Abstract
Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary conditions are exact solutions of the electrical impedance equation, performing a brief comparison with the finite element method. Finally, we discuss the possible contributions of these results to the field of the electrical impedance tomography.
1. Introduction
The study of the forward Dirichlet boundary value problem for the electrical impedance equation in the plane, is fundamental for understanding its inverse problem, commonly known as electrical impedance tomography, first correctly posed in mathematical form by Calderon [1] in 1980. It is remarkable that, for more than twenty years after the problem was stated, the mathematical complexity of (1) could provoke that many experts considered impossible to obtain its general solution in analytic form [2], even for the simplest cases of , excluding the constant case.
This perception changed when, independently, Kravchenko in 2005 [3] and Astala and Päivärinta [4] in 2006 noticed that the two-dimensional case of (1) was completely equivalent to a special class of Vekua equation [5].
Many other important results were obtained soon after. Indeed, Kravchenko et al. published in 2007 what can be considered the first general solution of (1) in analytic form [6], when possesses a certain form, employing Taylor series in formal powers [7]. Moreover, when the conductivity is separable variables, the real parts of the formal powers conform a complete set for approaching solutions of the forward Dirichlet boundary value problem of (1) in the plane [8].
The main objective of this work is to start a discussion about the application of the Pseudoanalytic function theory when is not, in general, a separable-variables function, which would allow the study of conductivity cases more interesting in physics and engineering. As a matter of fact, even it is not clear yet how to extend the proof provided in [8], about the completeness of the set of formal powers for the cases when is piecewise separable variables, the numerical calculations will show that a variation of the technique, originally posed for purely mathematical problems, could well serve for analyzing more general cases, providing quite acceptable results with compared when classical methods.
In other words, this work states that if the values of the electrical conductivity are known at every point within a bounded domain in the plane, it will be always possible to introduce a piecewise separable-variables function, such that we can use it to obtain a set of base functions for approaching solutions of the forward Dirichlet boundary value problem of (1), employing pseudoanalytic functions. This would be true for a certain class of bounded domains, but this class will be wide enough to include many interesting cases for the applied sciences. Nonetheless, on behalf of briefness, we will only study the unit circle. The reader will appreciate that most of the results will be valid for more domains.
In order to prove the veracity of the last assessments, we will consider a set of conductivity examples, separable variables, and nonseparable variables, for which we can obtain exact solutions, to be imposed as boundary conditions. The effectiveness of the approaches will be estimated by comparing them with the boundary conditions, employing a standard Lebesgue measure.
Finally, we will discuss how this numerical technique could contribute to the study of the inverse Dirichlet boundary value problem of (1) in the plane, also known as the Electrical Impedance Tomography problem.
2. Preliminaries
According to the Pseudoanalytic function theory posed by Bers [7], let the pair of complex-valued functions and fulfill the condition: where represents the complex conjugation of : and denotes the standard imaginary unit . This condition implies that the functions and are linearly independent, therefore any complex-valued function can be expressed by their linear combination: where and are purely real-valued functions. On the light of this idea, L. Bers introduced the concept of the -derivative of in the form This derivative will exist if and only if the following equality is true: Hereafter, we will use the notations and . These pairs of partial differential operators are usually introduced with the factor , but in this work it will be somehow more convenient to work without it.
Introducing the notations: the derivative presented in (4), that we will refer to as the -derivative of , can be rewritten as whereas condition (5) will become
A pair of complex functions satisfying condition (2) will be called a generating pair, and the functions introduced in (6) will be referred to as the characteristic coefficients of the generating pair . As a matter of fact, the expression (8) is known as the Vekua equation [5], and every function , solution of (8), will be called -pseudoanalytic.
The following statements were originally posed in [7]. We present them with certain modifications, in order to better analyze the special class of Vekua equation corresponding to the electrical impedance equation (1) in the plane.
Theorem 1. The functions and , constituting the generating pair of the form (2), are -pseudoanalytic, and their -derivatives vanish identically:
Theorem 2 (see [7, 9]). Let be a nonvanishing function, defined within some domain , and let It is easy to verify that and conform a generating pair (2), whose characteristic coefficients, according to (6), are Therefore, for this special class of generating pairs, the corresponding Vekua equations will have the form
Definition 3. Let and be two generating pairs of the form (10), and let their characteristic coefficients fulfill the condition Thus the generating pair will be called a successor pair of , as well will be called a predecessor of .
Definition 4. Let the elements of the set be all generating pairs, and let every be a successor of . Hence, the set (14) will be called a generating sequence. If , we will say that is embedded into (14). Moreover, if there exists a number such that , we will say that the generating sequence (14) is periodic, with period .
L. Bers also introduced the concept of the -integral of a complex function . The reader can find the conditions for warranting its existence and a detailed description of its properties in [7, 9]. Since the functions employed in this work are, by definition, -integrable, we will only present a certain set of those properties.
Definition 5. Let be a generating pair with the form (10). Its adjoin pair is defined as
Definition 6. The -integral of a complex function (when it exists) is defined according to the expression where is a rectifiable curve going from up to , in the complex plain. In particular, the -integral of reaches But according to Theorem 1, the -derivatives of and vanish identically, hence (17) can be considered the -antiderivative of .
2.1. Formal Powers
Definition 7. The formal power belonging to the generating pair , with formal exponent , complex coefficient , center at , and depending upon the complex variable , is defined by the expression where and are constants that fulfill the condition The formal powers with higher formal exponents are defined according to the recursive formulas Notice the integral operators at the right hand side of the last equality are all -antiderivatives.
Remark 8. The formal powers possess the following properties: (1) when . (2)Every is -pseudoanalytic. (3)If , where and are real constants, we will have
Theorem 9. Every complex-valued function , solution of the Vekua equation (8), can be expanded in terms of the so-called Taylor series in formal powers: where the absence of the subindex “” indicates that all formal powers belong to the same generating pair.
Remark 10. Since every complex-valued function , solution of (8), can be expressed in the form (22), it is possible to assert that (22) is an analytic representation of the general solution for the Vekua equation (8).
2.2. The Electrical Impedance Equation in the Plane
As it has been previously posed in several works (see, e.g., [3, 9, 10]), when the conductivity function can be expressed in terms of a separable-variables function by introducing the notations the two-dimensional electrical impedance equation (1) can be rewritten precisely as a Vekua equation of the form (12). Moreover, its corresponding generating pair is embedded into a periodic generating sequence, with period , such that when is an even number, and when is odd.
Therefore, based upon the statements posed in Definition 7, possessing a generating sequence will allow us to approach a set of formal powers: within a bounded domain , and, subsequently by virtue of Remark 8, we will be able to approach any formal power , at any point .
Because the present work intends to contribute to the construction of a novel theory for the electrical impedance tomography problem, we will focus our attention into a classic domain , the unit disk with center at .
2.3. A Complete Orthonormal System
In [8] Campos et al. posed a very important property of the formal powers.
Theorem 11 (see [8]). The set of real parts of the formal powers, with coefficients and , valued at the boundary of a bounded domain constitutes a complete system for approaching solutions of the forward Dirichlet boundary value problem for the electrical impedance equation (1) in the plane.
That is, any boundary condition can be approached asymptotically by the linear combination of the elements belonging to (29): where the coefficients and are all real constants.
Since the elements of the set (29) are, by definition, linearly independent [7], it is possible to perform a standard Gram-Schmidt orthonormalization process in order to obtain the set of functions defined on the boundary , such that when imposing a boundary condition , we will have
Because the set is orthonormal, the calculation of the coefficients can be performed by several classical methods. Particularly, we will employ the scalar product
2.4. A Basic Numerical Approach
Remembering that coincides with the perimeter of the unit circle, we can employ the numerical methods detailed in [11] for obtaining a system of formal powers (notice that, according to (25), ), defined at the boundary :
More precisely, let us consider the formal powers . Taking into account that the integral expressions introduced in (17) are path independent [7], we can choose the rectifiable curve to be a straight line segment, going from until some point of the unit circle. Thus, we can allocate points equidistantly distributed on such line, obtaining the set of complex numbers where is some angle associated to the radius. Given this set of points, and considering that the generating sequence corresponding to the pair (25) is periodic with period , we can numerically approach the formal powers employing, for example, a variation of the trapezoidal integration method according to the expressions where , and
Besides, we can introduce a set of angles , to be employed in (35), according to the formula for the iterative expressions (36) that can be performed at each angle . When the full procedure is complete, we will possess a set of formal powers , approached for radii, and with points per radius. Thus, by collecting the real parts of valued at the points , which are precisely those located at the boundary , we will obtain a numerical approximation of the set An identical procedure can be performed for obtaining the elements of Since the numerical approaching of the set (34) is complete, a standard Gram-Schmidt method will reach the orthonormal system (31) for approaching the boundary condition .
The effectiveness of this numerical approach has been analyzed in several works (see [8, 10]). Here we will only analyze two particular examples of separable-variables conductivity functions , before studying those that are not separable variables.
3. The Cases When the Conductivity Possesses a Separable-Variables Form
3.1. Example When Is an Exponential Function
Proposition 12. Let . Then the function will be a particular solution of (1).
Figure 1 plots an illustration of the exponential conductivity.
The numerical procedure described in Section 2.4 will be employed for approaching the forward Dirichlet boundary value problem when possesses the form of the proposition above, imposing as the boundary condition the exact solution
The error will be defined according to the Lebesgue measure where the addition within the integral expression corresponds to the approached solution (32).
Table 1 contains the relation between the error and the number of formal powers . The parameters and have been fixed at the value , since for this particular case, as displayed in the table, they do not significantly influence the diminution of the error when increasing.
3.2. The Case When Has a Lorentzian Form
Proposition 13. Let Then the function will be a particular solution of (1).
An illustration of the conductivity function (43) is displayed in Figure 2.
Once more, the exact solution (44) will be imposed as the boundary condition. The numerical results are shown in Table 2. This, indeed, is a more interesting case, since the number of formal powers strongly influences the accuracy of the approach. The increment of the number of points per radius and the number of radii , as in the previous case, do not significantly increase the accuracy.
3.3. Construction of a Piecewise Separable-Variables Conductivity Function
Consider a bounded domain (in this case the unit circle) and divide it into a finite number of subsections , taking care that the point to be considered the center of the formal powers (see Definition 7) does not reside onto the boundary of two subsections. For simplicity, let us make the division by employing a finite set of parallel lines to the -axis, equidistant to one another, and let us locate .
Supposing that the values of the electrical conductivity are defined for every point within the domain , let us locate a straight line crossing every subsection, that does not intersect the bounding parallel lines of its corresponding subsection. Indeed, such lines can be simply parallel to the bounding ones.
The following step is to collect finite sets of conductivity values, corresponding to a number of points located at the crossing lines. For every line, the quantity of collected values must be big enough to warrant that an interpolating process, in this case performed with straight lines, will adequately approach all the remaining conductivity values defined over the line points.
Since we have already assumed that every crossing line will be parallel to the subsection-bounding lines, and in consequence to the -axis, all collected points corresponding to the same crossing line will possess the same -coordinate. Let us propose that the conductivity within every subsection can be represented according to the expression where denotes the -coordinate that is common to all points along the crossing line, is an interpolating function that approaches the collected conductivity values, and is a real positive constant such that , for all .
Applying this idea in every subsection, the conductivity within the bounded domain can be approached by means of the piecewise-defined function: Here represents the first -coordinate found within the domain when broaching the -axis from up to , whereas represents the last one. The pairs of -axis parallel lines , where , are given by the common -coordinates belonging to every pair of lines delimiting the subsections. Notice the piecewise function (46) is separable variables.
Thus, according to Section 2.2, it immediately follows that whereas For the generating pair we will have
We will use these piecewise-defined functions to perform the numerical procedure described in Section 2.4. The first case to analyze will be the exponential conductivity studied in Section 3.1, imposing as the boundary condition the exact solution presented in Proposition 12.
The second case will be the Lorentzian conductivity treated in Section 3.2, whose boundary condition will be the exact solution shown in Proposition 13. The results of these cases are exposed in Tables 3 and 4. For both, we can observe that the accuracy is strongly related with the number of sections and the number of collected values per section , as well as with the number of formal powers. Once more, the increment of the number of points per radius and the number of radii do not significantly improve the accuracy.
Beside, since the tables presented in the previous subsection have shown that the accuracy of the method is not considerably improved when neither the number of points per radius nor the number of radii increase, we will fix both values hereafter.
Tables 3 and 4 show that the piecewise separable-variables conductivity function (46) can be positively employed for numerically approaching solutions of this boundary value problem, because even the magnitudes of the errors are considerable bigger than those obtained when employing the original , their magnitudes are still acceptable when compared with other classical numerical methods.
More precisely, employing a standard finite element method technique for solving forward Dirichlet boundary value problems of elliptic partial differential equations in the plane, when considering the exponential conductivity and imposing the boundary condition , the error was , utilizing nodes in the mesh, corresponding to triangular elements. For the Lorentzian conductivity , with the boundary condition , employing identical number of nodes in the mesh, the resulting error was .
A more detailed description of the behavior of the method is provided in Tables 5, 6, and 7.
Notice that the comparison with the finite element method is given as a basic reference. Indeed, the number of nodes in the mesh was taken at , because it is the closest value to the number of points located in obtained when we multiply .
4. Analysis of Conductivity Functions That Are Not Separable Variables
This section is fully dedicated to study conductivity functions that do not possess a separable-variables form, but for which we are able to obtain exact solutions of (1), in order to impose them as boundary conditions.
4.1. The Nonseparable-Variables Exponential Case
Proposition 14. Let the conductivity function Then, a particular solution of (1) will be
Figure 3 illustrates the conductivity funcion of (50).
Tables 7 and 8 contain the information about the behavior of the error when changing the values of , , and . In this case the diminution of the number of sections and diminution of the samples per section do influence the accuracy, but their influence is not so significant as the diminution of the formal powers .
The error obtained when employing the finite element method approach was , employing points in the mesh.
4.2. The Nonseparable-Variables Lorentzian Case
Proposition 15. Let the conductivity function have the form An exact solution for (1) is
The conductivity function with the form (52) is illustrated in Figure 4.
Table 9 shows an abnormal behavior of the error when decreasing the number of sections and samples . That is, it does not follow a clear pattern. But more important is to remark that the finite element method reported an error , which implies that is better situated for analyzing this class of conductivity functions. Table 10 states that the diminution of the number of formal powers decreases the accuracy.
4.3. The Nonseparable-Variables Polynomial Case
Proposition 16. Let one assume that the conductivity function has the form thus the function will be a solution of (1).
In Figure 5 it is illustrated the polynomial conductivity function of (54).
Tables 11 and 12 show a regular behavior of the error , because it decreases when the values and do. This case possesses a singular characteristic when compared to the cases above: The diminution of the number of formal powers does not provoke a considerable loss of the accuracy. This exception will be remarked since it has not been detected in other class of conductivity functions.
On the other hand, the finite element method proved to provide a very similar accuracy for analyzing this case, because the obtained error was .
4.4. The Nonseparable-Variables Sinusoidal Case
Proposition 17. Let one consider the sinusoidal conductivity One can verify by direct substitution that the function is a solution of (1).
The Figure 6(a) illustrates the conductivity (56), whereas the Figure 6(b) illustrates the case (58).
| (a) |
| (b) |
Tables 13 and 14 show that the behavior of is not fully regular, because a diminution of the number and does not necessarily imply a reduction of the accuracy, whereas the diminution of the formal powers provokes significant changes of the error . For this case, the finite element method reported an error .
Notice that the exact solution (57) that was imposed as the boundary condition cannot be employed when a constant factor multiplies the argument of the trigonometric function, because the tangent would not be defined at certain points. Nevertheless, one last experiment is worth of research.
Let us suppose that the conductivity function possesses the form and let us approach the solution to the forward problem keeping as the boundary condition the expression (57). Tables 15 and 16 display the biggest error found in the present work, even the behavior of could be considered regular, because it decreases while , , and do. The reader will appreciate that the accuracy is, in general, unsatisfactory for applications. Indeed, the finite element method obtained an error , hence it is quite better situated for analyzing this kind of conductivity functions.
5. Conclusions
The comparison of the numerical method posed in this work and the finite element method is far to be complete. Thus the selected examples will be considered a basic reference from which many other trials must be adequately performed, since they could not be included in these pages because they constitute independent research topics. For instance, a detailed description of the computational complexity [12] of both methods would provide a point of view less dependent on the platform on which the methods are programmed; thus we could focus our attention into the basic properties.
Still, we will remark an important characteristic of the pseudoanalytic functions-based method. It provides information about the solutions of the forward Dirichlet boundary value problem for (1), that is not clear how to obtain when utilizing the finite element method.
More precisely, the numerical approach based upon the pseudoanalytic analysis reaches a full set of functions defined at the boundary, that might contain relevant data if we desire to examine the electrical impedance tomography problem.
For instance, let us consider the base functions approached for the case when , as examined in (50). The absolute values of the coefficients , employed for approaching the boundary condition , are displayed in the semilogarithmic graphic of Figure 7. This calculation was performed considering sections, samples per section, radii, points per radii, and formal powers; thus we will have base functions , defined at the boundary .
Notice that in Figure 7 the values of the coefficients possess a particular distribution, even we will not point an evident pattern. But since the set is an orthonormal base, it can be used to approach any other boundary condition. In particular, if we use this base for approaching the condition that indeed corresponds to the conductivity , we will observe that the values of the coefficients , plotted in Figure 8, behave in a very different way.
As a complementary example, let us consider the base functions corresponding to a Lorentzian conductivity studied in Section 4.2. Figure 9 displays the absolute values of the coefficients when we impose the boundary condition , as posed in (53).
But if we impose a boundary condition of the form upcoming from (51), we can appreciate, as plotted in Figure 10, that the absolute values of the obtained coefficients are, once more, very different to those of Figure 9.
This implies that the behavior of the values of the coefficients could well serve as a criterion for understanding if some conductivity function can effectively provoke a certain electric potential distribution at the boundary of same domain , when physical measurements provide the boundary condition.
Taking into account that most of the algorithms dedicated to approach solutions of the electrical impedance tomography problem recursively solve the forward problem [2], this novel information will be used to introduce changes in ; thus the boundary condition is fulfilled with a better convergence, and with less computational time.
Even it is not possible yet to estimate when this new method will be employed to approach solutions for the electrical impedance tomography problem, we will remark that the main contribution of this work is to provide a new technique for fully applying the elements of the modern Pseudoanalytic function theory into the analysis of conductivity functions that originated from physical experimental models, which will certainly provide additional information for better understanding the inverse Dirichlet boundary value problem of (1) in the plane.
In other words, before this proposal, the mathematical advances were exclusively improving the study of separable-variables conductivity functions, rarely useful for representing physical cases. These pages have shown that, under certain limits, any conductivity function is susceptible to be analyzed employing pseudoanalytic functions, whether they possess an exact mathematical form or not. As a matter of fact, it is enough that the conductivity values are fully defined over the points within a bounded domain , with a smooth enough boundary . The research will continue to better understand the limiting cases and the certain conditions of smoothness of the boundary; thus the posed ideas can be adequately employed in the applied sciences.
Disclosure
The numerical methods used along this work were fully developed in GNU C/C++ Compiler, employing a [email protected] GHz, on SLACKWARE 13.37 LINUX operating system. The experimental procedures showed that the numerical results can vary when using different platforms based on and processor architecture, or compilers between other operating systems, including different LINUX distributions or Registered Trade Mark operating systems. If the reader wishes to perform his own numerical trials, please contact the authors to obtain the resource codes.
Acknowledgments
The authors would like to acknowledge the support of CONACyT Projects 106722 and 81599, Mexico. Marco Pedro Ramirez-Tachiquin is grateful for the support of HILMA S.A. de C.V., Mexico; Ariana Guadalupe Bucio Ramirez would like to acknowledge the support of CONACyT and UPIITA, IPN; Rogelio Adrian Hernandez-Becerril and Cesar Marco Antonio Robles Gonzalez would like to acknowledge the support of CONACyT and ESIME, IPN.