A New Meshless Method for Solving 3D Inverse Conductivity Issues of Highly Nonlinear Elliptic Equations

Abstract

In this research, the 3D inverse conductivity issues of highly nonlinear elliptic partial differential equations (PDEs) are investigated numerically. Even some researchers have utilized several schemes to overcome these multi-dimensional forward issues of those PDEs; nevertheless, an effective numerical algorithm to solve these 3D inverse conductivity issues of highly nonlinear elliptic PDEs is still not available. We apply two sets of single-parameter homogenization functions as the foundations for the answer and conductivity function to cope with the 3D inverse conductivity issue of highly nonlinear PDEs. The unknown conductivity function can be retrieved by working out another linear system produced from the governing equation by collocation skill, while the answer is acquired by dealing with a linear system to gratify over-specified Neumann boundary condition on a fractional border. As this new computational approach is based on a concrete theoretical foundation, it can result in a deeper understanding of 3D inverse conductivity issues with symmetry and asymmetry geometries. The homogenization functions method is rather stable, effective, and accurate in revealing the conductivity function when the over-specified Neumann data with a large level of noise of 28%.

1. Introduction

For inverse conductivity issues of nonlinear elliptic partial differential equations (PDEs), Calderón [1] discussed the inverse boundary value problem (BVP), which originated from the problem of electrical prospection. Furthermore, the Calderón problem is also called the inverse conductivity problem. He was the first to fabricate the phrase of an inverse BVP for studying the electrical impedance tomography (EIT) technique by mathematical modeling because the conductivity occurs as a variable coefficient of diffusion in an elliptic PDE. It is an inverse issue of the Neumann to Dirichlet mapping by realizing the voltage and current on the boundary. Computerized tomography is a standard instrument in the non-destructive testing (NDT) of materials and medical diagnostics right now. In addition, the well-known schemes of magnetic resonance imaging and X-ray, the last four decades have testified to an increasing interest in the new imaging skill of EIT because of its low cost and simple implementation. A comprehensive review of EIT was shown in Borcea [2]. In the medical and industrial fields, we often encounter those difficult ill-posed inverse problems. There have been many papers in the engineering and mathematical literature resolving such various themes as the electrical impedance tomography, the approaches of numerical reestablishment, the stability and identifiability of solutions for conductivity, mathematical modeling of electrodes, schemes of numerical establishment, and the design of measurement devices [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

In general, we can utilize the NDT to deal with the corrosion of metals. Identifying the impedance coefficient may be an effective method to reveal the region of corrosion, and one may possibly evaluate the level of corrosion in accordance with the exterior estimations and then employ the solution of the inverse issue by the EIT. The Robin coefficient quantifies the level of corrosion of some inaccessible part of the boundary in a simpler corrosion model [22]. The model issue was to retrieve the Robin coefficient through the present flux and potential surveys on fractionally accessible data in a finite annulus.

Nevertheless, merely a few papers are interested in inverse conductivity issues of nonlinear elliptic PDEs. Liu and Atluri [23] have addressed the Calderón inverse issue to retrieve the unknown conductivity function by utilizing the iterative scheme of Lie-group type together with a finite-strip method. The advantages of the current algorithm were that no a priori information about the functional form of conductivity function was essential, and no added measurement of data inside the area was needed. Later, Liu and Liu [24] established a homogenization function to gratify Cauchy boundary conditions on a smooth 2D boundary of a simply connected area. This issue for choosing the inner boundary of the Poisson equation in an arbitrary doubly-connected plane area was addressed, which retrieved an unknown inner boundary of a rigid inclusion under the over-specified Cauchy data on the accessible outer boundary. The accuracy and robustness of the current homogenization boundary function method (HBFM) were assessed through several numerical experiments by comparing the exact inner boundary to the retrieved one under a large noisy effect. Then, Liu and Wang [25] utilized the HBFM to deal with a nonlinear inverse Cauchy issue of nonlinear elliptic type PDE in an arbitrary doubly-connected plane area. The unknown Dirichlet data on an inner boundary were retrieved by over-specifying the Cauchy data on an outer boundary. A homogenization function was derived from annihilating the Cauchy data on the outer boundary, and then a homogenization skill generated a transformed equation in terms of a transformed variable whose outer Cauchy boundary data were homogeneous. Besides, they examined several numerical instances and obtained good results even with a large, noisy disturbance.

For the mathematical analysis of the Calderón problem, Caro et al. [26] proved the corresponding global evaluations in all dimensions higher than three. The evaluations were based on the construction of solutions to the Schrödinger equation by complicated geometrical optics developed in the anisotropic setting by Dos Santos Ferreira et al. [27] to deal with the Calderón issue in certain admissible geometries. Later, Piiroinen and Simon [28] acquired a probabilistic formulation of Calderón’s inverse conductivity issue. This formulation came in three equivalent versions and each of them may yield both a novel perspective as well as a novel set of (probabilistic) tools when it came to studying questions related to the unique determinability of conductivities from boundary data. After that, Alessandrini et al. [29] contemplated the electrostatic inverse BVP, also known as EIT, for the situation where the conductivity was a piecewise linear function on the area, and they demonstrated that a Lipschitz stability evaluation for the conductivity in terms of the local Dirichlet-to-Neumann (DN) map held true. Apart from that, Muñoz [30] displayed the uniqueness of the conductivity for the quasilinear Calderón’s inverse issue. Under some structural suppositions on the direct issue, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique DN map. This approach of proof considered some complex-valued, linear test functions based on the point of the boundary of the area and a linearization of the DN map placed at these special sets of solutions. However, they did not demonstrate some examples to prove their algorithm.

For the applications of EIT, de Moura et al. [31] used EIT in multiphase flow for identifying flow patterns and void-fraction calculation. An image made of the conductivity of the domain was formed by combining an inverse issue and measurements. For that issue, they presented a recursive Gauss–Newton optimization sequential importance resampling (GNOSIR) filter aimed to reconstruct vertical slug flow. Their results demonstrated that the GNOSIR filter improved the evaluation when there was a sudden and fast evolution of the inclusion’s contour between states; however, they only coped with 2D problems. Later, Duran et al. [32] coped with the EIT image reconstruction by using convolutional neural networks (CNN), which was that the resulting image of the convolution process was smaller than the original input image. This issue was usually addressed by introducing padding, which was the addition of layers in the borders of the original input image. Their work presented the use of a doubly periodic padding, which was relevant for toroidal image issues such as the electric potential distribution measured by utilizing the EIT. The CNN was trained by employing a database generated by numerical calculations. Their resulting image reconstructions were presented for different noisy potential inputs; however, they merely dealt with 2D problems. After that, Wagner et al. [33] explored the evaluation of spatial strain distributions by elastoresistive thin-film sensors utilizing 2D Electrical Impedance Tomography. Hence, spatially-limited strains and the subsequent conductivity changes in a thin-film surface sensor were modeled and experimentally tested. Then, the inverse reconstruction of both simulated and experimental estimations by the EIT was compared and analyzed.

This study is classified as follows. Section 2 demonstrates the highly nonlinear issue statement and establishes the single-parameter homogenization functions in Section 3. In Section 4, we interpret the homogenization functions method to apply the inverse conductivity issues. Three numerical experiments of the inverse conductivity issue of highly nonlinear elliptic equations are displayed in Section 5. Finally, we illustrate the conclusions in Section 6.

2. Problem Statement and the Homogenization Functions Method

The purpose of this study is to construct an effective homogenization functions scheme that possesses the merit of easy numerical implementation and owns good flexibility when applied to the 3D inverse conductivity issue of highly nonlinear elliptic PDE for engineering and medical images in a cuboid $\mathsf{\Omega }:=\{(x,y,z)\in (0,d)\times (0,e)\times (0,f)\}$:

$$\begin{array}{l}\sigma (x,y,z)[v_{xx}(x,y,z)+v_{yy}(x,y,z)+v_{zz}(x,y,z)]+v_x(x,y,z){\sigma }_x(x,y,z)+v_y(x,y,z){\sigma }_y(x,y,z)\\ +v_z(x,y,z){\sigma }_z(x,y,z)+F[v(x,y,z)]=P(x,y,z), (x,y,z)\in \mathsf{\Omega }, \end{array}$$

(1)

$$\begin{array}{l}v(0,y,z)=g_1(y,z), v(d,y,z)=g_2(y,z),\\ v(x,0,z)=g_3(x,z), v(x,e,z)=g_4(x,z),\\ v(x,y,0)=g_5(x,y), v(x,y,f)=g_6(x,y),\end{array}$$

(2)

where v(x, y, z) is electrical potential, F is a first high-order nonlinear operator, P(x, y, z) is a presented source function and σ(x, y, z) is an unknown electrical conductivity function to be retrieved under the over-specified Neumann data on the bottom:

$$v_z(x,y,0)=g_7(x,y),$$

(3)

Considering this issue, we assume that the Dirichlet data of σ(x, y, z) on boundaries can be estimated:

$$\begin{array}{l}\sigma (0,y,z)=h_1(y,z), \sigma (d,y,z)=h_2(y,z),\\ \sigma (x,0,z)=h_3(x,z), \sigma (x,e,z)=h_4(x,z),\\ \sigma (x,y,0)=h_5(x,y), \sigma (x,y,f)=h_6(x,y),\end{array}$$

(4)

The issue in Equations (1)–(4) is fundamentally highly nonlinear since σ(x, y, z) and v(x, y, z) are both unknown functions. The current inverse coefficient issue is to retrieve the electrical conductivity coefficient function σ(x, y, z) inside the cuboid with the aid of the estimations of the potential on the entire boundary and the present flux on a partially accessible boundary in a finite cuboid.

Illustrating the concept of homogenization function [34], we contemplate a 3D boundary value problem (BVP):

$$S[v(x,y,z)]=P(x,y,z), (x,y,z)\in \mathsf{\Omega },$$

(5)

$$\begin{array}{l}v(0,y,z)=g_1(y,z), v(d,y,z)=g_2(y,z),\\ v(x,0,z)=g_3(x,z), v(x,e,z)=g_4(x,z),\\ v(x,y,0)=g_5(x,y), v(x,y,f)=g_6(x,y),\end{array}$$

(6)

where S is a second high-order nonlinear operator, P(x, y, z) is a presented source function, and g_i, i = 1, …, 6 are presented boundary data.

Through

$$\begin{array}{ll}{E}^1(x,y,z)=& \left(1-\frac{x}{d}\right)\left[g_1(y,z)-\left(1-\frac{y}{e}\right)g_3(0,z)-\frac{y}{e}{g}_4(0,z)\right]\\ & +\frac{x}{d}\left[g_2(y,z)-\left(1-\frac{y}{e}\right)g_3(d,z)-\frac{y}{e}{g}_4(d,z)\right]\\ & +\left(1-\frac{y}{e}\right)g_3(x,z)+\frac{y}{e}{g}_4(x,z). \end{array}$$

(7)

Knowing that

$$E^1(0,y,z)=g_1(y,z), E^1(d,y,z)=g_2(y,z).$$

(8)

Let

$$\begin{array}{ll}E(x,y,z)=& E^1(x,y,z)+\left(1-\frac{z}{f}\right)\left[g_5(x,y)-E^1(x,y,0)\right]\\ & +\frac{z}{f}\left[g_6(x,y)-E^1(x,y,f)\right].\end{array}$$

(9)

We can employ the compatibility data of boundary conditions as follows:

$$\begin{array}{l}{g}_5(0,y)=E^1(0,y,0)=g_1(y,0), g_6(0,y)=E^1(0,y,f)=g_1(y,f),\\ g_5(d,y)=E^1(d,y,0)=g_2(y,0), g_6(d,y)=E^1(d,y,f)=g_2(y,f),\end{array}$$

(10)

it can be verified easily

$$\begin{array}{l}{E}^1(0,y,z)=g_1(y,z), E^1(d,y,z)=g_2(y,z),\\ E^1(x,0,z)=g_3(x,z), E^1(x,e,z)=g_4(x,z),\\ E^1(x,y,0)=g_5(x,y), E^1(x,y,f)=g_6(x,y).\end{array}$$

(11)

Thus, we can acquire a 3D homogenization function:

$$\begin{array}{ll}E(x,y,z)=& \left(1-\frac{x}{d}\right)\left[g_1(y,z)-\left(1-\frac{y}{e}\right)g_3(0,z)-\frac{y}{e}{g}_4(0,z)\right]\\ & +\frac{x}{d}\left[g_2(y,z)-\left(1-\frac{y}{e}\right)g_3(d,z)-\frac{y}{e}{g}_4(d,z)\right]\\ & +\left(1-\frac{y}{e}\right)g_3(x,z)+\frac{y}{e}{g}_4(x,z)\\ & +\left(1-\frac{z}{f}\right)(g_5(x,y)-\left(1-\frac{x}{d}\right)\left[g_1(y,0)-\left(1-\frac{y}{e}\right)g_3(0,0)-\frac{y}{e}{g}_4(0,0)\right]\\ & -\frac{x}{d}\left[g_2(y,0)-\left(1-\frac{y}{e}\right)g_3(d,0)-\frac{y}{e}{g}_4(d,0)\right]\\ & -\left(1-\frac{y}{e}\right)g_3(x,0)-\frac{y}{e}{g}_4(x,0))\\ & +\frac{z}{f}(g_6(x,y)-\left(1-\frac{x}{d}\right)\left[g_1(y,f)-\left(1-\frac{y}{e}\right)g_3(0,f)-\frac{y}{e}{g}_4(0,f)\right]\\ & -\frac{x}{d}\left[g_2(y,f)-\left(1-\frac{y}{e}\right)g_3(d,f)-\frac{y}{e}{g}_4(d,f)\right]\\ & -\left(1-\frac{y}{e}\right)g_3(x,f)-\frac{y}{e}{g}_4(x,f)).\end{array}$$

(12)

Because of the attribute of E(x, y, z) and with the aid of the variable transformation from v(x, y, z) to w(x, y, z) = v(x, y, z)-E(x, y, z), we can transform the elemental 3D BVP with non-homogeneous boundary data to a one with homogeneous boundary data for w(x, y, z):

$$S[w(x,y,z)]=P(x,y,z)-S[E(x,y,z)], (x,y,z)\in \mathsf{\Omega },$$

(13)

$$w(0,y,z)=w(d,y,z)=w(x,0,z)=w(x,e,z)=w(x,y,0)=w(x,y,f)=0.$$

(14)

Hence, E(x, y, z) is a significant part of the homogenization function to demolish the non-zero boundary data.

3. A Single-Parameter Homogenization Function

We address the inverse conductivity issue of highly nonlinear elliptic PDE by utilizing the homogenization functions. For this aim, we bring in the normalized coordinates:

$$\overline{x}=\frac{x}{d},\overline{y}=\frac{y}{e},\overline{z}=\frac{z}{f}.$$

(15)

We can generalize Equation (12) by bringing in the mth order shape functions:

$${\epsilon }_m(\overline{x})={\overline{x}}^m,{\epsilon }_m(\overline{y})={\overline{y}}^m,{\epsilon }_m(\overline{z})={\overline{z}}^m,$$

(16)

where

$${\epsilon }_m(0)=0,{\epsilon }_m(1)=1,\forall m\in N.$$

(17)

To supersede x/d, y/e, and z/f in Equation (12) by ${\epsilon }_m(\overline{x}), {\epsilon }_m(\overline{y}) \mathrm{and} {\epsilon }_m(\overline{z}),$ we can acquire the mth order homogenization function as follows:

$$\begin{array}{ll}{E}^v(m,x,y,z)& =[1-{\epsilon }_m(\overline{x})]\{g_1(y,z)-[1-{\epsilon }_m(\overline{y})]g_3(0,z)-{\epsilon }_m(\overline{y})g_4(0,z)\}\\ & +{\epsilon }_m(\overline{x})\{g_2(y,z)-[1-{\epsilon }_m(\overline{y})]g_3(d,z)-{\epsilon }_m(\overline{y})g_4(d,z)\}\\ & +[1-{\epsilon }_m(\overline{y})]g_3(x,z)+{\epsilon }_m(\overline{y})g_4(x,z)+[1-{\epsilon }_m(\overline{z})\left]\right\{g_5(x,y)-[1-{\epsilon }_m(\overline{x})]\{g_1(y,0)\\ & -[1-{\epsilon }_m(\overline{y})]g_3(0,0)-{\epsilon }_m(\overline{y})g_4(0,0)\}-{\epsilon }_m(\overline{x})\{g_2(y,0)-[1-{\epsilon }_m(\overline{y})]g_3(d,0)\\ & -{\epsilon }_m(\overline{y})g_4(d,0)\}-[1-{\epsilon }_m(\overline{y})]g_3(x,0)-{\epsilon }_m(\overline{y})g_4(x,0)\}\\ & +{\epsilon }_m(\overline{z})\{g_6(x,y)-[1-{\epsilon }_m(\overline{x})]\{g_1(y,f)-[1-{\epsilon }_m(\overline{y})]g_3(0,f)-{\epsilon }_m(\overline{y})g_4(0,f)\}\\ & -{\epsilon }_m(\overline{x})\{g_2(y,f)-[1-{\epsilon }_m(\overline{y})]g_3(d,f)-{\epsilon }_m(\overline{y})g_4(d,f)\}\\ & -[1-{\epsilon }_m(\overline{y})]g_3(x,f)-{\epsilon }_m(\overline{y})g_4(x,f)\}.\end{array}$$

(18)

By utilizing Equation (17), we can verify

$$\begin{array}{l}{E}^v(m,0,y,z)=g_1(y,z), E^v(m,d,y,z)=g_2(y,z),\\ E^v(m,x,0,z)=g_3(x,z), E^v(m,x,e,z)=g_4(x,z),\\ E^v(m,x,y,0)=g_5(x,y), E^v(m,x,y,f)=g_6(x,y),\end{array}$$

(19)

Thus, we can choose E^v(k, x, y, z) as the foundations of v(x, y, z) and assume

$$v(x,y,z)={\displaystyle \sum _{m=1}^{p_1}{a}_m{E}^v(m,x,y,z)},$$

(20)

where

$${\displaystyle \sum _{m=1}^{p_1}{a}_m}=1,$$

(21)

we can acquire a simple solution of v(x, y, z) and hence the solution of σ(x, y, z) of the nonlinear inverse conductivity issue in the cuboid can be accomplished in the next part. Equation (21) is used to promise that v(x, y, z) in Equation (20) can gratify the boundary conditions in Equation (2).

The data g₇(x, y) = v_z(x, y, 0) is over-specified in Equation (3), and by taking the z-differential of Equation (20) and introducing z = 0, we can acquire

$${\displaystyle \sum _{m=1}^{p_1}{a}_m{E}_z^v(m,x,y,0)=}{g}_7(x,y),$$

(22)

beginning with which we can produce linear equations by the collocation of points x_k and y_k, k = 1,…, q₁:

$${\displaystyle \sum _{m=1}^{p_1}{a}_m{E}_z^v(m,x_k,y_k,0)=}{g}_7(x_k,y_k), k=1,\dots ,q_1.$$

(23)

Dealing with the 2q₁ + 1 linear Equations (21) and (23), we can decide the p₁ coefficients a_m, m = 1,…, p₁, and then Equation (17) is used to work out v(x, y, z) in the entire area.

4. Numerical Algorithm for Inverse Conductivity Issues

We can produce another mth order 3D homogenization function for σ(x, y, z) by

$$\begin{array}{ll}{E}^{\sigma }(m,x,y,z)& =[1-{\epsilon }_m(\overline{x})]\{h_1(y,z)-[1-{\epsilon }_m(\overline{y})]h_3(0,z)-{\epsilon }_m(\overline{y})h_4(0,z)\}\\ & +{\epsilon }_m(\overline{x})\{h_2(y,z)-[1-{\epsilon }_m(\overline{y})]h_3(d,z)-{\epsilon }_m(\overline{y})h_4(d,z)\}\\ & +[1-{\epsilon }_m(\overline{y})]h_3(x,z)+{\epsilon }_m(\overline{y})h_4(x,z)+[1-{\epsilon }_k(\overline{z})\left]\right\{h_5(x,y)-[1-{\epsilon }_k(\overline{x})]\{h_1(y,0)\\ & -[1-{\epsilon }_m(\overline{y})]h_3(0,0)-{\epsilon }_m(\overline{y})h_4(0,0)\}-{\epsilon }_m(\overline{x})\{h_2(y,0)-[1-{\epsilon }_m(\overline{y})]h_3(d,0)\\ & -{\epsilon }_m(\overline{y})h_4(d,0)\}-[1-{\epsilon }_m(\overline{y})]h_3(x,0)-{\epsilon }_m(\overline{y})h_4(x,0)\}\\ & +{\epsilon }_m(\overline{z})\{h_6(x,y)-[1-{\epsilon }_m(\overline{x})]\{h_1(y,f)-[1-{\epsilon }_m(\overline{y})]h_3(0,f)-{\epsilon }_m(\overline{y})h_4(0,f)\}\\ & -{\epsilon }_m(\overline{x})\{h_2(y,f)-[1-{\epsilon }_m(\overline{y})]h_3(d,f)-{\epsilon }_m(\overline{y})h_4(d,f)\}\\ & -[1-{\epsilon }_m(\overline{y})]h_3(x,f)-{\epsilon }_m(\overline{y})h_4(x,f)\}.\end{array}$$

(24)

which automatically gratifies the specified data in Equation (4).

By utilizing

$$\sigma (x,y,z)={\displaystyle \sum _{m=1}^{p_2}{b}_m{E}^{\sigma }(m,x,y,z)},$$

(25)

$${\displaystyle \sum _{m=1}^{p_2}{b}_m}=1,$$

(26)

back substituting v(x, y, z) in Equation (20) into Equation (1) and collocating $q_2^3$ points in the area Ω, we can produce $q_2^3$ linear equations:

$$\begin{array}{ll}& {\displaystyle \sum _{m=1}^{p_2}\{[v_x(x_i,y_j,z_r)E_x^{\sigma }(m,x_i,y_j,z_r)+v_y(x_i,y_j,z_r)E_y^{\sigma }(m,x_i,y_j,z_r)+v_z(x_i,y_j,z_r)E_z^{\sigma }(m,x_i,y_j,z_r)}\\ +& E^{\sigma }(m,x_i,y_j,z_r)[v_{xx}(x_i,y_j,z_r)+v_{yy}(x_i,y_j,z_r)+v_{zz}(x_i,y_j,z_r)]\}{b}_m\\ =& P(x_i,y_j,z_r)-F[v(x_i,y_j,z_r)],i,j,r=1,\dots ,q_2.\end{array}$$

(27)

Coping with $q_2^3+1$ linear Equations (26) and (27), we can decide the p₂ coefficients b_m, m = 1,…, p₂, Equation (25) is used to retrieve σ(x, y, z) in the total area. This is a new algorithm of the 3D inverse conductivity issue of highly nonlinear elliptic PDE by employing the homogenization functions method (HFM). While E^v(m, x, y, z) plays the role of the foundations to broaden the solution σ(x, y, z), we do not require to resolve nonlinear equations.

5. Numerical Experiments

We will utilize the homogenization functions method to solve inverse conductivity issues of highly nonlinear elliptic PDEs through three examples from the second-order nonlinear issue to the highly nonlinear problem. We are interested in the stability of our scheme while the measured data are contaminated by random noise for varied problems. We can estimate the stability by increasing the varied random noise in the measured data:

$${\widehat{g}}_7(x_i,y_j)=g_7(x_i,y_j)+sR(i,j),$$

(28)

where R(i, j) are random numbers between [−1, 1], and s is the intensity of noise.

Besides, we deliberate the relative root-mean-square-error defined by

$$e_m(\sigma )=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^M{\displaystyle {\sum }_{j=1}^M{\displaystyle {\sum }_{r=1}^M{[{\sigma }_m(x_i,y_j,z_r)-\sigma (x_i,y_j,z_r)]}^2}}}}{{\displaystyle {\sum }_{i=1}^M{\displaystyle {\sum }_{j=1}^M{\displaystyle {\sum }_{r=1}^M{\sigma }^2(x_i,y_j,z_r)}}}}}$$

(29)

to estimate the accuracy of the retrieved conductivity function, in which we compare the numerically retrieved conductivity σ_m and the exact conductivity σ at M × M × M grid points (x, y, z), i, j, r = 1,…, M. We also employ the maximum absolute error (MAE_m) to estimate the accuracy of the numerical result, which is defined by

$${\mathrm{MAE}}_m:=\underset{i,j,r=1,\dots ,M}{\max }\left|{\sigma }_m(x_i,y_j,z_r)-\sigma (x_i,y_j,z_r)\right|.$$

(30)

All the computational algorithms were implemented to the Fortran code on the Microsoft Developer Studio platform in OS Windows 10 (64 bit) with i3-4160 3.60 GHz CPU and 16 GB memory.

5.1. Example 1

We deliberate the following 3D second-order nonlinear issue with a large domain

$$\begin{array}{l}\sigma (x,y,z)[v_{xx}(x,y,z)+v_{yy}(x,y,z)+v_{zz}(x,y,z)]+v_x(x,y,z){\sigma }_x(x,y,z)+v_y(x,y,z){\sigma }_y(x,y,z)\\ +v_z(x,y,z){\sigma }_z(x,y,z)+v^2(x,y,z)=P(x,y,z), (x,y,z)\in \mathsf{\Omega }, \end{array}$$

(31)

with an exact solution

$$v(x,y,z)={(x-5)}^2\exp (-2y)z^2, \sigma (x,y,z)=(x-{5)}^2+y^2+z^2, (x,y,z)\in (0,12)\times (0,3)\times (0,9).$$

(32)

We can obtain P(x, y, z) by inserting Equation (32) into Equation (31). Under the parameters: d = 12, e = 3, f = 9, p₁ = 2, q₁ = 3, p₂ = 3, q₁ = 2, and a noise with s = 0.4. We utilize the HFM to obtain the quite accurate solutions, as displayed in Figure 1, with the maximum absolute value of v being 3416.15 and the maximum absolute error (MAE_m) of v being $4.69\times {10}^{-2}$, and the MAE_m of σ is $9.53\times {10}^{-5}$. Note that the maximum result of σ is 139.0, and the result e_m(σ) = 6.86 × 10⁻⁷ is very small. For this instance, the CPU time is 3.18 s. From this instance, we find that these results are very accurate, time-saving, and stable even under the noise effect and large domain. Besides, we claim that the proposed algorithm can address the electrical conductivity coefficient function σ(x, y, z) accurately inside the cuboid with the aid of the estimations of the potential on the entire boundary and the present flux on a partially accessible boundary in a bounded cuboid.

5.2. Example 2

We ponder another 3D moderate nonlinear issue with the moderate domain as follows:

$$\begin{array}{l}\sigma (x,y,z)[v_{xx}(x,y,z)+v_{yy}(x,y,z)+v_{zz}(x,y,z)]+v_x(x,y,z){\sigma }_x(x,y,z)+v_y(x,y,z){\sigma }_y(x,y,z)\\ +v_z(x,y,z){\sigma }_z(x,y,z)-v^3(x,y,z)+6v_x(x,y,z)v_y(x,y,z)v_z(x,y,z)=P(x,y,z), (x,y,z)\in \mathsf{\Omega },\end{array}$$

(33)

with an exact solution

$$\begin{array}{l}v(x,y,z)=\sin x+\cos y+\exp \left(\mathrm{z}\right)+x^3+y^2+z^3,\\ \sigma (x,y,z)=2\exp (x+y+z), (x,y,z)\in (0,1)\times (0,2)\times (0,2).\end{array}$$

(34)

We can acquire P(x, y, z) by inserting Equation (34) into Equation (33). Under the parameters: d = 1, e = f = 2, p₁ = 2, q₁ = 5, p₂ = 2, q₂ = 4, and a noise with s = 0.4. We employ the HFM to acquire the accurate solutions, as demonstrated in Figure 2, with the maximum absolute value of v being 19.8, the maximum absolute error (MAE_m) of v being 0.218, and the MAE_m of σ is 0.237. Note that the maximum result of σ is 296.83, and the result e_m(σ) = 1.49 × 10⁻³ is rather small. For this case, the CPU time is only 2.83 s. From this example, we reveal that these numerical results are accurate, effective, and stable even under the noise effect and moderate area.

5.3. Example 3

Next, we contemplate the following 3D highly nonlinear problem

$$\begin{array}{l}\sigma (x,y,z)[v_{xx}(x,y,z)+v_{yy}(x,y,z)+v_{zz}(x,y,z)]+v_x(x,y,z){\sigma }_x(x,y,z)+v_y(x,y,z){\sigma }_y(x,y,z)\\ +v_z(x,y,z){\sigma }_z(x,y,z)+v^7(x,y,z)+v_x^2(x,y,z)+v_y^2(x,y,z)+v_z^2(x,y,z)=P(x,y,z), (x,y,z)\in \mathsf{\Omega },\end{array}$$

(35)

with an exact solution

$$\begin{array}{l}v(x,y,z)=\cos x\cos y\sin z+x^2{y}^2\cosh z,\\ \sigma (x,y,z)=\sinh z\sin (\sqrt{2}x)\cosh y+\cosh z\cos \left(\sqrt{2}x\right)\sinh y, (x,y,z)\in (0,1)\times (0,1)\times (0,1).\end{array}$$

(36)

We can obtain P(x, y, z) by inserting Equation (36) into Equation (35). Under the parameters: d = e = f = 1, p₁ = 2, q₁ = 9, p₂ = 3, q₂ = 8, and a noise with s = 0.5, which is a large level of noise 28%. We use the HFM to procure the accurate solutions, as illustrated in Figure 3, with the maximum absolute value of v being 1.79, the maximum absolute error (MAE_m) of v being $1.77\times {10}^{-2}$, and the MAE_m of σ is $0.181$. Note that the maximum result of σ is 23.08, and the result e_m(σ) = 1.06 × 10⁻² is quite small. For this numerical experiment, the CPU time is 4.52 s. From this numerical experiment, we exhibit that these numerical results are effective, accurate, and stable even under the large noise influence.

6. Conclusions

We addressed the 3D inverse conductivity issues of highly nonlinear elliptic equations by the homogenization functions method (HFM) in this article. The significant concept was constructed the single-parameter homogenization functions to produce practical foundations to broaden the solution and the conductivity function in the entire area. Although the 3D inverse conductivity issue is highly nonlinear, we still can effortlessly distinguish the unknown conductivity function by dealing with two sets of linear equations. Furthermore, to the author’s best knowledge, there is no study in the literature that the numerical algorithms for those three issues can supply more accurate results than the present results even under a large level of noise of 28%. The current method can be broadened to deal with the multi-dimensional inverse nonlinear steady-state PDEs, as demonstrated in Figure 4, and will be figured out in the near future.