L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coecients in Heterogeneous Media in Diusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Estimation of Spatially

Dependent Coecients

in Heterogeneous Media

in Diusive Heat Transfer

Problems

ARTICLE HISTORY

Received 24 February 2024

Accepted 19 April 2024

Lucas Lopes da Silva Costa

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

costa.lucas@iprj.uerj.br

ORCID: 0000-0003-1940-0875

Eduardo Cunha Classe

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

eduardo.classe@iprj.uerj.br

ORCID: 0000-0002-2405-3946

Lucas da Silva Asth

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

lucas.asth@iprj.uerj.br

ORCID: 0000-0002-6189-1068

Luiz Alberto da Silva Abreu

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

luiz.abreu@iprj.uerj.br

ORCID: 0000-0002-7634-7014

Diego Campos Knupp

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

diegoknupp@iprj.uerj.br

ORCID: 0000-0001-9534-5623

Leonardo Tavares Stutz

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

ltstutz@iprj.uerj.br

ORCID: 0000-0003-3005-765X

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

Estimation of Spatially Dependent Coefficients in

Heterogeneous Media in Diffusive Heat Transfer

Problems

Lucas Lopes da Silva Costa

Postgraduate Program in

Computational Modeling (Universidade

do Estado do Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

costa.lucas@iprj.uerj.br

ORCID: 0000-0003-1940-0875

Luiz Alberto da Silva Abreu

Dept. de Engenharia Mecânica e

Energia (Universidade do Estado do

Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

luiz.abreu@iprj.uerj.br

ORCID:

0000-0002-7634-7014

Eduardo Cunha Classe

Postgraduate Program in

Computational Modeling (Universidade

do Estado do Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

eduardo.classe@iprj.uerj.br

ORCID

: 0000-0002-2405-3946

Diego Campos Knupp

Dept. de Engenharia Mecânica e

Energia (Universidade do Estado do

Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

diegoknupp@iprj.uerj.br

ORCID:

0000-0001-9534-5623

Lucas da Silva Asth

Postgraduate Program in

Computational Modeling (Universidade

do Estado do Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

lucas.asth@iprj.uerj.br

ORCID:

0000-0002-6189-1068

Leonardo Tavares Stutz

Dept. de Engenharia Mecânica e

Energia (Universidade do Estado do

Rio de Janeiro)

Polytechnic Institute of Rio de Janeiro

Nova Friburgo, Brazil

ltstutz@iprj.uerj.br

ORCID : 0000-0003-3005-765X

Abstract— This article addresses the solution to the inverse

problem in a one-dimensional transient partial differential

equation with a source term, commonly encountered in heat

transfer modeling for diffusion problems. The equation is utilized

in a dimensionless form to derive a more general solution that is

applicable in various contexts. The Transition Markov Chain

Monte Carlo (TMCMC) method is utilized to estimate spatially

variable thermophysical properties within the equation. This

approach involves transitioning between probability densities,

gradually refining the prior distribution to approximate the

posterior distribution. The results indicate the effectiveness of the

TMCMC method in addressing this inverse problem, and it offers

a robust methodology for estimating spatially variable

coefficients.

Keywords—Inverse Problem, Transition Markov Chain

Monte Carlo (TMCMC), Heterogeneous Media, Estimation of

Variable Coefficients, Heat Conduction

I. INTRODUCTION

The identification of thermophysical properties is a

fundamental process in various fields of science and

engineering, where understanding these properties is

essential to comprehend material behavior or identify them.

Properties like thermal conductivity, density, and specific

heat directly influence how a material responds to

temperature changes [4]. When modeling the heat transfer

process, these properties can be expressed through

parameters within partial differential equations [3] [5] [10].

This, in turn, paves the way for varied approaches in

estimating these parameters, ranging from direct methods to

indirect approaches, each carrying its own advantages and

disadvantages.

Within direct methods, direct experimental

measurements on thermophysical properties of material

samples are conducted. While recognized for their

precision, these methods often prove to be costly, time-

consuming, and in certain cases, intrusive to the material

under analysis.

On the other hand, indirect methods offer an attractive

alternative. They do not demand direct measurements of

thermophysical properties but instead explore relationships

between these properties and other variables that can be

more easily measured [1][11]. However, indirect methods

often rely on assumptions and models to establish these

relationships, introducing uncertainties in the estimation.

One particular approach that has gained prominence is

the utilization of Bayesian frameworks, such as the

Transitional Markov Chain Monte Carlo (TMCMC)

method, to estimate thermo-physical properties. The

distinctive feature of Bayesian methods is the incorporation

of prior information, i.e., prior knowledge about the

properties in question [11]. TMCMC, for instance,

constructs a probability distribution that takes into account

both experimental data and prior information, resulting in

more reliable estimates and quantified uncertainties [9] [11].

The aim of this work is to demonstrate the utilization of

the TMCMC technique for computing unspecified

parameters in a differential equation, proposing three

distinct models of their spatial variation. The obtained

results demonstrate the effectiveness of the TMCMC

method in solving the inverse problem, providing a robust

methodology for this type of problem. Furthermore, this

work may validate the use of TMCMC as a reliable and

versatile tool for parameter estimation in different contexts,

paving the way for more advanced applications, such as

characterizing new materials with different thermal

properties.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coefficients in Heterogeneous Media in Diffusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

II. M

ETHODOLOGY

In this section, the methodology employed in this study

will be presented. In the subsequent subsections, the

mathematical formulation of the physical problem will be

explained, and the intricacies of Transition Markov Chain

Monte Carlo (TMCMC) will be explored. Introduced by [8],

this approach draws inspiration from the adaptive

Metropolis-Hastings technique and employs Monte Carlo

principles through Markov Chains. A comprehensive

overview of the TMCMC method will be provided,

including discussions on its fundamental principles and

procedural steps. The aim of this exposition is to provide a

clear understanding of how the TMCMC method operates,

especially in the context of estimating coefficients in

solving inverse problems.

A. Mathematical Formulation

In this section, the mathematical formulation underlying

the physical phenomenon of heat transfer within a material

of length L=10 will be delved into. This investigation

considers Neumann boundary conditions coupled with a

constant initial condition. The primary objective of this

section is to model the dynamic evolution of temperature,

represented as T(x,t), across space and time.

󰇛

󰇜

∂T(x,t)

∂t



∂

∂x

󰇡

k(x)

∂T(x,t)

∂x

󰇢

󰇛

󰇜

(1a)

In this context, k(x) represents the thermal conductivity

coefficient, a measure characterizing an intrinsic ability of a

material to conduct heat. In turn, the coefficient w(x),

known as the thermal diffusion coefficient, incorporates the

inherent thermal diffusivity property of the material in

question. The term p(x) refers to an internal heat source

within the material. The spatial domain is defined in the

interval 0 < x < L, while time is restricted to positive values,

t > 0, where L denotes the physical extent of the material.

These parameters are expressed in terms of Neumann

boundary conditions:

∂T(x,t)

∂x

󰇻

x=0

= 0

(1b)

∂T(x,t)

∂x

󰇻

x=L

= 0

(1c)

These expressions characterize the rates of heat transfer

at the material boundaries, and the initial condition is

established as shown below, where T

is a constant

representing the initial temperature distribution within the

material.

T(x,0)



(1d)

This study examines Equation (1) in three distinct

scenarios: firstly, when both coefficients k(x) and w(x) are

kept constant; secondly, when they are modeled as linear

functions; and finally, when they follow exponential

functions. The primary aim of these analyses is to assess the

Transitional Markov Chain Monte Carlo (TMCMC) method

ability to accurately estimate these parameters.

Transitional Markov Chain Monte Carlo (TMCMC)

The Transitional Markov Chain Monte Carlo (TMCMC)

method, as proposed by [8], draws inspiration from the

adaptive Metropolis-Hastings method as suggested by [6],

and it is grounded on the Monte Carlo methodology through

Markov Chains. The main idea is to avoid direct sampling

of difficult probability distributions by sampling from a

series of intermediate distributions that converge to the

posterior distribution [8].

This method inherits the advantages of Adaptive

Metropolis-Hastings (AMH), which is suitable for very

sharp, flat, and multimodal probability density functions

(PDFs), and is particularly efficient in high-dimensional

PDFs. Additionally, the TMCMC method has the capability

to automatically select intermediate PDFs, enhancing its

versatility and effectiveness in sampling complex

distributions [8].

The posterior distribution is calculated using Bayes'

theorem, described by Equation (2) [9], as shown below:

π(P





π(P)π(Y| P) (2)

But, as mentioned earlier, the TMCMC method avoids

computing the distribution in this way, in order to employ a

series of intermediate distributions as follows:

(P)



󰇛

󰇜

π󰇛󰇜

 (3)

The steps for the TMCMC algorithm are outlined as

follows [7]:

1. Samples {P₀,₁, P₀,₂, ..., P₀,ₙ} are acquired from the

prior distribution f₀(P) = π(P) using Monte Carlo

simulation. The process initiates with p₀ set to 0, and steps

2 and 3 are repeated for j = {0, 1, 2, ...}.

2. Likelihood distributions π(Y | Pⱼ,₁), ..., π(Y | Pⱼ,ₙ) are

computed, and the weights wⱼ,ₖ = π(Y | Pⱼ,ₖ)^(pⱼ₊₁ - pⱼ) are

determined. The selection of pⱼ₊₁ ensures that the coefficient

of variation (COV) of the importance weights {wⱼ,₁, ...,

wⱼ,ₙ} equals 100%. Additionally, normalized weights {wⱼ,₁,

..., wⱼ,ₙ} are calculated.

3. Based on the normalized weights {wⱼ,₁, ..., wⱼ,ₙ},

candidates are randomly chosen from {Pⱼ,₁, Pⱼ,₂, ..., Pⱼ,ₙ}. A

new candidate is proposed according to the distribution

N(Pⱼ,ₖ, Σⱼ), forming the sequence {Pⱼ₊₁,₁, Pⱼ₊₁,₂, ..., Pⱼ₊₁}. The

covariance matrix Σⱼ is defined by an equation.

∑

= β

∑

j=1

j,k

󰇣P

j,k





×P

j,k









󰇤 (4a)

with



∑

l=1

j,1

. P

j,1

∑

l=1

j,l

 (4b)

The parameter β is a factor that scales the distribution of the

covariance matrix proposal [8].

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

III. R

ESULTS

In this section, the study conducted a comparative

analysis of the parameters w(x) and k(x) estimated in the

Partial Differential Equation (PDE), as mentioned earlier.

Three distinct variations were considered: constant, linear,

and exponential. To obtain experimental measurements in

the direct problem, three spatial measurement points were

selected: x=2.5, x=5.0, and x=7.5, resulting in a total of 101

measurements for each sensor within the analyzed time

interval. The standard deviations of the measurement errors

σ were defined as 0.5 for the constant case, 1.0 for the linear

case, and 1.5 for the exponential case. Therefore, these

experimental measurements are now referred to as actual

measurements. It is worth noting that these measurement

errors were chosen to be proportional to the measured

temperature, specifically around 2%. This choice was based

on the averaging of experimental measurements for each

model.

The Transition Markov Chain Monte Carlo (TMCMC)

method was employed to simultaneously obtain estimates of

these parameters, and the results were compared for each

variation. The study was conducted with a total of 20,000

samples for the Constant model, 50,000 for the Linear

model, and 50,000 for the Exponential model and β = 0.1 in

all three situations. In order to simulate a source with

characteristics of a smooth step curve, the following

mathematical formulation for p(x) was used.

 p

(

)

[

1+e

100

(

0.5L

)

]

(6)

The specific formulations and characteristics of the

analyzed models for w(x) and k(x) are detailed in the

following sections, accompanied by their respective

mathematical formulations and corresponding results.

It is important to emphasize that all results presented in

this work were generated using the computational platform

Wolfram Mathematica 12.0, operating on a desktop

equipped with a Central Processing Unit (CPU) AMD

Ryzen Threadripper1950x clocked at 4 GHz and 64 GB of

DDR4 type RAM. The adopted operating system is

Windows 10 in its 64-bit version.

A. Model with Constant Coefficients

Firstly, the TMCMC method was applied to estimate the

parameters of a model with constant coefficients. In the

direct problem, k(x) = 1 and w(x) = 1 were used. Table I

below shows the exact values of the coefficients, as well as

the results obtained after the method was applied.

TABLE I. ESTIMATED RESULTS VIA TMCMC – MODEL WITH

CONSTANT COEFFICIENTS

Parameter

Exact

Value

Estimated

Standard

Deviation

Error(%)

1.00 1.00056 0.00142 0.056

1.00 1.00641 0.01269 0.641

The table analysis reveals that the method was effective

in parameter estimation, resulting in reduced relative errors

and standard deviations. Fig. 1 depicts a comparison

between estimated values, represented by the blue curve,

and actual measurements denoted by red points, along with

the 95% confidence interval depicted by the blue shaded

region. On the other hand, Fig. 2 presents the residual

analysis of the three utilized sensors, along with their

corresponding linear regression.

Fig. 1. Temperature Measurements with 95% confidence interval -

Model with Constant Coefficients

Fig. 2. Residual analysis – Model with Constant Coefficients

Through the analysis of the graphs, it is evident that the

estimated measurements exhibit high agreement with the

actual measurements. The residual analysis reveals that the

differences between these measurements are close to zero

across the entire domain, as evidenced by the linear

regression. Fig.3 and Fig. 4 illustrate the histogram of the

estimates for the parameters w(x) and k(x). It is important

to note that all estimated samples were normalized by their

respective exact values, rendering the histogram

dimensionless.

Fig. 3. Histogram of the w

estimated parameter - model with constant

coefficients

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coefficients in Heterogeneous Media in Diffusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Fig. 4. Histogram of the k

estimated parameter - model with constant

coefficients

The means of the estimated values are close to the exact

values, as expected. It is noteworthy that the estimation for

the parameter w(x) was more accurate than for the

parameter k(x).

B. Model with Linear Coefficients

Similarly to the model with constant coefficients, Table

II presents the values used in solving the direct problem for

the case of linear coefficients in the form w

(

)

x+w

and k

(

)

x+k

.The corresponding estimates, standard

deviations, and relative errors are also indicated.

TABLE II. ESTIMATED RESULTS VIA TMCMC - MODEL

LINEAR WITH LINEAR COEFFICIENTS

Parameter

Exact

Value

Estimated

Standard

Deviation

Error(%)

0.09 0.08992 0.00185 0.091

0.10 0.10021 0.00857 0.215

0.09 0.09099 0.00552 1.101

0.90 0.09117 0.02360 8.832

Except for the parameter k

, all estimates yielded

relative errors of less than 3%. Fig. 5 illustrates the

comparison between the measurements of estimated values,

represented by the blue curve, and actual measurements

denoted by red points, along with the 95% confidence

interval depicted by the blue shaded region. Meanwhile,

Fig. 6 displays the residual analysis between these two

measurements and the linear regression of the points.

Fig. 5. Temperature Measurements with 95% confidence interval –

Model with Linear Coefficients

Fig. 6. Residual analysis - Model with Linear Coefficients

Once again, a remarkable resemblance is observed

between the estimated measurements and the actual

measurements. However, it is noticeable that for the linear

case, the confidence interval encompasses all the conducted

measurements. The residual analysis demonstrates that the

differences between the measurements are close to zero

across the entire domain, as shown by the linear regression.

Fig. 7, Fig. 8, Fig. 9 and Fig. 10 displays the histograms of

parameter estimates for this case.

Fig. 7. Histogram of the w

estimated parameter - model with linear

coefficients

Fig. 8. Histogram of the w

estimated parameter - model with linear

coefficients

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

Fig. 9. Histogram of the k

estimated parameter - model with linear

coefficients

Fig. 10. Histogram of the k

estimated parameter - model with linear

coefficients

The means of the estimated values approach the exact

values, as expected, reinforcing the reliability of the

TMCMC method. It is worth noting that the estimate for the

parameter w(x) reveals superior precision compared to the

parameter k(x), suggesting the need for a more in-depth

analysis to comprehend the underlying causes of this

discrepancy.

C. Model with Exponential Coefficients

Finally, Table III showcases the values and estimates of

the parameters associated with the case of exponential

coefficients in the form w

(

)

and k

(

)

TABLE III. ESTIMATED RESULTS VIA TMCMC - MODEL WITH

LINEAR COEFFICIENTS

Parameter

Exact

Value

Estimated

Standard

Deviation

Error (%)

0.10 0.10159 0.00157 1.595

0.25 0.24738 0.00249 1.046

0.10 0.10066 0.00226 0.663

0.25 0.24637 0.00487 1.450

Once again, the estimates resulted in significantly

reduced relative errors and standard deviations. Fig. 11

illustrates the comparison graphs between the

measurements with the estimated parameters, represented

by the blue curve, and the actual measurements denoted by

red points, along with the 95% confidence interval depicted

by the blue shaded region. Meanwhile, Fig. 12 displays the

residual analysis between these measurements with the

linear regression of the points.

Fig. 11. Temperature Measurements with 95% confidence interval -

Model with Exponential Coefficients

Fig. 12. Residual analysis - Model with Exponential Coefficients

Similar to the previous cases, the estimated

measurements exhibit high agreement with the actual

measurements. The residual analysis confirms that the

differences between these measurements are close to zero

across the entire domain. Figs.13, 14, 15 and 16 display the

histograms of parameter estimates for this case.

Fig. 13. Histogram of the w

estimated parameter - model with

exponential coefficients

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coefficients in Heterogeneous Media in Diffusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

Fig. 14. Histogram of the w

estimated parameter - model with

exponential coefficients

Fig. 15. Histogram of the k

estimated parameter - model with

exponential coefficients

Fig. 16. Histogram of the k

estimated parameter - model with

exponential coefficients

It is important to note that despite this change in

distribution, the TMCMC method demonstrated estimating

the parameters with lower relative error compared to the

previous linear cases. This observation underscores the

relative capability of the method in dealing with exponential

coefficients, even with the loss of uniformity in histograms,

indicating a relative precision in estimating these

parameters.

IV. C

ONCLUSION

Throughout this study, the evaluation of the TMCMC

method efficacy in estimating coefficient parameters was

conducted across three distinct scenarios. The analysis of

the obtained tables and histograms reveals variability in the

method efficiency based on the analyzed case. Notably, it

was found that the method faced more significant challenges

in estimating parameters for k(x) in the second scenario,

corresponding to a linear model. Despite this additional

complexity, the relative error consistently remained below

9%.

A detailed analysis of the generated histograms allows

for a deeper understanding of the results. In all investigated

scenarios, a notable precision was observed in estimating

the parameters. In the constant model case, the value

distribution showed a well-defined Gaussian shape,

centered around the exact value, demonstrating highly

accurate estimation. However, in the linear case, there was

a more significant dispersion in the probable values,

especially considering the parameters associated with k(x).

Lastly, in the third case, an even higher precision compared

to the linear case was highlighted, along with the presence

of distributions that appeared to be bimodal, indicating the

occurrence of two peaks of probable values for the k(x)

parameters, something that warrants further investigation.

These results offer a comprehensive insight into the

applicability and performance of the TMCMC method in

parameter estimation, highlighting its nuances across

different model configurations. The achieved accuracy,

even in the face of specific challenges, underscores the

robustness and potential of this method for parameter

analyses and inferences across various contexts.

CKNOWLEDGMENT

The present work was supported by the Coordenação de

Aperfeiçoamento de Pessoal de Nível Superior - Brasil

(CAPES) - Funding Code 001.

The authors also acknowledge the financial support

from the CNPq and FAPERJ funding agencies.

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ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

10.5281/zenodo.12171420

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

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ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

AUTHORS

Lucas Lopes da Silva Costa. (São Sebastião do Alto, Rio de

Janeiro, Brazil, July 11th, 1999). B.Sc. in Applied and Computational

Mathematics from Fluminense Federal University (UFF), Brazil, 2021.

Currently pursuing a Master's degree in Computational Modeling

at the State University of Rio de Janeiro UERJ), which began in

2022, aiming to deepen knowledge in Applied Mathematics and

Scientiﬁc Computing. During his undergraduate studies, he actively

participated in the Mathematics of Epidemics extension project

(PEB) and contributed as a student member in course coordination.

Alongside his master's studies, he is also pursuing a Teaching Degree

in Mathematics at UFF, demonstrating a strong commitment to

pedagogical training. Additionally, he gained practical experience as

an intern in the Education sector with the Municipal Government of

Macuco - RJ, applying theoretical knowledge in a real-world context

between 2021 and 2022. His research interests include mathematical

modeling, computational simulation, and educational methodologies

in mathematics.

Eduardo Cunha Classe, Nova Friburgo, Rio de Janeiro, Brazil, April

29, 1998. B.Sc. in Mechanical Engineering from the State University

of Rio de Janeiro (UERJ), 2021. During his undergraduate studies, he

participated in a scientiﬁc initiation project for the characterization of

corrosion resistance of stainless steel alloys, was a member of the Baja

SAE program, and a member of the SPE student chapter. Currently

pursuing a Master's degree in Computational Modeling at the same

university, which began in 2022, aiming to deepen his knowledge in

heat transfer and computational modeling.

Lucas da Silva Asth (Nova Friburgo, Brazil, November 25th, 1996).

B.Sc. in Mechanical Engineering from the State University of Rio de

Janeiro, Brazil, 2021. MSc (2024) and currently pursuing a Ph.D. in

Computational Modeling at the State University of Rio de Janeiro

(UERJ). his research interests include heat and mass transfer, structural

dynamics, inverse problems and optimization.

Lucas L. da Silva Costa

Eduardo Cunha Classe

Lucas da Silva Asth

L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coecients in Heterogeneous Media in Diusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

AUTHORS

Luiz A. S. Abreu was born in Nova Friburgo, Brazil, on December 15th,

1982. He obtained his B.Sc. in mechanical engineering from the Rio

de Janeiro State University (UERJ) in 2009, his M.Sc. in mechanical

engineering from the Federal University of Rio de Janeiro (UFRJ) in

2011 and his D.Sc. in mechanical engineering from the same university

in 2014. He has been an aliate member of the ABCM—Brazilian

Society of Mechanical Sciences and Engineering since 2016. He has

advised or co-advised over 12 DSc and MSc theses, most of them

in the Graduate Program in Computational Modeling (PPGMC) at

UERJ. He is the author of over 20 articles published in major scientiﬁc

journals and conference proceedings and 5 book chapters. His

research interests include the solution of inverse problems, as well

as the use of meshfree, numerical, analytical, and hybrid numerical–

analytical methods for solving direct problems, mainly in mechanical

engineering applications.

Diego C. Knupp (Nova Friburgo, Brazil, November 9th, 1984). B.Sc. in

Mechanical Engineering from the State University of Rio de Janeiro,

Brazil, 2009. MSc (2010) and DSc (2013) in Mechanical Engineering

from the Federal University of Rio de Janeiro, Brazil. Currently

Professor of Mechanical Engineering at the State University of Rio

de Janeiro at the Polytechnique Institute - IPRJ/UERJ, heads the

Laboratory Patricia Oliva Soares of Experimentation and Numerical

Simulation in Heat and Mass Transfer, LEMA. Author of around 200

articles in major journals and conferences and one book published

abroad. Advisor of over 18 DSc and MSc thesis, his research interests

include hybrid methods, bioheat transfer, structural dynamics, inverse

problems and optimization.

Prof. Knupp has been aliate member of the Brazilian Academy of

Sciences (2019-2023) and currently serves as associate editor for the

Annals of the Brazilian Academy of Sciences journal.

Leonardo Tavares Stutz, Brazilian, born in Nova Friburgo, Rio de

Janeiro. He is currently Associate Professor at the Polytechnique

Institute (IPRJ) of the State University of Rio de Janeiro (UERJ),

Brazil, since 2006. He has Bachelor’s degree (1997), Master degree

(1999) and a Ph.D. (2005) in Mechanical Engineering from the

Federal University of Rio de Janeiro (UFRJ), Brazil. He is the author

of over forty articles published in scientiﬁc journals and conference

proceedings. He supervised over fourteen mater’s dissertations and

doctoral thesis in the Graduate Program in Computational Modeling

(PPGMC) at IPRJ. His research interests include Structural Dynamics

and Vibration, Vibration Damping, Inverse Problems, Parameter

Estimation, Bayesian Inference and Computational Modelling. Much

of his work has been on structural damage identiﬁcation problems

and on viscoelastic parameter estimation problems.

Luiz Alberto da Silva Abreu

Diego Campos Knupp

Leonardo Tavares Stutz

L. Costa, E. Classe, L. Asth, L. Abreu, D. Knupp and L. Stutz,

“Estimation of Spatially Dependent Coecients in Heterogeneous Media in Diusive Heat Transfer Problems”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.