retrieved) which corresponds to the minimum of a (cost) function. Few theoretical results have been proved explicitly concerning this algorithm, and then essentially in one and two dimensions [START_REF] Lagarias | Convergence properties of the Nelder-Mead Simplex method in low dimensions[END_REF]. Nevertheless, the Simplex algorithm has been frequently employed, particularly in parameter estimation problems [START_REF] Devlin | A simple and powerful method of parameter estimation using simplex optimization[END_REF], [START_REF] Johnson | Parameter estimation by least-squares methods[END_REF], [START_REF] Sebaa | Application of the Biot model to ultrasound in bone: inverse problem[END_REF], [START_REF] Buchanan | Recovery of the parameters of cancellous bone by inversion of effective velocities, and transmission and reflection coefficients[END_REF], [START_REF] Yeh | Characterization of mechanical and geometrical properties of a tube with axial and circumferential guided waves[END_REF], and even in problems involving noisy data [START_REF] Tomick | On convergence of the Nelder-Mead simplex algorithm for unconstrained stochastic optimization[END_REF], [START_REF] Scotti | Multiparameter identification of a lossy fluid-like object from its transient acoustic response[END_REF]. Like other optimization schemes, the Simplex algorithm is iterative by nature, with each iteration requiring a certain number of (cost) functional computations. The number of iterations and function computations increases with: a) the precision with which one wants to locate the minimum of the (cost) functional, b) the distance of the initial values of the parameters to their target values, and depends on the topological nature of the cost functional. In the present study, the dimension of the search space (of real variables) is one and the function to be minimized appears (by the previously-evoked theoretical considerations) to be convex (i.e., such as to admit only one minimum). In f minsearch, the maximum number of function evaluations and maximum number of iterations can be chosen, via the parameters MaxF unEvals and MaxIter (designated by MF E and MI in the following respectively, by the user. This appears to us to be a better strategy than choosing the precision, because aiming at a high precision may require a prohibitive amount of function evaluations and/or iterations. Each call to this function returns a value called exitf lag that describes the exit condition of f minsearch. exitf lag = 1 when f minsearch converged to a solution, exitf lag = 0 when MF E or MI was reached and no further computations are made during this call, exitf lag = -1 when the algorithm is terminated by the output function. Consequently, it is important to choose MF E and MI large enough for f minsearch to converge to a bona fide solution (corresponding to an authentic minimum of the cost functional as indicated by exitf lag = 1) for each call to f minseach.

The minimum of the cost functional (in the real parameter space explored by the Simplex scheme, anchored at the user-defined starting value of the to-be-retrieved parameter), is found for R = r. r can depend on the starting value, so that it is advisable to carry out the inversion in N s stages, each of which involves a different starting value. Thus, if R (n) 1 is the n-th starting value, then the N s -stage Simplex scheme generates N s candidate retrievals r(n)

1 ; n = 1, 2, ..., N s . The question is then how to select the most appropriate solution (this procedure is hereafter termed regularization) among the multitude of (N s -stage) retrievals. Let min K (N )(n) be the value of the minimum of the cost function found by the Simplex scheme at the n-th stage. The first step in the regularization procedure consists in rejecting solutions that are not admissible i.e., a solution that is negative, since it was assumed that ε ′ 1 > 0 (or ε ′′ 1 > 0). Very large positive retrievals are not rejected on the grounds that they are indicative of retrieval instability and therefore of some use. At this point, the values of n corresponding to admissible retrievals form the set N a ⊂ N s . The second step stems from the observation that the result of the multistage Simplex minimization scheme can lead to one or more r

(n) 1 that are not within the initial search interval [R b 1 , R e 1 ]
; a logical option is to reject such retrievals (otherwise, why impose initial search intervals other than to constrain the retrievals?), by assigning a large value of min K (N )(n) to them. Note that this action furnishes a means for telling us to enlarge

[R b 1 , R e 1 ]
. The last step in the regularization proceeds as follows: the final retrieved value of r 1 is designated by r1 and is chosen, amongst the remaining candidate retrievals {r (n) 1

; n ∈ N a }, to be the one corresponding to the minimum of the set {min K (N )(n) ; n ∈ N a }. This regularization procedure is incorporated into our MATLAB code to numerically solve the inverse problem.

We found that the Simplex scheme gives rise to exactly the same results as the companion matrix-eigenvalue scheme outlined in sect. 5. Moreover, it gives rise to exactly the same results as the mathematical solutions presented in sects. 4.3, 4.5, 4.4, 4.6 in a great variety of test cases. Thus, the numerical minimization of a cost function via the Simplex scheme is a robust, sound method for solving an inverse parameter-retrieval problem such as ours.

7 Results concerning the retrieval error as a function of nuisance parameter uncertainties 7.1 Overview of the evaluation of retrieval error for variable nuisance parameter uncertainty

All the following numerical results pertain to the choice (the true parameters of which are):

ε ′ 1 = 2, ε ′′ 1 = 0.1, ε 0 = 1, a = 0.1, e i = 1, θ i = 2 • , b = 0.2
, MF E = MI = 1000 (when the Simplex scheme is employed). Moreover, θ b = 0 • , θ e = 360 • , and N = 3 unless indicated otherwise.

Six general cases of nuisance parameter uncertainty are possible:

1) one nuisance parameter is uncertain, the five others are equal to their true values; 2) two nuisance parameters are uncertain, the four others are equal to their true values; 3) three nuisance parameters are uncertain, the three others are equal to their true values; 4) four nuisance parameters are uncertain, the two others are equal to their true value; 5) five nuisance parameters are uncertain, the one other is equal to their true value; 6) all six nuisance parameters are uncertain.

Due to the large number of possibilities, we consider representative samples of cases 1), 2) and 5) only. The offered graphs are composed of three panels, the left-hand one of which relates Ẽ′ 1 or Ẽ′′ 1 to a variable nuisance parameter G, the central one of which relates

ε ε ′ 1 = Ẽ′ 1 -ε ′ 1 ε ′ 1 or ε ε ′′ 1 = Ẽ′′ 1 -ε ′′ 1 ε ′′ 1
to δ g = G-g g , and the right-hand one of which relates K (3) (for the retrieved parameter) to δ g . Moreover, the circles refer to the retrievals obtained by the

Retrieval of E ′

1 : variable uncertainty of one nuisance parameter, all other nuisance parameters are certain This figure shows that |ε ε ′ 1 | is very small, which means that the retrieval of ε ′ 1 is insensitive to uncertainty of the nuisance parameter E ′′ 1 .

Fig. 6 concerns the effect of variable uncertainty of the nuisance parameter G = G 4 = E 0 .

0.9 The left-hand panel of Fig. 7, relative to variable uncertainty of the nuisance parameter

G = G 2 = A, is the numerical equivalent of the portion A ∈ [A + , ∞[, E ′ 1 ∈ [E 0 , ∞[ in fig.
3 translating the analytic properties of the exact mathematical solution of the inverse problem. This figure shows that, at A = .09 and .11, the retrieval error is of the same order as the nuisance parameter uncertainty since

|ε ε ′ 1 | |δ a |.
The left-hand panel of Fig. 8, relative to variable uncertainty of the nuisance parameter 4 translating the analytic properties of the exact mathematical solution of the inverse problem. 0.9 This figure shows that, at E i = .9 and 1.1, the retrieval error far exceeds the nuisance parameter uncertainty since |ε ε ′ 1 | >> |δ e i | = .1, this being (in the neighborhood of δ e i = .1) a manifestation of nuisance parameter uncertainty-induced retrieval instability. This figure shows that the retrieval error is quite small with respect to Θ i uncertainty since

G = G 3 = E i , is the numerical equivalent of the portion E i ∈ [E i 0 , E i + [, E ′ 1 ∈ [0, ∞[ in fig.
ε ε ′ 1 ≤ 1.26 × 10 -3 over the whole Θ i interval [1 • , 3 • ].
The left-hand panel of Fig. 10, relative to variable uncertainty of the nuisance parameter

G = G 5 = B, is the numerical equivalent of the portion B ∈ [B 0 , B + [, E ′ 1 ∈ [E ′ 0 , ∞[ in fig.
2 translating the analytic properties of the exact mathematical solution of the inverse problem.

Retrieval of E ′′

1 : variable uncertainty of one nuisance parameter, all other nuisance parameters are certain This figure shows that, at E i = .9, the retrieval error is of the same order as the nuisance parameter uncertainty since |ε ε ′′ 1 | |δ e i |, whereas, at E i = 1.1, the retrieval error far exceeds the nuisance parameter uncertainty since |ε ε ′′ 1 | >> |δ e i | = .1, this being a manifestation of severe instability in the neighborhood of δ e i = .1. This figure shows that the retrieval error is quite small with respect to Θ i uncertainty since

G = G 1 = E ′ 1 . 1.8 2 2.2 2.4 2
ε ε ′′ 1 ≤ 1.53 × 10 -3 over the whole Θ i interval [1 • , 3 • ].

Retrieval of E ′

1 : Variable uncertainty of one nuisance parameter, fixed uncertainty of another nuisance parameter, all other nuisance parameters are certain Figs. 17, 18 concern the effect of variable uncertainty of the nuisance parameter G = G 1 = E 0 when E i is -10% and +10% uncertain respectively. These two figures should be compared to fig. 6 relative to the case when E i is certain. Figs. [START_REF] Hasar | Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs[END_REF], 20 concern the effect of variable uncertainty of the nuisance parameter G = G 4 = Θ i when B is -10% and +10% uncertain respectively. These two figures should be compared to fig. 9 relative to the case when B is certain. 7.5 Retrieval of E ′′ 1 : Variable uncertainty of one nuisance parameter, fixed uncertainty of another nuisance parameter, all other nuisance parameters are certain Figs. 29, 30 concern the effect of variable uncertainty of the nuisance parameter G = G 1 = E 0 when E i is -10% and +10% uncertain respectively. These two figures should be compared to fig. 12 relative to the case when E i is certain. Figs. 31, 32 concern the effect of variable uncertainty of the nuisance parameter G = G 4 = Θ i when B is -10% and +10% uncertain respectively. These two figures should be compared to fig. 15 relative to the case when B is certain. 7.8 Table of the influence of uncertainty regarding five nuisance parameters, the sixth nuisance parameter being certain, on the accuracy of the retrieval of ε ′

In table 3 we give the numerical values of Ẽ′ 1 and ε ε ′ 1 (obtained by numerical minimization, via the Simplex scheme, of K (3) ) in all of the 32 possible cases in which five nuisance parameter are δ = ±10% uncertain at a time. This table reveals that in sixteen of these cases, |ε ε ′ 1 | > 5|δ|, and in eight of these cases, |ε ε ′ 1 | >> |δ| (due to combined nuisance parameter uncertainty-induced instability).

Table of the influence of uncertainty regarding five nuisance

parameters, the sixth nuisance parameter being certain, on the accuracy of the retrieval of ε ′′ 1

In table 4 we give the numerical values of Ẽ′′ 1 and ε ε ′′ 1 (obtained by numerical minimization, via the Simplex scheme, of K (3) ) in all of the 32 possible cases in which five nuisance parameters are δ = ±10% uncertain at a time. This table reveals that in 22 of these cases, |ε ε ′ 1 | > 5|δ|, and in 12 of these cases, |ε ε ′ 1 | >> |δ| (due to combined nuisance parameter uncertainty-induced instability). The comparison of table 4 with table 3 shows that the pattern of retrieval error of E ′′ 1 is substantially the same as that of E ′ 1 .

  1 . . . . . . . . . . . . . . . 4.5.2 Comments on the exact and approximate solutions for E ′′ 1 . . . . . . . 4.6 Finding the imaginary part of the permittivity (E ′′ ) of the cylinder by minimizing K (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Inversion by solving numerically the quartic equations 6 Inversion by a numerical scheme for finding the minimum of the cost functional 6.1 Basic retrieval scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Simplex minimization of the cost functional . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Problem configuration in the xy plane.

1 .

 1 second relation is a linear equation (involving the previously-obtained u The third relation is a linear equation (involving the previously-obtained u

Figure 2 :

 2 Figure 2: Graphical representation of the analytical properties of E ′ 1 (B).

Figure 3 : 25 4. 3 . 6

 32536 Figure 3: Graphical representation of the analytical properties of E ′ 1 (A).

Fig. 5 Figure 5 :

 55 Fig. 5 concerns the effect of variable uncertainty of the nuisance parameter G = G 1 = E ′′ 1 .

Figure 6 :

 6 Figure 6: The abscissas in the left-hand panel represent E 0 . The abscissas in the central and right-hand panels represent δ ε 0 . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figure 7 :

 7 Figure 7: The abscissas in the left-hand panel represent A. The abscissas in the central and right-hand panels represent δ a . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figure 8 :

 8 Figure 8: The abscissas in the left-hand panel represent E i . The abscissas in the central and right-hand panels represent δ e i . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Fig. 9 Figure 9 :

 99 Fig. 9 concerns the effect of variable uncertainty of the nuisance parameter G = G 1 = Θ i .

Figure 10 :

 10 Figure 10: The abscissas in the left-hand panel represent B. The abscissas in the central and right-hand panels represent δ b . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Fig. 11

 11 Fig. 11 concerns the effect of variable uncertainty of the nuisance parameter G = G 1 = E ′ 1 .

Figure 11 : 47 Fig. 12 Figure 12 :

 11471212 Figure 11: The abscissas in the left-hand panel represent E ′ 1 . The abscissas in the central and right-hand panels represent δ ε ′ 1 . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Fig. 13 concernsFigure 13 :

 1313 Fig. 13 concerns the effect of variable uncertainty of the nuisance parameter G = G 2 = A.

Fig. 14 Figure 14 :

 1414 Fig. 14 concerns the effect of variable uncertainty of the nuisance parameter G = G 4 = E i .

Fig. 15 Figure 15 :

 1515 Fig. 15 concerns the effect of variable uncertainty of the nuisance parameter G = G 1 = Θ i .

Fig. 16 concernsFigure 16 :

 1616 Fig. 16 concerns the effect of variable uncertainty of the nuisance parameter G = G 5 = B.

Figure 17 :Figure 18 :

 1718 Figure17: E i = .9 (δ e i = -.1). The abscissas in the left-hand panel represent E 0 . The abscissas in the central and right-hand panels represent δ ε 0 . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K(3) in the right-hand panel. circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Figure 19 :

 19 Figure 19: B = .18 (δ b = -.1). The abscissas in the left-hand panel represent Θ i . The abscissas in the central and right-hand panels represent δ θ i . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Figure 20 :Figs. 21 ,Figure 21 :

 202121 Figure 20: B = .22 (δ b = .1). The abscissas in the left-hand panel represent Θ i . The abscissas in the central and right-hand panels represent δ θ i . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figure 22 :

 22 Figure 22: E 0 = 1.1 (δ ε 0 = .1). The abscissas in the left-hand panel represent A. The abscissas in the central and right-hand panels represent δ a . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figs. 23 ,Figure 23 :Figure 24 :

 232324 Figs.[START_REF] Isakov | Inverse obstacle problems[END_REF], 24 concern the effect of variable uncertainty of the nuisance parameter G = G 3 = E i when A is ±10% uncertain. These two figures should be compared to fig.8relative to the case when A is certain.

Figs. 25 ,Figure 25 :Figure 26 :

 252526 Figs.[START_REF] Lagarias | Convergence properties of the Nelder-Mead Simplex method in low dimensions[END_REF], 26 concern the effect of variable uncertainty of the nuisance parameter G = G 5 = B when E i is -10% and +10% uncertain respectively. These two figures should be compared to fig.10relative to the case when E i is certain.

Figs. 27 Figure 27 :

 2727 Figs. 27 and 28 concern the effect of variable uncertainty of the nuisance parameterG = G 5 = E ′′1 when B is -10% and +10% uncertain respectively. These two figures should be compared to fig.5relative to the case when B is certain.

Figure 28 :

 28 Figure 28: B = .22 (δ b = .1). The abscissas in the left-hand panel represent E ′′ 1 . The abscissas in the central and right-hand panels represent δ ε ′′ 1 . The ordinates represent the retrieved E ′ 1 in the left-hand panel, ε ε ′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figure 29 :Figure 30 :

 2930 Figure29: E i = .9 (δ e i = -.1). The abscissas in the left-hand panel represent E 0 . The abscissas in the central and right-hand panels represent δ ε 0 . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K(3) in the right-hand panel. The circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Figure 31 :

 31 Figure 31: B = .18 (δ b = -.1). The abscissas in the left-hand panel represent Θ i . The abscissas in the central and right-hand panels represent δ θ i . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Figure 32 :

 32 Figure 32: B = .22 (δ b = .1). The abscissas in the left-hand panel represent Θ i . The abscissas in the central and right-hand panels represent δ θ i . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figs. 33 ,Figure 33 :

 3333 Figs.[START_REF] Nesvadba | Database of physical properties of agro-food materials[END_REF], 34 concern the effect of variable uncertainty of the nuisance parameter G = G 2 = A when E 0 is ±10% uncertain. These two figures should be compared to fig.13relative to the case when E 0 is certain.

Figure 34 :

 34 Figure 34: E 0 = 1.1 (δ ε 0 = .1). The abscissas in the left-hand panel represent A. The abscissas in the central and right-hand panels represent δ a . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

Figs. 35 ,Figure 35 :Figure 36 :Figure 37 :

 35353637 Figs.[START_REF] Preis | Time-domain analysis of quasistatic electric fields in media with frequency-dependent permittivity[END_REF], 36 concern the effect of variable uncertainty of the nuisance parameter G = G 3 = E i when A is ±10% uncertain. These two figures should be compared to fig.14relative to the case when A is certain.

Figs. 39 ,

 39 Figs.[START_REF] Sebaa | Application of the Biot model to ultrasound in bone: inverse problem[END_REF], 40 concern the effect of variable uncertainty of the nuisance parameter G = G 5 = E ′ 1 when B is -10% and +10% uncertain respectively. These two figures should be compared to fig.11relative to the case when B is certain.

Figure 39 :

 39 Figure 39: B = .18 (δ b = -.1). The abscissas in the left-hand panel represent E ′ 1 . The abscissas in the central and right-hand panels represent δ ε ′ 1 . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles refer to the retrieval obtained by resolution of the quartic equation via the mathlab function roots.

Figure 40 :

 40 Figure 40: B = .22 (δ b = .1). The abscissas in the left-hand panel represent E ′ 1 . The abscissas in the central and right-hand panels represent δ ε ′ 1 . The ordinates represent the retrieved E ′′ 1 in the left-hand panel, ε ε ′′ 1 in the central panel, and K (3) in the right-hand panel. The circles have the same signification as in the previous figure.

7

  Results concerning the retrieval error as a function of nuisance parameter uncertainties 7.1 Overview of the evaluation of retrieval error for variable nuisance parameter uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Retrieval of E ′ 1 : variable uncertainty of one nuisance parameter, all other nuisance parameters are certain . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Retrieval of E ′′ 1 : variable uncertainty of one nuisance parameter, all other nuisance parameters are certain . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Retrieval of E ′ 1 : Variable uncertainty of one nuisance parameter, fixed uncertainty of another nuisance parameter, all other nuisance parameters are certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Retrieval of E ′′ 1 : Variable uncertainty of one nuisance parameter, fixed uncertainty of another nuisance parameter, all other nuisance parameters are certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Table of the influence of uncertainty regarding one nuisance parameter, all other nuisance parameters being certain, on the accuracy of the retrieval of ε ′ 1 7.7 Table of the influence of uncertainty regarding one nuisance parameter, all

in the left-hand panel, ε ε ′′ 1 in the central panel, and K[START_REF] Abelès | Methods for determining optical parameters of thin films[END_REF] in the right-hand panel. The circles have the same signification as in the previous figure.This figure shows that, at A = .09 and .11, the retrieval error is of the same order as the nuisance parameter uncertainty since |ε ε ′′ 1 | |δ a |, but with the manifestation of an instability in the neighborhood of δ a = -0.1.

Perturbation solutions of E ′′ (±) 1

for small ε ′′ 1 Let us return to (194) which can be written as

wherein:

and

We expand U j and u j in series of powers of the supposedly-small quantity

to find, as in sect. 4.3.1,

0 + U

1 + U

(1)

= 0 ,

= 0 . (203)

We now address the problem of the U (k) j . To do this, we employ the expressions for O 4 and O 3 in (191):

so that:

Under the (small-δ) hypothesis (implicit in the perturbation method)

we found in sect. 4.3.1:

Consequently:

whence

The solution of (201) is

The next step in the perturbation procedure is to obtain u

(1)

1 . This is done via (202) from which we obtain

or, on account of (218), u

The next step in the perturbation procedure is to obtain u

1 from u (0) 1 and u

1 . This is done via (203). But, instead of writing the explicit expression for u

(2) 1 , we recall that here u 1 stands for E ′′ 1 , and to second order in ε ′′

or, on account of (218) and neglecting terms of order ε ′′ 2 (due to the assumed smallness of ε ′′ 1 , the approximate solution for E ′′ 1 is

with the understanding that, on account of what the same type of perturbation analysis as in sect. 4.3.1 can show, the (only-admissible) exact solution for ε ′′ 1 is

In fact, we employed the MATLAB function roots, which is based on the companion matrix and eigenvalue evaluation algorithm, to find the roots of our two quartic equations. It turned out that these roots are identical to the mathematically-exact roots we found previously in sects. 4.3 and 4.5 (for N = ∞), the same applying to the case N ≥ 3. As in these two sections, we choose the "right" root (one E ′ 1 and one for E ′′ 1 ) by imposing the physical constraints:

6 Inversion by a numerical scheme for finding the minimum of the cost functional 6.1 Basic retrieval scheme

The direct problem model, by which the data is generated, involves the (true) parameter set p = {p 1 , p 2 , ...}, a subset r of which is to be retrieved in the inverse problem context. At present, r reduces to the single parameter

The remaining parameters of p form the set of nuisance parameters g.

Although we saw previously that an exact, explicit solution to our inverse problem could be found, at present we want to solve this problem in a numerical manner (as is usually the case in inverse problems) and see to what extent the so-obtained solution coincides with the exact solution.

The numerical inversion is carried out by means of a (retrieval) model involving the parameter set P whose components are qualitatively the same as the corresponding components of p. During the inversion, the real (or imaginary) part of the permittivity takes on (variable

), while the remaining parameters of the retrieval model, forming the set G, are fixed, but (as explained earlier) not necessarily equal to the corresponding parameters of the set g. Now, the basic inversion scheme consists in searching, in iterative, numerical manner, for the real R = r that minimizes the discrepancy (the measure of which is a cost functional K (N ) ) between trial electric potential fields (resulting from the trial parameter R 1 and the nuisance parameter set G) and the true electric potential field (resulting from the true parameter r 1 and the nuisance parameter set g).

Note that if, as will be assumed in the examples presented hereafter, there exists some uncertainty in the nuisance parameters, G will not be identical to g. In fact, this possible (model) discordance will affect the accuracy of the retrieval.

Simplex minimization of the cost functional

Most inverse problems are solved by numerical minimization of a cost functional for the simple reason that these inverse problems cannot be solved in a mathematical, explicit, exact

Table of the influence of uncertainty regarding one nuisance

parameter, all other nuisance parameters being certain, on the accuracy of the retrieval of ε ′ 1

In table 1 we give the numerical values of Ẽ′ 1 and ε ε ′ 1 (obtained by numerical minimization, via the Simplex scheme, of K (3) ) in all of the 12 possible cases in which only one nuisance parameter is δ = ±10% uncertain at a time.

Table of the influence of uncertainty regarding one nuisance

parameter, all other nuisance parameters being certain, on the accuracy of the retrieval of ε ′′ 1

In table 2 we give the numerical values of Ẽ′′ 1 and ε ε ′′ 1 (obtained by numerical minimization, via the Simplex scheme, of K (3) ) in all of the 12 possible cases in which only one nuisance parameter is δ = ±10% uncertain at a time. 1 shows that the pattern of retrieval error of E ′′ 1 is substantially the same as that of E ′ 1 , as predicted in theoretical manner previously.

Conclusion

The inverse problem we set out to solve was the retrieval of one of the seven parameters (either the real (ε ′ 1 ) or imaginary (ε ′′ 1 ) part of the permittivity of a cylinder) that enter into a 2D quasistatic electricity configuration. The exact solution of the forward problem was obtained by separation of variables and employed to furnish the data serving as the input to the inverse problem. The retrieval model also relied on the separation of variables solution, but with one of the seven true parameters thereof replaced by a variable ε ′ 1 or ε ′ 1 and the remaining six (called nuisance) parameters by more or less well-known values.

We solved the inverse problem in four manners: 1) by mathematically searching for the minimum of the cost functional K relative to continuous data on a measurement circle, 2) by mathematically searching for the minimum of the cost functional K (N ) relative to discrete data registered at N sensors on the measurement circle, 3) by numerically searching (via the Simplex algorithm) for the minimum of K (N ) , 4) by numerically solving the quartic eqution precluding the mathermatical solution. The first two manners led to exact, mathematicallyexplicit solutions which lend themselves to a complete mathematical analysis of the way in which the retrieval error varies as a function of the nuisance parameter uncertainties. The second manner led to the result that N = 3 sensors, equispaced over the angular range [0, 2π[, are necessary and sufficient to provide the required data for the inversion. The second, third and fourth manners led to identical numerical results and the latter were identical to those of the first manner for N ≥ 3. It was shown, in addition to the existence and uniqueness of the inverse problem solution, that the latter is unstable with respect to uncertainties concerning the nuisance parameters A, E i and B, acting individually or in combination. These instabilities manifest themselves by extremely-large retrieval error in the neighborhoods of certain values of these nuisance parameters. It was also shown that, even quite far from these neighborhoods, the retrieval error |ε ε 1 | can be much larger than a generic nuisance parameter uncertainty |δ g |. Finally, it was found numerically, in agreement with the theory, that the pattern of retrieval error of ε ′′ 1 is much the same as that of ε ′ 1 , notably as concerns the instability issue; moreover the relative retrieval error for M uncertain parameters turned out to be roughly proportional to M (outside of the instability regions, and for both the real and imaginary parts of the permittivity).

This investigation underlines the necessity, in parameter-retrieval inverse problems, to take account of nuisance parameter uncertainty in order to evaluate the accuracy of the retrieved parameter(s). In our study, only one parameter was retrieved at a time, while from one to five parameters were uncertain. It may be possible to reduce the global retrieval error by retrieving two or more parameters at a time while considering the remaining parameters to be uncertain, but this issue is out of the scope of the present study.