Abstract
In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part and a homogeneous Dirichlet condition on an unknown part of the boundary of a bounded domain, . We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of and if it preserves its properties with varying the eigenvalues.
1. Introduction
In the paper, we consider the mathematical modeling employed in electrostatic imaging during nondestructive testing. This analysis engenders an inverse boundary value problem for a Laplace equation. This problem entails the identification of an unknown domain, , within a conducting host medium with constant conductivity based on known Cauchy data on the boundary of the medium, . For detailed examples, please consult [1].
We suppose that is a simply connected domain in with smooth boundaries and The boundary is assessed by imposing a voltage pattern at a number of electrodes attached to the boundary and measuring the resultant current through the electrodes.
The inverse problem we are concerned with is to determine the unknown part, , of the boundary of the bounded domain , based on available Cauchy data on : where denotes a solution of the Laplace equation: satisfying the condition: where denotes a continuous positive function on , denotes a continuous negative function on , and denotes the outer normal vector unit.
My thesis (please consult [2]) is aimed at identifying via Cauchy data on using methods based on conformal mapping and the minimization function. The results (please consult [3]) were published in AMO (Advanced Modeling and Optimization) in .
Here, we aim to compare the results obtained using conformal mapping and the fixed-point method and observe the variations in the shape of with varying eigenvalues.
2. Methods
2.1. Conformal Mapping Method
The conformal mapping method was presented in [3].
This paper is primarily aimed at presenting iterative methods to solve the inverse problem by using ideas from conformal mapping theory that have been developed over the last decade. For detailed references, please consult [1, 4, 5].
However, in our approach, we consider a spectral problem on the simply connected domain in with smooth boundary satisfying (1).
The family of holomorphic functions corresponding to the eigenvalues of the spectrum was identified via the fixed-point method and Nyström method.
The proposed reconstruction of the boundary curve based on available Cauchy data on the accessible boundary, , is based on a conformal map . Thus, we consider , the boundary of the complex disk with radius one centered at the origin, and map onto via . We introduce the following notations.
Further, and are parameterized as follows: where is a continuous differentiable periodic from onto satisfying the property that is injective. This property implies that
We can normalize the mapping by prescribing and define the bijective boundary correspondence function by setting
It is clear that the values of on the boundary produce a unique solution of the Cauchy problem, given by
Therefore, the operator is well defined.
The Cauchy-Riemann equations give rise to nonlinear ordinary differential equations via Bessel equations for the values on this part of the boundary which is to be determined via the fixed-point method.
(10) is followed by the solution of several ill-posed Cauchy problems with respect to the holomorphic function by using a regularized power series expansion to retrieve the unknown part of the boundary curve using .
2.2. Position of Problem
The inverse problem is thereby reduced to the determination of a holomorphic function that is a solution of with
Using the polar coordinates of the solution of the Dirichlet problem on the disk with boundary conditions and the solution of the Bessel equation of order , see [3], we have with where denotes the Bessel function of order and corresponds to .
Therefore, by we determine a family of holomorphic functions, , such that with is well defined. We present the details of this algorithm, including a convergence analysis result in the following sections.
3. Results and Discussion
3.1. Presentation of the Fixed Point Method
A fixed point is a solution of equation where is an operator defined from a nonempty set onto itself. The principal condition for a nonlinear operator to admit a fixed point is for it to be a strict contraction.
The determination of is based on the Banach fixed-point theorem, where a contracting nonlinear application on a complete metric space admits a fixed point [1].
General methods of construction of approximate conformal maps can be found in survey works by [1, 5, 6].
3.1.1. Nonlocal Differential Equation
In this section, we attempt to determinate , satisfying (15), using the fixed point method.
Let denote the Dirichlet-Neumann operator on , which maps onto the normal derivative of , defined as follows: with the respective boundary values given by
The boundary traces on and are, respectively, given by
However, we have the nonlocal differential equation for the conformal map , which is a critical element in the inverse algorithm.
The boundary correspondence function, , is a solution of As , equations (20) and (21) are well defined.
Based on the available Cauchy data, , we have (12) the fundamental solution of (11).
Let us discuss some definitions and properties related to fixed points before employing the fixed-point methods to determinate .
3.1.2. Fixed-Point Theorems (See [1])
Theorem 1. (Fixed-Point theorem of Banach)
Let be a nonlinear operator where is a complete metric space. We assume that is a strict contraction. Then, admits a unique fixed point (see [7]).
Proof of Theorem 2. Consider any point and iteratively define
Then,
If , then
Hence is a Cauchy sequence in and, as is complete, there exists such that in Thus, it is evident that
Hence, is fixed point for and the fact that is a strict contraction ensures its uniqueness.
The following is the version of the Banach fixed-point theorem and its corollary that is used in the subsequent proofs.
Theorem 3. Let be a nonempty complete metric space. If is such that, for at least one integer is a strict contraction, then admits an unique fixed point.
Theorem 4. Let be a complete metric space and let be a -almost contraction. Then, (1)For any , the Picard iteration , converges to some ;(2)The following estimate holds
3.1.3. Fixed-Point Equation and Iteration
Let be an operator defined as follows:
We assume that is not identically zero.
Further, we introduce an operator defined by with
By integrating (32) and considering that and , we obtain
Using the boundary conditions of , we can express
It is easy to verify that is nonlinear and contracting.
Now, we use the following theorem (see [1]).
Theorem 5. Let be a pair of Cauchy data obtained from (8).
Then, in terms of the holomorphic map and its boundary correspondence function , the function is a fixed point for .
We obtain a fixed point defined by successive iteration of beginning with an arbitrary approximation , for all . Therefore, Theorem 3.4 induces the following iteration:
The solution of the Cauchy problem is required to identify the boundary function from the function . For this purpose, let us express using the following expansion:
Using , we obtain using the following expansion:
The following convergence result justifies the procedure of iteration (31).
Theorem 6 (see [1]). Let be the unit disk and be a continuous positive function defined on it. Then, successive approximations converge provided that the initial estimate of the boundary function is sufficiently close to the exact function .
Proof (see [4]). Theorem 6 is proved by first establishing that in the case where . Thereafter, a continuity argument is applied to extend the estimate to the norm of for in a neighborhood of the fixed point defined in Theorem 3.4.
Proof of Theorem 7. We calculate the derivate of the nonlinear operator . We begin by estimating . We have
Therefore, we have,
Using a continuity argument to estimate the norm of near the fixed point of Theorem 3.4, we have for all Thus, and .
The approximation of near the fixed point of the theorem is
We now determinate a numerical approximation of the conformal map , for all .
3.1.4. Numerical Approximation of the Conformal Map
In this section, we use an ordinary differential equation of the boundary correspondence function to obtain an approximation of the conformal map, , for all .
Let us introduce two boundary correspondence functions, and , defined from to
On , equation (21) becomes
Here, we use the length of arc determined by .
By removing the identity, that is obtained by integrating (41). Then, we have for , the equation where we have for convenience with operator defined in (35).
For the iterative solution of (43), we have the following convergence results:
Theorem 8. The successive approximations defined by (45) converge from the initial approximation provided that is close enough to .
When we obtain , that is, , the boundary values of on to be used for the iteration of (45) are given by
with
So, on ,
We can express the expansion of in the form
which is numerically stable for . By construction, considering (40), we have and .
Now, we describe the numerical algorithm that we will implement in this paper. We use a method involving integral equations on . The integral equation obtained is resolved via the Nyström method.
We give ourselves a regular mesh,
The approximations of at the nodes of mesh are calculated using the previous approximate values
The iterations allow approximations of the normal derivatives via trigonometric interpolation.
The integration interpolation polynomial on is used to associate to , via (43), with .
Beginning with identity, we have
which corresponds to
or more precisely
We substitute in the integral via the Nyström method using the interpolation polynomial .
3.1.5. Nyström Method (See [8])
This method, called the quadrature method, is the application of numerical methods of calculating integrals to obtain a linear system. We write where denotes the interpolation polynomial of Lagrange of degree , for the pair of values where we use with the polynomials are Lagrange polynomials of associated to .That is,
Explicitly, we can express (57) in the form
Therefore, we can set for the approximations. Hence,
If , then
After fixing , we calculate using equation (45), for all .
The estimate of allows us to calculate using (29), (35), and (48) and, subsequently, the expression of .
We use
The traces of the curves corresponding to the equations defined below provide an approximate domain.
We attempt to determine for . The approach for the other case is similar.
We know that (59) is equivalent to
After calculation, we have or
3.1.6. Convergence Results of
We set , as defined above.
For , we have a sequence defined in by
The Picard theorem ensures the convergence of .
We can write
Thus, we have
Finally, we can deduce the following result:
Proposition 9. Let be a -contraction and be a sequence of functions defined by If verifies this recurrence relation Then, a constant depending on and .
Proof of Proposition 10. The Picard theorem on the convergence of .
We can write
So,
This yields the result.
Further, we set . We have
It is evident that does not depend on .
Therefore, according to (20) and (21), the map is then defined by
with The coefficient is determined using the available Cauchy data and eigenvalues.
For all , so is holomorphic.
In the following argument, we pose because they are determined by eigenvalues.
is analytic, continuous on ; it realizes a conformal mapping from onto , according to Theorem (see [2] page ). Further, it realizes a conformal mapping (univalent) from the domain onto the domain .
Hence, for all and all , there exists an univalent holomorphic function , which realizes a conformal mapping from the domain onto the domain , which is a solution of problem (19), defined by
For all with is determined using the available Cauchy data and eigenvalues.
Proposition 11. Consider the problem where is a spiral defined on by the following equations: Then, is a part of the spiral, , image of under defined for all and .
Proof of the Proposition 12. It suffices to determine the integral and decimal parts of .
3.1.7. Numerical Simulations
We determine the approximate shape of the unknown part of the boundary corresponding to the two first eigenvalues.
Let be the map defined in (78).
We put and where is defined to be negative.
First, we determine of corresponding to the first and second eigenvalues, after approximating the corresponding shapes of . Using the boundary values, we have previously established,
The coefficients and are determined based on the Cauchy data corresponding to each eigenvalue of the Laplacian operator. Hence, they are obtained by
Therefore, the calculation yields that corresponding to :
3.1.8. Conformal Map Corresponding to First Eigenvalue
Corresponding to the first eigenvalue , we have
Therefore, we obtain the result for (1) and
Figure 1 is obtained by applying and in (86). Figure 1(a) shows the shapes of and in an orthogonal mark with graphic unit 20 cm, and Figure 1(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us the enlargement of shapes of and . (2) and
(a)
(b)
Figure 2 is obtained by applying and in (86). Figure 2(a) shows the shapes of and in an orthogonal mark with graphic unit 20 cm, and Figure 2(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. It is this right part which gives us also the enlargement of shapes of and .
(a)
(b)
It is to be noted that, in each case ( or ), is sinusoïdal, is a part of spiral, and converges to a finite limit. Therefore, is asymptotic to a circle and is disk shaped.
3.1.9. Conformal Map Corresponding to the Second Eigenvalue
Corresponding to the second eigenvalue , we have
Thus, we conclude the result for and .
Figure 3 is obtained by applying and in (87). Here too, Figure 3(a) shows the shapes of and in an orthogonal mark with graphic unit 20 cm, and Figure 3(b) shows these same shapes in an orthogonal mark with graphic unit 5 cm. Figure 3(a) gives us also the enlargement of shapes of and .
(a)
(b)
Here too, is sinusoïdal, is a part of spiral, and converges to a finite limit.
3.1.10. Comparison Results for and
The result for the first eigenvalue obtained via the minimization functional method for and in our previous study, and the result obtained using the fixed-point method for the same data has been depicted in the following diagram.
In Figure 4, we can observe that (i) is sinusoïdal in both these studies(ii) is a part of spiral in the present study and is a part of spiral with a cusp, in the study in 2014(iii)Both methods yielded a spiral shape for
(a)
(b)
3.1.11. Variations of with respect to
The following diagram allows us to compare the shapes of corresponding the results in the first and the second study for the first eigenvalues.
In Figure 5, we obtain results of the variations of the shape of with respect to . Figure 5(a) shows the variation of the shapes of by applying and in (86) and in (87), i.e., for the first and second eigenvalues. Figure 5(b) shows the variation of the shapes of by applying and in (86) and in (87), i.e., for the first and the second eigenvalue.
(a)
(b)
3.2. Results
All previous figures are obtained using “GeoGebra Geometry.”
Figures 1(b), 2(b), and 3(b) give us the enlargement of shapes of and found, and they allow us to see their exact shapes. Also, these exact shapes allow us to make a good interpretation of the results.
So, the simulations yield the following results: (1)The shape of is a spiral. When the value of increases, its spiral shape is accentuated, and it is constituted of identical connected parts. Further, when the eigenvalue increases, the length of is observed to increase accompanied by a change in its spiral shape (see Figure 5).(2)However, the shape of is a spiral corresponding to all variations of and the eigenvalues. Its shape does not vary and it converges to a finite limit, which is circular in shape.(3)Figure 4(a) shows the results of the previous study, which was conducted in 2014, for and , and Figure 4(b) shows the results of the present study for and . We can observe that the shape of is spiral in both the results; however, the present study provides a more precise shape of the spiral that is asymptotic to a circle. This shape can justify that is a disk.
Therefore, the result obtained using the fixed-point method is more precise and further identifies as a disk.
4. Conclusion
In summary, preserves the spiral shape of and its properties but it varies those of based on variations in eigenvalues.
Therefore, is a spiral, which is asymptotic to a circle and is identified to be a disk.
This result can be applied to the one-dimensional case, but it is more interesting to study this phenomenon in dimensions three and four.
In an inverse problem with similar conditions, it is more relevant to use conformal mapping and the fixed-point method to yield interesting numerical results. The general case that is a disk remains an open problem.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there is no conflict of interest.
Acknowledgments
We would like to thank Editage (https://www.editage.com) for English language editing services.