Choice of the regularization parameter for the Cauchy problem for the Laplace equation

Magda Joachimiak (Faculty of Environmental Engineering and Energy, Poznan University of Technology, Poznan, Poland)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 16 January 2020

Issue publication date: 25 August 2020

4262

Abstract

Purpose

In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy.

Design/methodology/approach

The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied.

Findings

Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm Nmax.

Practical implications

Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary.

Originality/value

The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

Keywords

Citation

Joachimiak, M. (2020), "Choice of the regularization parameter for the Cauchy problem for the Laplace equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 30 No. 10, pp. 4475-4492. https://doi.org/10.1108/HFF-10-2019-0730

Publisher

:

Emerald Publishing Limited

Copyright © 2020, Magda Joachimiak.

License

Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode


Nomenclature

a

= multinomial coefficient of the function of distribution of temperature T̃(w);

c

= multinomial coefficient of the function of distribution of temperature T (x, y);

E (α)

= functional, energy integral;

Jα

= regularizing functional (Jα = J + α2I);

m

= number of Chebyshev nodes on the y-axis;

n

= number of Chebyshev nodes on the x-axis;

N1–1

= degree of the polynomial describing unknown distribution of temperature on the Γ1 boundary;

Nmax

= norm;

q

= heat flux density, W/m2K;

T

= temperature, K;

T̃

= temperature, function dependent on the Chebyshev node;

w

= Chebyshev node;

Wi

= Chebyshev polynomial of the first kind of i-th degree;

x, y

= Cartesian coordinates;

[x]n

= integer part of the division of number x by n; and

x mod n

= remainder of the division of number x by n.

Greek symbols

α

= regularization parameter;

δ

= error;

δM

= error of measurement data (Morozov principle);

γ

= multinomial coefficient, pertains to the sought temperature distribution on the boundary Γ1;

Γ

= boundary of the domain Ω, (Γ = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4); and

Ω

= calculation domain.

Subscript

A

= analytical solution;

c

= calculated value;

m

= measured value;

ran

= randomly disturbed value; and

Γi

= on the boundary Γi (for i = 1, 2, 3 and 4).

1. Introduction

Inverse problems are ill-posed in the Hadamard (1902) sense. It means that a slight disturbance to measurement data results in significant errors of the obtained results (Joachimiak and Ciałkowski, 2017, 2014, 2018; Nowak, 2017). Therefore, problems of such type need to be regularized. There are many methods used to regularize inverse problems. Among them, there is the Tikhonov regularization (Beck and Woodbury, 2016; Chen et al., 2019; Djerrar et al., 2017; Frąckowiak et al., 2019a; Laneev, 2018; Marin, 2010, 2016; Niu et al., 2014; Sun, 2016; Tikhonov and Arsenin, 1977; Yaparova, 2016), the Tikhonov–Philips regularization (Joachimiak et al., 2019a), the discrete Fourier transform (Frąckowiak and Ciałkowski, 2018; Wróblewska et al., 2015) and SVD algorithm (Hasanov and Mukanova, 2015). In her article, Cheruvu (2017) applied the wavelet regularization of Laplace’s equation in the arbitrarily shaped domain. The solution to the Cauchy problem for the Laplace’s equation was also sought with the use of the iterative Tikhonov-type method (Delvare and Cimetière, 2017). Han et al. (2011) in the article presented numerical tests concerning the solution to the Cauchy problem for Laplace’s equation with the use of the energy regularization method. Obtained results were compared with the Tikhonov regularization for which the regularization parameter was chosen based on the Morozov principle. In the paper of Liu and Wang (2018), the Cauchy problem for the Laplace’s equation was solved with the use of the method of fundamental solutions and the energy regularization technique to choose the source points. The Laplace’s equation was also solved with the use of iterative algorithms (Frąckowiak et al., 2015a, 2015b), of the Trefftz method (Ciałkowski and Frąckowiak, 2002; Ciałkowski and Grysa, 2010; Grysa et al., 2012; Hożejowski, 2016; Lin et al., 2018), of the method of fundamental solution (Kołodziej and Mierzwiczak, 2008; Mierzwiczak et al., 2015; Mierzwiczak and Kołodziej, 2011) and of the collocation method (Joachimiak et al., 2016). In many cases, the regularization of the inverse problem concerns the problem of choosing the regularization parameter. The regularization parameter can be chosen based on the Morozov principle (Chen et al., 2019; Han et al., 2011; Joachimiak et al., 2019a; Marin, 2016; Morozov, 1984; Sun, 2016) or using the L-curve method (Jin and Zheng, 2006; Marin and Munteanu, 2010; Marin, 2005). In the study of Marin (2011), the optimal regularization parameter was sought based on the generalized cross-validation criterion. Currently, research work focuses on finding new methods of regularization (Cheng and Feng, 2014; Zhuang and Chen, 2017) and on the modification of already known and used methods (Yang et al., 2015; Zheng and Zhang, 2018). Because of a wide application of inverse problems in engineering problems, such as the cooling of the blades in gas turbines (Frąckowiak et al., 2017; Frąckowiak et al., 2019b; Frąckowiak et al., 2011), analysis of the boiling heat transfer in minichannels (Hożejowska et al., 2009; Maciejewska and Piasecka, 2017), analysis of thermal and thermo-chemical treatment (Joachimiak et al., 2019b) or monitoring of power boilers operation (Taler et al., 2016, 2017), developing methods for regularization of these problems and investigating the process of choosing the regularization parameter are very significant.

In this article, the solution to the Cauchy problem for the Laplace’s equation was investigated with the use of the Chebyshev polynomials. To regularize the solution to the inverse problem, the modification of the Tikhonov and of the Tikhonov–Philips regularizations, described in the article (Joachimiak et al., 2019a) was applied. The choice of the regularization parameter was made based on the Morozov principle, the minimum energy integral criterion and the L-curve method.

2. Calculation model

In many technical applications, it is impossible to measure temperature on the boundary of the heated component of the device or machine, such as the combustion chamber, the inner side of the body of a turbine or a boiler. Then, the distribution of temperature can be determined by finding the solution to the inverse problem. Based on the distribution of temperature on the part of the boundary (TΓ2,TΓ3,TΓ4, fig. 1) and, additionally, knowing the heat flux density on the part of the boundary (qΓ3, fig. 1) one can determine the distribution of temperature on the boundary, where it is impossible to measure this temperature (TΓ1, fig. 1). Such a posed problem is the Cauchy problem, particularly sensitive to errors in measurement and in the calculation. In the stationary thermal field, the heat equation is reduced to the Laplace’s equation (for the non-linear case, the Kirchhoff’s substitution transforms the heat equation into the Laplace’s equation).

(1) 2Tx2+2Ty2=0

Laplace’s equation is solved in the domain Ω = {(x, y) ∈ ℝ2: −1 ≤ x ≤ 1, −1 ≤ y ≤* 1} with the following boundary conditions (Figure 1):

(2) T(x,y=1)=TΓ2(x)for1x1
(3) T(x=1,y)=TΓ3(y)for1y1
(4) T(x=1,y)n=qΓ3(y)for1y1
(5) T(x,y=1)=TΓ4(x)for1x1

It was assumed that the solution can be noted as the linear combination of the Chebyshev polynomials (Paszkowski, 1975).

(6) T(xi,yj)=p=0n1q=0m1cpqWp(xi)Wq(yj)

To solve the Cauchy problem, the collocation method was used. It was assumed that there are n points along the x-axis and m points along the y-axis (including points on the boundary). Collocation points being inside the interval (−1, 1) are the Chebyshev nodes (Paszkowski, 1975). Nodes were renumbered, which enables the temperature function to be noted in the following equation (7):

(7) T˜(wl)=k=1mnakW[k1]m(x(l1)modn+1)W(k1)modm(y[l1]n+1)
where the coefficients ak (k = 1, 2, …, mn) are unknown. Sought temperature distribution on the boundary Γ1 was assumed as the linear combination of the Chebyshev polynomials (Paszkowski, 1975).
(8) TΓ1(y)=h=1N1γhWh1(y)

Coefficients ak (k = 1, 2, …, mn) are expressed by the values of coefficients γi (i = 1, 2, …, N1). Hence, the determination of the temperature distribution is reduced to the determination of coefficients γi. To do so, the functional of the following form was minimized.

(9) J=Γ2(Tc,Γ2Tm,Γ2)2dΓ2+Γ3(Tc,Γ3Tm,Γ3)2dΓ3+Γ3(Tc,Γ3nqm,Γ3)2dΓ3+Γ4(Tc,Γ4Tm,Γ4)2dΓ4
where c in subscript denotes the calculated value, while m denotes the measured value. The integral in equation (9) on the boundary Γ2 can be noted in the following equation (10):
(10) JΓ2=Γ2(Tc,Γ2Tm,Γ2)2dΓ2=i=1nΓ2i(Tc,Γ2iTm,Γ2i)2dΓ2i

Applying numerical integration we have:

(11) JΓ2=i=1nΔxi(Tc(xi,1)Tm(xi,1))2=i=1n(Δxi(Tc(xi,1)Tm(xi,1)))2
where Δx1=x2x12, Δxi=xi+1xi12 for i = 2, 3, …, n−1 and Δxn=xnxn12. Having inserted the equation (7) into the equation (11), we obtained:
(12) JΓ2=i=1n(Δxi(k=1mnakW[k1]m(xi)W(k1)modm(1)Tm(xi,1)))2

Solving the direct problem, where the temperature on boundaries Γ1, Γ2, Γ3 and Γ4 was known, was reduced to solving the matrix equation

(13) Ax=b
what was described in detail in the paper (Joachimiak et al., 2019a). Based on the solution to the direct problem, constants ak [Equation (12)] are of the following equation (14):
(14) ak=Fk+h=1N1γhHh,k
where Hh,k=j=2m1A˜k,jnWh1(yj), while Ãk,jn (k = 1, 2, …, mn; j = 2, 3, …, m − 1) are elements of the matrix A−1. After substituting equation (14) into equation (12) we obtained:
(15) JΓ2=i=1n(Δxi(k=1mn(Fk+h=1N1γhHh,k)W[k1]m(xi)W(k1)modm(1)Tm(xi,1)))2=i=1n(h=1N1γhk=1mnΔxiHh,kW[k1]m(xi)W(k1)modm(1)+k=1mnΔxiFkW[k1]m(xi)W(k1)modm(1)Tm(xi,1))2=i=1n(h=1N1γhC1(i,h)D1(i))2

We would like the integral JΓ2 to have a value equal to zero or as close to zero as possible, hence, we equate the squared expression [Equation (15)] to zero. Hence, we have that:

(16) i=1,2,,nh=1N1γhC1(i,h)=D1(i)

We obtain n linear equations of the following equation (17):

(17) {C1(i,h)}{γh}=D1(i)
for i = 1, 2, …, n and h = 1, 2, …, N1. It can be reduced to the matrix equation.
(18) [C1,n]{γ}={D1,n}

Similarly, for other integrals [Equation (9)] we obtained:

(19) JΓ3=Γ3(Tc,Γ3Tm,Γ3)2dΓ3=i=1m(h=1N1γhC2(i,h)D2(i))2
(20) Jq,Γ3=Γ3(Tc,Γ3nqm,Γ3)2dΓ3=i=1m(h=1N1γhC3(i,h)D3(i))2
(21) JΓ4=Γ4(Tc,Γ4Tm,Γ4)2dΓ4=i=1n(h=1N1γhC4(i,h)D4(i))2

After JΓ3, Jq,Γ3 and JΓ4 had been equated to zero, equations of the following forms were obtained:

(22) i=1,2,,mh=1N1γhC2(i,h)=D2(i)
(23) i=1,2,,mh=1N1γhC3(i,h)=D3(i)
(24) i=1,2,,nh=1N1γhC4(i,h)=D4(i)

Based on equations (16) and (22)-(24), an oversized system of linear equations was obtained as the matrix equation, which would be solved with the use of the SVD algorithm:

(25) [[C1,n][C2,m][C3,m][C4,n]]{γ}={{D1,n}{D2,m}{D3,m}{D4,n}}

what can be noted in the shorter form:

(26) [BM]{γ}={BW}

Because of a great sensitivity of results to disturbances to measurement data, the Cauchy problem was regularized. The regularizing functional of the following form was assumed:

(27) Jα=J(γ)+α2I(γ)=[BM]{γ}{BW}2+α2Γ1((T˜)2+(T˜y)2)dΓ1

Regularization term can be noted as the sum of integrals.

(28) α2I(γ)=α2Γ1((T˜)2+(T˜y)2)dΓ1=α2i=1m1Γ1i((T˜)2+(T˜y)2)dΓ1i
where Γ1=i=1m1Γ1i. Performing numerical integration using the trapezoidal rule, we obtained:
(29) α2I(γ)=i=1m1α2yi+1yi2[(T˜(1,yi+1))2+(T˜(1,yi+1)y)2+(T˜(1,yi))2+(T˜(1,yi)y)2]

On the boundary Γ1 we have:

(30) T˜(1,yi)=k=1mnakW[k1]m(1)W(k1)modm(yi)=k=1mn(Fk+h=1N1γhHh,k)W[k1]m(1)W(k1)modm(yi)==k=1mnFkW[k1]m(1)W(k1)modm(yi)+h=1N1γhk=1mnHh,kW[k1]m(1)W(k1)modm(yi)==A1(i)+h=1N1γhA2(i,h)
(31) T˜(1,yi)y=k=1mnFkW[k1]m(1)W(k1)modm'(yi)+h=1N1γhk=1mnHh,kW[k1]m(1)W(k1)modm'(yi)==A3(i)+h=1N1γhA4(i,h)

Hence,

(32) α2I(γ)=i=1m1{αyi+1yi2(A1(i+1)+h=1N1γhA2(i+1,h))}2+i=1m1{αyi+1yi2(A3(i+1)+h=1N1γhA4(i+1,h))}2+i=1m1{αyi+1yi2(A1(i)+h=1N1γhA2(i,h))}2+i=1m1{αyi+1yi2(A3(i)+h=1N1γhA4(i,h))}2

Each of the components [Equation (32)] was equated to zero. The equation of the following form was obtained:

(33) i=1,2,,m1αyi+1yi2(A1(i+1)+h=1N1γhA2(i+1,h))=0
(34) i=1,2,,m1αyi+1yi2(A3(i+1)+h=1N1γhA4(i+1,h))=0
(35) i=1,2,,m1αyi+1yi2(A1(i)+h=1N1γhA2(i,h))=0
(36) i=1,2,,m1αyi+1yi2(A3(i)+h=1N1γhA4(i,h))=0

Equations (33)-(36) can be reduced to the following system of equations.

(37) α[CM]{γ}=α{CW}
where α is the regularization parameter. When the equation (26) and regularization [Equation (37)] are included, the following system of equations is obtained:
(38) [[BM]α[CM]]{γ}={{BW}α{CW}}

The solution to the system of equations (38) was sought in the least-squares sense with the use of the SVD algorithm.

3. Choice of the regularization parameter

To determine unknown regularization parameter α, the Morozov principle, the minimum of energy integral criterion and L-curve method were applied. For the solution obtained with the use of the Morozov principle, the mean (Morozov_A) and the maximal (Morozov_B) errors of the heat flux density δM on the boundary Γ3 were evaluated. Interval halving method was used to determine zero of the function FM (α) defined by the following equation (39):

(39) [BM]{γ}{BW}2δM2=FM

Unknown regularization parameter α was also sought based on the minimization of the functional (energy integral) of the following equation (40):

(40) E(α)=Ω(T(α))2dΩ,T(α)C2(Ω)
where (T(α))2=(T(α)x)2+(T(α)y)2. The minimum of the energy integral corresponds to satisfying the Laplace’s equation (with respective boundary conditions), which is discussed in the paper (Gelfand and Fomin, 1979). Therefore, we choose the parameter α for which minαE(α) occurs, i.e. the derivative E'=dEdα reverses the sign. Domain Ω was divided into rectangular domains with the use of equidistant nodes and next into domains being right-angled triangles. The integral value was calculated with the use of the finite element method. Value ∇T was determined based on the form of the solution equation (6). Values γh were obtained by solving the equation (38).

On the basis of the solution of the equation (38), the L-curve was drawn as the correlation between ‖[BM]{γ} − {BW}‖ on the x-axis and ‖[CM]{γ} − {CW}‖ on the y-axis. We sought for the regularization parameter α with which corresponded the point of the L-curve locating on the curvature of this line. To evaluate the choice of the regularization parameter α, the following norm was defined:

(41) Nmax=max|TΓ1_CTΓ1_A|max|TΓ1_A|

4. Numerical examples

Calculations were made in the domain Ω for the function.

(42) f1=ln((xa)2+(yb)2),q1,Γ3=2(1a)(1a)2+(yb)2

and

(43) f2=Re(1z(a+bi))=xa(xa)2+(yb)2,q2,Γ3=(1a)2+(yb)2[(1a)2+(yb)2]2

We assumed such values of constants a and b that singularities would be outside the calculation domain Ω and that the values of gradients would change significantly within this domain (a = 1.3, b = 1.3, b = 1.1). Values of the norm Nmax [Equation (41)], not including the regularization [Equation (26)], without disturbance (δran = 0) and with random disturbance to the heat flux density up to 0.01q (δran = 0.01) and to 0.02q (δran = 0.02) are summarized in Table I. Disturbance q is an additive function with the uniform distribution. A slight disturbance to measurement data results in a significant error of the sought temperature on the boundary Γ1. Hence, it is necessary to regularize the inverse problem [Equation (38)] and to choose the regularization parameter α properly.

4.1 Example 1

Calculations were made for the function f1 [Equation (42)]. Heat flux density was disturbed randomly to 0.02q (δran = 0.02). Regularization parameter α was chosen with the use of the Morozov principle, the minimum of energy integral criterion and the L-curve method.

Figure 2 presents the course of the energy integral and its derivative depending on the parameter α. To solve the Cauchy problem, the authors applied such value of the regularization parameter α for which the energy integral E (α) took the minimal value, which meant that the derivative of the energy integral E'=dEdα reversed the sign. The value of α was 5.13 × 10−4 (Table II). The Cauchy problem was also regularized for the regularization parameter α amounting to 4.64 × 10−1, which was determined based on the L-curve course (Figure 3).

To choose the regularization parameter with the Morozov principle, the values of mean and maximal error δM were evaluated for the heat flux on the boundary Γ3. The respective values were obtained: 0.008 and 0.02 (Table II). Next, zero of the function FM (α) was calculated as per the equation (39). The respective values of the regularization parameter were obtained: 1.98 × 10−6 and 4.0799 × 10−2 (Table II).

The lowest value of the norm Nmax amounting to 6.18 × 10−2 (Table II) for the function f1 was obtained for the case of choosing the regularization parameter with the use of the Morozov principle for the maximal error of the heat flux δM (Morozov_B). This criterion brought satisfying results, as did the choice of the regularization parameter made with the use of the minimum energy integral criterion (Nmax = 9.796 × 10−2). When the L-curve method was used, the obtained results were considerably worse (Nmax = 2.22 × 10−1). For the Morozov principle, for the mean error δM of the heat flux (Morozov_A), the highest value of the norm Nmax amounting to 50.42 was obtained. Distributions of temperature on the boundary Γ1 resulting from the analytical solution (AS) and from the solution of the Cauchy problem are presented in Figure 4.

4.2 Example 2

Calculations were made for the function f2 [Equation (43)]. Heat flux density was disturbed randomly to 0.02q (δran = 0.02). The best results were obtained for the choice of the regularization parameter made with the use of the minimum energy integral criterion (Nmax = 4.77 × 10−2), and the worse results were obtained for the L-curve method (Nmax = 0.361). Values of the regularization parameter and of the norm Nmax, being the measure of the quality of the parameter α choice, for the function f2 are summarized in Table III. Distributions of temperature on the boundary Γ1 for the AS and for the solution to the Cauchy problem with regularization are presented in Figure 5. Distribution of temperature on the boundary Γ1 obtained with the use of the minimum of energy integral criterion slightly diverges from the AS.

To examine thoroughly the criterion for the regularization parameter selection with the use of the minimum of energy integral, calculations were performer also for the following functions:

(44) f3=cosxcoshy+sinxsinhy
(45) f4=exsiny
(46) f5=x33xy2+e2ysin2xexcosy
(47) f6=ex2y2sin2xy
which were chosen based on publications (Liu et al., 2018; Conde Mones et al., 2017; Fu et al., 2013; Sun, 2017). Values of the regularization parameter and of the norm Nmax for functions f3f6 are summarized in Table IV. For the function f6 and the disturbance to the heat flux density δran = 0.02 and δran = 0.05 the minimum of energy integral was not achieved. Distributions of temperature on the boundary Γ1 being sought are presented in Figure 6. For functions f3f5 the disturbance δran = 0.05 was taken into account, and for the function f6 it was δran = 0.01.

5. Conclusion

This paper presents the solution of the Cauchy problem for Laplace’s equation. Obtained distributions of temperature on the boundary Γ1 were analyzed in terms of the dependence on the method for choosing the regularization parameter. The best results were obtained for the choice of the regularization parameter made with the use of the minimum of energy integral criterion and the Morozov principle (δM is the maximal error for the heat flux on the boundary Γ3). The advantage of the application of the minimum energy integral criterion is a unique determination of the regularization parameter α for which E (α) has minimal value. Regularization made with the use of the minimum energy integral criterion gives satisfying results. However, its disadvantage is the fact that not for all calculation examples the minimum energy integral was determined. For the Morozov principle, the obtained results (distribution of temperature on the boundary Γ1) depend on calculation or evaluation of the value of the heat flux δM error, what is a disadvantage of this method. Inexact evaluation of the measurement data error can result in obtaining the distribution of temperature on the boundary Γ1, which is subject to great uncertainty. The choice of the parameter α made with the use of the L-curve method gave the worst results. Smooth L-curve course was obtained, what was related to the problem with the unique determination of the regularization parameter α using this method.

Figures

Calculation domain

Figure 1.

Calculation domain

Energy integral (E) and the derivative of the energy integral (E′) depending on the value of the regularization parameter α (function f1)

Figure 2.

Energy integral (E) and the derivative of the energy integral (E′) depending on the value of the regularization parameter α (function f1)

L-curve with the point for which the regularization parameter α was chosen (function f1)

Figure 3.

L-curve with the point for which the regularization parameter α was chosen (function f1)

Distribution of temperature on the boundary Γ1 obtained from the AS and form the Cauchy problem when the regularization parameter α was chosen with the use of the Morozov principle (Morozov_B), the minimum of energy integral criterion (E) and the L-curve method (L-curve) for the function f1

Figure 4.

Distribution of temperature on the boundary Γ1 obtained from the AS and form the Cauchy problem when the regularization parameter α was chosen with the use of the Morozov principle (Morozov_B), the minimum of energy integral criterion (E) and the L-curve method (L-curve) for the function f1

Distribution of temperature on the boundary Γ1 obtained from the AS and form the solution to the Cauchy problem when the regularization parameter α was chosen with the use of the Morozov principle (Morozov_A and Morozov_B), the minimum of energy integral criterion (E) and of the L-curve method (L-curve) for the function f2

Figure 5.

Distribution of temperature on the boundary Γ1 obtained from the AS and form the solution to the Cauchy problem when the regularization parameter α was chosen with the use of the Morozov principle (Morozov_A and Morozov_B), the minimum of energy integral criterion (E) and of the L-curve method (L-curve) for the function f2

Distribution of temperature on the boundary Γ1 obtained from the AS and form the solution to the Cauchy problem when the regularization parameter α was chosen with the use of the minimum of energy integral criterion (E) for functions f2 − f6

Figure 6.

Distribution of temperature on the boundary Γ1 obtained from the AS and form the solution to the Cauchy problem when the regularization parameter α was chosen with the use of the minimum of energy integral criterion (E) for functions f2f6

Values of the norm Nmax for calculations without regularization (α = 0), without disturbance (δran = 0) and with random disturbance to the heat flux density up to 0.01q (δran = 0.01) and 0.02q (δran = 0.02)

Error of heat flux density
disturbed randomly
Nmax
f1 f2
δran = 0 2.78·10−2 0.40047
δran = 0.01 21978535 2814929
δran = 0.02 43957070 5629859

Values of the measurement data error δM, of the regularization parameter α and of the norm Nmax for the choice of the regularization parameter α made using the Morozov principle (Morozov_A and Morozov_B), the minimum of energy integral criterion (E) and the L-curve method (L-curve) for the function f1

Method of the choice of the
regularization parameter
δM α Nmax
Morozov_A 0.008 1.98 × 10−6 50.42
Morozov_B 0.02 4.0799 × 10−2 6.18 × 10−2
E 5.13 × 10−4 9.796 × 10−2
L-curve 4.64 × 10−1 2.22 × 10−1

Values of the measurement data error δM, of the regularization parameter α and of the norm Nmax for the choice of the regularization parameter α made using the Morozov principle (Morozov_A and Morozov_B), the minimum of energy integral criterion (E) and of the L-curve method (L-curve) for the function f2

Method of the choice of the
regularization parameter
δM α Nmax
Morozov_A 0.002 1.84 × 10−3 0.115
Morozov_B 0.004 6.702 × 10−3 0.174
E 3.37 × 10−4 4.77 × 10−2
L-curve 3.998 × 10−1 0.361

Values of the regularization parameter and of the norm Nmax for functions f3f6 with the disturbance to the heat flux density δran from 0.01 to 0.05

δran = 0.01 δran = 0.02 δran = 0.05
Function α Nmax α Nmax α Nmax
f3 2.98 × 10−2 2.38 × 10−3 2.89 × 10−2 4.49 × 10−3 2.803 × 10−2 1.107 × 10−2
f4 4.99 × 10−4 3.46 × 10−2 1.0 × 10−3 3.98 × 10−2 1.0 × 10−3 6.406 × 10−2
f5 4.99 × 10−4 3.46 × 10−2 1.0 × 10−3 3.98 × 10−2 1.0 × 10−2 2.16 × 10−1
f6 1.0 × 10−3 2.33 × 10−1 No minimum No minimum

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Corresponding author

Magda Joachimiak can be contacted at: magda.joachimiak@put.poznan.pl

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