Genetic Algorithm-Based Optimization Approach for Solving a Class of

Inverse Problems with Tikhonov Regularization

JAMAL DAOUDI1, CHAKIR TAJANI2∗

1,2SMAD Team, Department of Mathematics

Polydisciplinary Faculty of Larache

Abdelmalek Essaadi University

MOROCCO

Abstract: -In this paper, we are interested in solving the data completion problem for the Laplace equation. It

consists to determine the missing data on the inaccessible part of the boundary from overspecified conditions in

the accessible part. Knowing that this problem is severely ill-posed, we consider its formulation as an optimization

problem using Tikhonov regularization. Then, we consider an optimization approach based on adapted Real Coded

Genetic Algorithm (RCGA) to minimize the cost function and recover the missing data. The performed numerical

simulations, with different domains, illustrate the accuracy and efficiency of the proposed method with an adequate

regularization parameter, in addition to the good agreement between the numerical solutions and different noise

level of the given data.

Key-Words: Inverse Problem, Tikhonov regularization , Genetic Algorithm, Data completion problem,

Optimization

Received: July 9, 2023. Revised: September 28, 2023. Accepted: October 23, 2023. Published: November 14, 2023.

1 Introduction

According to [1], two problems are said to be ”in-

verses” of each other if the formulation of one of them

has implications for the other. This definition is some-

what arbitrary, but it does give the two problems a

symmetrical role. A more operational way to define

an inverse problem is that it consists to determine

causes knowing effects. In other words, an inverse

problem is the opposite of a direct problem, which

consists of deducing effects given known causes.

The data completion problem for the Laplace

equation is a important type of inverse problem which

aims to retrieve Cauchy data for the inaccessible part

of a boundary using measurements obtained from the

accessible one. This problem arises in various appli-

cations. For instance, in Electrocardiography (ECG),

[2], where the objective is to estimate the electrical

activity within the heart based on surface measure-

ments, assuming constant conductivity. In Electroen-

cephalography (EEG), [3], the goal is to estimate the

current source generated by neuronal activity, consid-

ering the electrical conductivity of tissue and mea-

surements taken at specific points on the head’s sur-

face. In fissure detection, [4], the aim is to pinpoint

and characterize cracks in a material using measure-

ments obtained at the material’s boundaries. In cor-

rosion detection, [5], the emphasis is to deduce the

shape of the corrosive boundary based on the col-

lected data.

The data completion problem for the Laplace

equation is known to be severely ill-posed problem. It

does not satisfy the conditions of the well-posed prob-

lem, [6], as it is sensitive to measurement errors and

can lead to unstable and inaccurate results. To rem-

edy this, we need to introduce regularization methods.

Among the well-known methods for inverse prob-

lems, we can cite; sequential function specification

methods, [7], iterative methods [8], [9], and Tikhonov

regularization, [10]. These methods try to find a bal-

ance between accuracy and stability.

In inverse problems, we aim to find the model pa-

rameters that best explain the observed data. This can

be done by defining an objective function which mea-

sures the discrepancy between the model output and

the data, and then minimizing that function. There

are two main classes of optimization methods: De-

terministic optimization methods are best suited for

continuously differentiable objective functions with a

limited number of local extrema. In contrast, stochas-

tic optimization methods involve random sampling of

the objective function across the entire feasible space

and can be applied to both differentiable and non-

differentiable objective functions.

Genetic algorithms (GAs) represent a subset of

metaheuristics, which includes artificial bee colony,

[11], particle swarm optimization, [12], ant colony

optimization, [13], and the Bat algorithm, [14],

among others. These methodologies draw inspiration

from natural phenomena and have demonstrated their

effectiveness in addressing a multitude of challenges

spanning various domains. Although, GAs do not of-

fer a guarantee of identifying the absolute best solu-

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tion, they excel in identifying solutions that closely

approach the optimal outcome within a reasonable

time frame.

GAs have demonstrated notable success in tack-

ling a range of inverse problems. For instance, they

have been applied to estimate interaction mechanisms

among system components, [15], solve groundwater

source identification challenges, [16], address non-

linear IHCP scenarios, [17], compute minimizers of

Tikhonov functionals, [18], detect the shape of un-

known boundary segments in the solution domain

for the Laplace equation, [19], identification of the

Robin coefficient, [20], solve support vector

machines (SVMs), [21], reconstruct epicardial

potentials from body surface potentials, [22], identify

isotropic inclu-sions through a single boundary

measurement, [23], solve IHCP by minimizing a

modified version of Tikhonov’s functional, [24],

and address the inverse problem of aerodynamic

shape optimization, [25].

The main feature of our approach is the use of an

optimization approach based on a Real Coded Genetic

Algorithm (RCGA) as a powerful optimization tool

capable of handling complex, non-monotonic objec-

tive functions. In [26], the authors addressed the data

completion problem for the Laplace equation by re-

constructing the Dirichlet condition in the inaccessi-

ble part of the boundary of a regular domain, without

resorting to Tikhonov regularization. In this study, we

extend this approach to two scenarios: regular and

irregular domains, aiming to estimate the unknown

Dirichlet and Neumann conditions. In addition, we

assess the stability of the proposed approach by ex-

amining the optimal choice of the regularization pa-

rameter. Finally, we conduct a thorough analysis of

the solution’s stability under varying levels of noise.

The rest of this paper is organized as follows: Sec-

tion 2 provides the mathematical formulation of the

problem, including the differences between the for-

ward and inverse problem, and its formulation as an

optimization one. Section 3 provides a brief overview

of GAs, in addition to the proposed approach based on

adapted RCGA . Section 4 presents numerical results

and discussion. Finally, Section 5, summarizes the

main results and provides concluding thoughts.

2 Mathematical formulation

2.1 Forward and inverse problem

We consider the following Elliptic Cauchy problem:







−∆u= 0 in Ω

u=fon Γc

∂nu=gon Γc

(1)

where Ω⊂R2is an open bounded set with Lips-

chitz continuous boundary Γ, which consists of tow

disjointed parts Γ = Γc ∪ Γi, with Γc is the accessi-

ble part of the boundary and Γi is the inaccessible

part (Fig.1).

Figure 1: Example of the domain

The functions fand gare the given Dirichlet and

Neumann conditions, and ∂nuis the normal deriva-

tive of the unknown function u.

Forward problem: the aim of the forward prob-

lem is to find u, by solving the Cauchy problem for

Laplace equations from knowledge of the Cauchy

data uD=u/Γior the uN=∂nu/Γi.

Inverse problem: the aim of the inverse problem

is to estimate the Cauchy data (uD, uN)on the inac-

cessible part of the boundary Γifrom the knowledge

of the Cauchy data fand gon the accessible part of

the boundary Γc.

2.2 Optimization problem

Since the uDand uNon the boundary Γiis to be de-

termined, two direct problems are considered:

(PD) : 





−∆u= 0 in Ω

∂nu=gon Γc

u=uDon Γi

(2)

(PN) : 





−∆u= 0 in Ω

u=fon Γc

∂nu=uNon Γi

(3)

It is important to note that if uD∈H1/2 (Γi)and

g∈H−1/2 (Γc)(respectively, f∈H1/2 (Γc)and

uN∈H−1/2 (Γi)), then it exists a unique solution

u(uD, g)(respectively, u(uN, f)) for the direct prob-

lem Eq.(2) (respectively, Eq.(3)). Our goal is to find

uD(respectively, uN) that satisfies:

u(uD, g) =fon Γc

(respectively, ∂nu(uN, f) =gon Γc)(4)

In doing so, we attempt to minimize the two well-

posed least-squares functional JDand JNdefined

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by:

JD(u) = 1

2ku(uD, g)−fk2

L2(Γc)+α1

2kuDk2

L2(Γi)

(5)

JN(u) = 1

2k∂nu(uN, f)−gk2

L2(Γc)+α2

2kuNk2

L2(Γi)

(6)

where, α1and α2are the regularization parameters

and α1

2kuDk2

L2(Γi)and α2

2kuNk2

L2(Γi)are the well-

known Tikhonov regularization functional. The regu-

larization serves to stabilize the solution. The process

of minimizing JDand JNrepresents a delicate bal-

ance between achieving data fidelity and stabilizing

the solution. The selected values for α1and α2di-

rectly impact the stability of the solution.

3 Genetic Approach for the inverse

problem

3.1 Overview of Genetics Algorithms

Metaheuristics are a class of optimization algorithms

that are designed to solve complex optimization prob-

lems for which traditional methods might be imprac-

tical or ineffective, [27]. They provide a general

framework for exploring and searching large solution

spaces to find near-optimal solutions. Metaheuristics

are particularly useful when dealing with problems

that involve non-linearities, discontinuities, and other

challenging characteristics. Metaheuristic algorithms

are usually inspired by natural phenomena, social be-

haviors, or problem-specific strategies. They don’t

guarantee finding the global optimal solution but aim

to find good approximations in a reasonable amount

of time.

Genetic Algorithms (GAs) are a classic example

of metaheuristic algorithms based on the mechanisms

of natural selection and genetics, [28]. It combines a

”survival of the fittest” strategy with a random struc-

tured exchange of information. For a problem for

which a solution is unknown, a set of possible so-

lutions is created randomly. The characteristics (or

variables to be determined) are then used in gene se-

quences that will be combined with other genes to

form chromosomes and then individuals. Each solu-

tion is associated with an individual, and this individ-

ual is evaluated and classified according to its resem-

blance to the best, but still unknown, solution to the

problem. It can be shown that through a natural selec-

tion process inspired by Darwin’s theory of evolution,

this method gradually approaches a solution. Natu-

ral selection is a potent force in evolution, leading to

the development of intricate and well-adapted organ-

isms. It operates on the principle that organisms with

traits well-suited to their environment are more likely

to survive and reproduce. This dynamic ultimately

yields a population finely attuned to their surround-

ings. Genetic algorithms replicate this process by es-

tablishing a pool of potential solutions and favoring

the reproduction of the most successful ones. Selec-

tion is based on higher fitness scores, and as iterations

progress, the population refines, producing solutions

with increasingly superior fitness scores, approaching

an optimal solution, Fig.2 shows a flowchart of the se-

quential basic steps of a genetic algorithm.

Figure 2: Flowchart of genetic algorithm

GAs are a powerful tool for solving difficult prob-

lems. It is often used in practice and has been shown

to be effective in a wide variety of problems. How-

ever, it is important to note that the GA is not guar-

anteed to find the optimal solution to a problem. It is

also important to choose the parameters of the genetic

algorithm carefully, such as the size of the population

and the adapted crossover and mutation.

3.2 A REAL CODED GENETIC

ALGORITHM

In this article, we employ a real-coded genetic algo-

rithm (RCGA) to address the data completion prob-

lem for the Laplace equation. In a RCGA, each chro-

mosome represents a vector of real parameters, where

each gene corresponds to a real number, and each al-

lele corresponds to a real value.

We have chosen to use an RCGA because, as a gen-

eral rule, RCGA demonstrate superior performance

compared to binary-coded GAs when it comes to

high-precision optimization problems. Additionally,

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the encoding of variables as floating-point numbers

allows for a seamless implementation of finely-tuned

local optimization processes, [29], [30]. In this al-

gorithm, we incorporate a specialized operator that

leverages the floating-point representation of the so-

lution space, thereby enhancing the convergence rate

of the algorithm

3.3 Genetic operators

3.3.1 Arithmitic crossover

Crossover is a genetic operation that combines two

parents to create offspring. Crossover works by swap-

ping parts of the chromosomes between the parents.

However, crossover is not performed on every pair of

parents. The probability of crossover is controlled by

a parameter called the crossover probability pc.

Typically, parents are denoted as:

F(1) =F(1)

1, . . . , F(1)

n

F(2) =F(2)

1, . . . , F(2)

n(7)

Similar representation is also used for offspring:

C(1) =C(1)

1, . . . ., C(1)

n

C(2) =C(2)

1, . . . , C(2)

n(8)

The number of parents and offspring may vary

depending on the crossover operator considered. In

arithmetic crossover, [31], two parents produce two

offspring. Thus the parents defined by equation

Eq.(7), produce two offspring, which can be deter-

mined as follows:

C(1)

i=αiF(1)

i+ (1 −αi)F(2)

C(2)

i=αiF(2)

i+ (1 −αi)F(1)

(9)

The αiare uniform random numbers between 0 and

It should be noted that, we talk about the uniform

arithmetical crossover when the αiare constant.

However, in the non-uniform arithmetical crossover,

the αimay vary with the progression of generations.

3.3.2 Power mutation

The mutation operator introduces a potential modifi-

cation in selected elements of specific chromosomes.

The original chromosome, denoted by C, is defined

as:

C= (C1,C2, . . . , Ci, . . . , Cn)(10)

where, Cirefers to the element to be mutated. The

chromosome Cafter mutation is given by:

C0= (C1,C2, . . . , C0

i, . . . , Cn)(11)

In this new configuration, C0

irepresents the mutated

value of C.

The Power Mutation (PM), [32], for real-coded ge-

netic algorithm is one of the commonly used mutation

operators. This operator relies on a power distribu-

tion, defined by the following function:

f(x) = ξxξ−1,0≤x≤1(12)

The corresponding density function is:

F(x) = xξ,0≤x≤1(13)

In these equations, ξsignifies the distribution index.

This mutation serves to generate a solution C0in prox-

imity to a parent solution C. It starts by generating a

uniform random number tbetween 0 and 1. Addition-

ally, it creates a random number ζthat adheres to the

aforementioned distribution.

To compute the mutated solution, the following for-

mula is applied:

i=Ci+ζ·(ui− Ci)if t≥η

Ci−ζ·(Ci−li)if t < η (14)

where, liand uidenote the lower and upper bounds

of the decision variable, respectively. ηis a uniformly

distributed random number ranging from 0 to 1.

The extent of mutation is governed by the muta-

tion index ξ. When ξis small, one can expect less

disruption in the solution. Conversely, with larger ξ

values, greater diversity is achieved. The likelihood

of generating a mutated solution C0to the left (or right)

of Cis proportionate to the distance of Cfrom li(or

ui), ensuring that the mutated solution always remains

feasible.

3.4 Process to estimate the Cauchy data

In this section, we will outline the computational steps

involved in using the RCGA to address the consid-

ered inverse problem. The problem can be formally

stated as follows: Given the Cauchy data fand g

on the accessible part of the boundary Γc, our ob-

jective is to propose a polynomial approximation of

the Cauchy data uDand uNon Γiusing the RCGA.

The RCGA will systematically explore potential so-

lutions through iterative updates, evaluating each in-

dividual’s fitness using the Finite Element Method

(FEM) to derive corresponding fitness values. Ulti-

mately, the optimal set of parameters will be deter-

mined by selecting the individual with the highest fit-

ness value.

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Algorithm 1 Algorithm for solving the data comple-

tion problem for the Laplace equation via GA

1: Parameter setting:

•N: Population size,

•pc: Probability of Crossover,

•pm: Probability of Mutation,

•k: Maximum number of Generations,

•liand uiare the lower and the upper bounds,

respectively.

2: Initialize population: UD(0)p∼ U(li, ui)for

p= 0,· · · , N .

3: for it = 1,· · · , k do

4: for p= 0,· · · , N do

5: Solve the direct problem Eq.(15).

Pp:









∆up= 0 in Ω

up=UD(it−1)pon Γi

∂nup=gon Γc

(15)

6: Evaluate each individual using Eq.(5).

7: end for

8: Uit = [u0,· · · , uN]

9: Apply selection us=Se(Uit)

10: Apply crossover uc=Cr(us)using Eq.(9).

11: Apply mutation um=Mu(uc)using Eq.(4).

12: Sort and select the best individual

(UD(it))p=argmin (JD(um),JD(uc)).

13: end for

4 Numerical Results and Discussion

In this section, we present the numerical results

achieved through the genetic procedure introduced in

Section 3, coupled with the Finite Element Method

(FEM), for solving the two mixed well-posed prob-

lems, Eq.(2) and Eq.(3), corresponding to the Laplace

equation. This analysis encompasses two case regu-

lar and irregular 2D domains. Furthermore, we assess

the stability of the genetic procedure under data per-

turbation induced by noise.

When dealing with inverse problems in the real world,

the boundary data is obtained through experimental

measurements, which makes it susceptible to mea-

surement errors. In our testing scenarios, we use the

following equation to produce synthetic noisy data:

uper =uexact ×(1 + ν×θ)on Γc(16)

where θis a random number that follows a uniform

distribution between −1 and 1, and the level of noise

is determined by the parameter ν.

The genetic operators and the parameters used for

this genetic algorithm were taken to be as follows:

• Maximum number of Generations k= 200,

• Population size Np= 70,

• Rate of crossover Rc= 0.8,

• Non-uniform Arithmetic Crossover with pc=

0.9,

• Random selection,

• Power mutation with pm= 0.1.

4.1 Example 1: Irregular domain

In the first case, the numerical tests are made on a

unit square domain Ω = (0, 1)2 (Fig.3). The

boundary Γ is divided into tow parts:

Γi={(0, y) : 0 < y < 1},

Γc= Γ\Γi.(17)

Figure 3: Unit square.

Figure 4: The analytical solution in 2D.

The over-specified Cauchy data in Γcare extracted

from the analytic solution, given by:

uex(x, y) = cos(x)cosh(y) + sin(x)sinh(y)(18)

Fig.4 presents the analytical solution in the whole

domain.

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4.1.1 Choice of regulation parameter

Fig.5 and Fig.6 show the objective functions JDand

JNwith respect to various values of the regulariza-

tion parameters α1and α2, respectively. These results

are obtained for the data completion problem using

exact boundary data extracted from Eq. (18). These

figures reveal that, considering JDand JN, the op-

timal values for the regularization parameters α1and

α2are 1×10−5and 1×10−4, respectively, for uD

and uNon Γi.

Figure 5: The cost function JDfor various values of

α1.

Figure 6: The Cost function JNfor various values of

α2.

Additionally, Fig.7 and Fig.8 show that the objec-

tive functions JDand JNsignificantly decrease dur-

ing the initial iterations. This decrease indicates that

the algorithm is successfully approaching the min-

imum of the objective function. As the iterations

progress, the convergence rate of the objective func-

tion gradually slows down, but it eventually reaches

a low value by iteration k= 200. These observations

suggest that the obtained solution is highly accurate

and provides an excellent fit to the available data.

Figure 7: The convergence of the cost function JD

during the genetic process.

Figure 8: The convergence of the cost function JN

during the genetic process.

Fig.9 and Fig.10 show how the numerical solution

gets closer and closer to the analytical solution as the

number of iterations increases. Initially, the numeri-

cal solution is significantly different from the analyti-

cal solution, but the difference quickly decreases with

each iteration. This shows that the iterative method is

effective at solving the problem under study.

Figure 9: The numerical solution uDon Γifor a spe-

cific iterations.

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Figure 10: The numerical solution uN on Γi for a spe-

cific iterations.

4.1.2 Stability of the Proposed Method

Fig.11 and Fig.12 compare the numerical solution to

the analytical solution for different levels of noise in

the measurement data ν = 1%, 2%, 5%, 7%. As the

noise level increases, the numerical solution de-

viates slightly from the exact solution. However, the

difference between the numerical and exact solutions

remains small, even when the noise level reaches a

high value of 7%. This suggests that the numerical

solutions obtained for problem Eq.(18) in this study

are stable with respect to the amount of noise ν added

to the input data.

Figure 11: The numerical solution uDfor various lev-

els of noise.

Figure 12: The numerical solution uNfor various lev-

els of noise.

Fig.13 and Fig.14 show the cost function for differ-

ent noise levels, ν= 1%,2%, 5%, and 7%. The fig-

ures indicate that as the noise level increases, the cost

function also increases, which means that the fit to the

data becomes less precise. However, even with higher

noise levels, the cost function remains relatively low,

suggesting that the numerical solution still provides a

satisfactory fit to the data.

Figure 13: JDfor various levels of noise.

Figure 14: JNfor various levels of noise.

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4.2 Example 2: regular domain

In the second case, the numerical tests are performed

on an unit disc Fig.15.

The boundary of this domain is divided into two parts:

Γi=(x, y) : x2+y2= 1, y > 0, x > 0,

Γc= Γ\Γi.

(19)

Figure 15: Unit disc.

The over-specified Cauchy data in Γcare extracted

from the analytic solution:

uex(x, y) = cos(x)exp(y)(20)

Fig.16 shows the analytical solution in whole do-

main.

Figure 16: The analytical solution in 2D.

4.2.1 Choice of regulation parameter

Fig.17 and Fig.18 show the objective functions JD

and JNwith respect to various values of the regu-

larization parameters α1and α2, respectively, for the

data completion problem using exact boundary data

extracted from Eq. (20). The optimal value for the

regularization parameters α1and α2is 1×10−5for

uDand uNon Γi, considering both JDand JN.

Figure 17: Objective function JDfor various values

of α1.

Figure 18: Objective function JNfor various values

of α2.

Figure 19: The convergence of the cost function JD

during the genetic process.

Fig.19 and Fig.20 show that the objective func-

tions JDand JNdecrease significantly in the early

iterations. This indicates that the algorithm is suc-

cessfully converging to the minimum of the objective

function. As the iterations progress, the convergence

rate slows down, but the objective function eventually

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reaches a low value by iteration k= 200. These ob-

servations suggest that the obtained solution is highly

accurate and provides a good fit to the available data.

Figure 20: The convergence of the cost function JN

during the genetic process.

Figure 21: The numerical solution uDon Γifor a spe-

cific iterations.

Figure 22: The numerical solution uNon Γifor a spe-

cific iterations.

Fig.21 and Fig.22 illustrate the progressive con-

vergence of the numerical solution to the analytical

solution as the number of iterations increases. Ini-

tially, the numerical solution exhibits significant devi-

ation from the analytical solution, but this discrepancy

diminishes rapidly with each iteration. This demon-

strates the efficacy of the iterative method in effec-

tively resolving the problem under consideration.

4.2.2 Stability of the Proposed Method

Figure 23: The numerical solution uDfor various lev-

els of noise.

Fig.23 and Fig.24 compare the numerical solution to

the analytical solution for different noise levels in the

measurement data ν= 1%,2%,5%,7%. The numer-

ical solution deviates slightly from the analytical solu-

tion as the noise level increases. However, the differ-

ence between the numerical and analytical solutions

remains small, even when the noise level reaches a

high value of 7%. This suggests that the numerical

solutions obtained for problem Eq.(20) in this study

are stable with respect to the amount of noise νadded

to the input data.

Figure 24: The numerical solution uNfor various lev-

els of noise.

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Figure 25: JDfor various levels of noise.

Figure 26: JNfor various levels of noise.

Fig.25 and Fig.26 show the cost function for dif-

ferent noise levels, ν= 1%,2%, 5%, and 7%. The

figures indicate that the cost function increases as the

noise level increases. This means that the fit to the

data becomes less precise as the noise level increases.

However, even with higher noise levels, the cost func-

tion remains relatively low, suggesting that the nu-

merical solution still provides a satisfactory fit to the

data.

5 Conclusion

In this paper, we address the challenging ill-posed in-

verse problem associated with the Cauchy problem

for the Laplace equation. We propose a novel ap-

proach to solve this problem by converting it into

an optimization problem. We use the RCGA with

Tikhonov regularization to solve the optimization

problem. The effectiveness of our approach is eval-

uated through numerical experiments conducted on

both regular and irregular domains. The results

demonstrate the efficacy of RCGA, enhanced with

adapted genetic operators, in successfully solving the

Cauchy problem associated with the Laplace equa-

tion.

Acknowledgment:

The authors would like to thank the editors and the

anonymous reviewers for their comments and

suggestions.

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Contribution of Individual Authors to the

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search, at all stages from the formulation of the prob-

lem to the final findings and solution.

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article.

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