# A posteriori regularization method for the two-dimensional inverse heat conduction problem

• Wei Cheng , Yi-Liang Liu and Qi Zhao
From the journal Open Mathematics

## Abstract

In this article, we consider a two-dimensional inverse heat conduction problem that determines the surface temperature distribution from measured data at the fixed location. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A quasi-boundary value regularization method in conjunction with the a posteriori parameter choice strategy is proposed to solve the problem. A Hölder-type error estimate between the approximate solution and its exact solution is also given. The error estimate shows that the regularized solution is dependent continuously on the data.

MSC 2010: 65M30; 35R30; 35R25

## 1 Introduction

The inverse heat conduction problem (IHCP) arises from many physical and engineering problems such as nuclear physics, aerospace, food science, metallurgy, and nondestructive testing. It is well known that the IHCP is severely ill-posed in Hadamard’s sense [1], i.e., the solution does not depend continuously on the data, any small error in the measurement can induce an enormous error in computing the unknown solution. Therefore, some regularization techniques are needed to restore the stability of the solution to the problem [2,3, 4,5].

As we know, many authors have studied IHCPs with different regularization methods. These methods include the Fourier method [6,7,8], the Tikhonov method [9,10,11], the method of fundamental solutions [12,13], the mollification method [14,15,16], the wavelet-Galerkin method [17,18], the wavelet method [19,20, 21,22], the variational method [23], and so on. However, to the authors’ knowledge, most of the aforementioned methods focus on the one-dimensional IHCP. A few works based on numerical methods have been presented for the two-dimensional IHCP. This article will investigate the following two-dimensional IHCP:

(1.1) u t = u x x + u y y , 0 < x < 1 , y > 0 , t > 0 , u ( 0 , y , t ) = g ( y , t ) , y 0 , t 0 , u x ( 0 , y , t ) = 0 , y 0 , t 0 , u ( x , 0 , t ) = 0 , 0 x 1 , t 0 , u ( x , y , 0 ) = 0 , 0 x 1 , y 0 ,

where g denotes the temperature history at fixed x = 0 . We want to recover the temperature distribution u ( x , , ) for 0 < x < 1 from temperature measurement g δ ( y , t ) . In this article, we will apply a quasi-boundary value method to solve the problem (1.1) and provide a Hölder-type error estimate between the approximate solution and its exact solution.

The quasi-boundary value method is a regularization technique that replaces the boundary condition or final condition with a new approximate condition. This regularization method has been used for solving the backward heat conduction problem [24,25,26, 27], the Cauchy problem for elliptic equations [28,29], the IHCP [30], and the inverse source identification problem [31,32]. In this article, we will use a quasi-boundary value method to solve the ill-posed problem (1.1).

Kurpisz and Nowak [33] used the boundary element method for solving the two-dimensional IHCP. Qian and Fu [34] applied a quasi-reversibility method and a Fourier method to solve the two-dimensional IHCP (1.1) and gave some quite sharp error estimates for the regularized solution. A differential-difference regularization method was used to deal with the two-dimensional IHCP (1.1) [35]. Wei and Gao [36] solved a two-dimensional IHCP by a meshless manifold method, which is based on the moving least-square method and the finite cover approximation theory in the mathematical manifold. Bergagio et al. [37] used the iterative finite-element algorithm to solve two-dimensional nonlinear IHCPs. It is worth noting that most of the aforementioned works apply a priori regularization parameter choice rule, which usually depends on both the noise level and the a priori bound. In practice, the an a priori bound cannot be known exactly. In this article, we will apply a quasi-boundary value method combined with a posteriori regularization parameter choice rule to solve the ill-posed problem (1.1).

Some researchers are dealing with the error estimate under an a posteriori parameter choice strategy. Engl and Gfrerer [38] applied the a posteriori parameter choice for general regularization methods to solve linear ill-posed problems. Shi et al. [39] gave a posteriori parameter choice strategy for the convolution regularization method. Adler et al. [40] used an a posteriori parameter choice strategy for the weak Galerkin least squares method. Trong and Hac [41] applied a modified version of the quasi-boundary value method with a priori and a posteriori parameter choice strategies to solve time-space fractional diffusion equations. Duc et al. [42] gave a posteriori parameter choice strategy for the Tikhonov-type regularization to deal with the backward heat equations with a time-dependent coefficient.

The widely used method for the a posteriori parameter choice is Morozov’s discrepancy principle, i.e., matching the error of the approximate solution with the accuracy of the initial data of the ill-posed problem. This discrepancy principle is first seen in [43]. Then the discrepancy principle has been used for solving different problems. Scherzer [44] used it for the Tikhonov regularization for the nonlinear ill-posed problems. Bonesky [45] applied it to select the regularization parameter for the Tikhonov regularization method for the linear operator equation. Fu et al. [46] considered it for the Cauchy problem for the Helmholtz equation with application to the Fourier regularization method. Feng et al. [47] investigated a backward problem for a time-space fractional diffusion equation, and obtained the order optimal convergence rates by using Morozov’s discrepancy principle and an a priori regularization parameter choice rule. In this article, we will use Morozov’s discrepancy principle to select the regularization parameter for a quasi-boundary value regularization method.

In order to use the Fourier transform technique, we extend the functions u ( x , , , ) , g ( , ) , g δ ( , ) , to be whole real ( y , t ) plane by defining them to be zero everywhere in ( y , t ) , y < 0 , t < 0 . We assume that these functions are in L 2 ( R 2 ) and wish to determine the temperature distribution u ( x , , ) L 2 ( R 2 ) for 0 < x < 1 from the temperature measurement g δ ( , ) L 2 ( R 2 ) . We also use the corresponding L 2 norm as follows:

(1.2) f = R 2 f ( y , t ) 2 d y d t 1 2 .

Let

h ˆ ( ξ , η ) = 1 2 π R 2 h ( y , t ) e i ( ξ y + η t ) d y d t

be the Fourier transform of function h ( y , t ) L 2 ( R 2 ) . Using Fourier transform on both sides of (1.1) with respect to the variable y and t , we can obtain the formal solution of problem (1.1) in the frequency domain as:

(1.3) u ˆ ( x , ξ , η ) = g ˆ ( ξ , η ) cosh ( x ϑ ( ξ , η ) ) ,

then using the inverse Fourier transform on (1.3), we have the formal solution of problem (1.1):

(1.4) u ( x , y , t ) = 1 2 π R 2 e i ( ξ y + η t ) g ˆ ( ξ , η ) cosh ( x ϑ ( ξ , η ) ) d ξ d η ,

where ξ and η are the variables of Fourier transform on y and t , respectively, and

(1.5) ϑ ( ξ , η ) = ξ 2 + i η .

From (1.5), we obtain

(1.6) ϑ ( ξ ) = ξ 4 + η 2 + ξ 2 2 + i sign ( η ) ξ 4 + η 2 ξ 2 2 .

Due to the Parseval formula and (1.3), we have

(1.7) u ( x , , ) = u ˆ ( x , , ) = R 2 cosh ( x ϑ ( ξ , η ) ) 2 g ˆ ( ξ , η ) 2 d ξ d η 1 2 .

Note that, for fixed 0 < x 1 , cosh ( x ϑ ( ξ , η ) ) tends to infinity when ξ or η . Formula (1.7) implies a rapid decay of g ˆ ( ξ , η ) at high frequencies. But such decay is not likely to occur in the measured noisy data g δ ( y , t ) at x = 0 . Therefore, small perturbation of g δ ( y , t ) in high-frequency components can blow up and completely destroy the temperature u ( x , y , t ) , i.e., problem (1.1) is severely ill-posed. So an effective regularization method is necessary for solving the problem (1.1).

In fact, in practice the data function g ( y , t ) is given only by measurement and measurement errors exist in g ( y , t ) . We assume that the exact data g ( y , t ) and the noisy data g δ ( y , t ) satisfy the following noise level:

(1.8) g g δ δ .

The constant δ > 0 denotes a bound on the measurement error.

We also assume that there exists an a priori condition for problem (1.1):

(1.9) R 2 e ϑ ( ξ , η ) g ˆ ( ξ , η ) 2 d ξ d η 1 2 E ,

where E > 0 is constant.

The main aim of this article is to solve the two-dimensional IHCP (1.1) by using the a posteriori quasi-boundary value method. This article is organized as follows. In Section 2, we provide a quasi-boundary value regularization method to formulate a regularized solution and give a posteriori choice strategy of regularization parameter based on Morozov’s discrepancy principle. In Section 3, a Hölder-type error estimate between the approximate solution and its exact solution is presented under the a posteriori regularization parameter choice rule. The article ends with a brief conclusion in Section 4.

## 2 An a posteriori parameter choice strategy for a quasi-boundary value method and some auxiliary results

In this section, we solve the ill-posed problems (1.1) by a quasi-boundary value method and give some auxiliary results under an a posteriori regularization parameter choice strategy.

The quasi-boundary value method is a regularization technique by replacing the boundary condition or final condition with a new approximate condition. So we add a perturbation term in the boundary condition and consider the following boundary conditions instead:

(2.1) u ( 0 , y , t ) + α u ( 1 , y , t ) = g δ ( y , t ) , y 0 , t 0 ,

where α plays a role of regularization parameter and the noisy data g δ are the measured data of functions g .

Let u α δ ( x , y , t ) be the solution of the following regularized problem:

(2.2) v t = 2 v x 2 + 2 v y 2 , 0 < x < 1 , y > 0 , t > 0 , v ( 0 , y , t ) + α v ( 1 , y , t ) = g δ ( y , t ) , y 0 , t 0 , v x ( 0 , y , t ) = 0 , y 0 , t 0 , v ( x , 0 , t ) = 0 , 0 x 1 , t 0 , v ( x , y , 0 ) = 0 , 0 x 1 , y 0 .

By taking Fourier transform on both sides of (2.2) with respect to the variable y and t , we can obtain the following form:

(2.3) u ˆ α δ ( x , ξ , η ) = cosh ( x ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) g ˆ δ ( ξ , η ) .

Comparing formula (1.3) for the exact solution with formula (2.3) for its quasi-boundary value approximation, we can see that the regularization procedure consists in replacing g ˆ ( ξ , η ) with an appropriately filtered Fourier transform of noisy data g δ ( y , t ) . The filter in (2.3) attenuates the high frequencies in g ˆ δ ( ξ , η ) . From this, we can replace the original filter 1 1 + α cosh ( ϑ ( ξ , η ) ) with another filter 1 1 + α cosh ( ϑ ( ξ , η ) ) and introduce a new approximation u α , δ ( x , y , t ) of problem (1.1)

(2.4) u ˆ α , δ ( x , ξ , η ) = cosh ( x ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) g ˆ δ ( ξ , η ) .

We call u α , δ ( x , y , t ) given by (2.4) a quasi-boundary value approximation of the exact solution u ( x , y , t ) .

We define an operator K x : u ( x , , ) g ( , ) , x [ 0 , 1 ) . Then problem (1.1) can be rewritten as the following operator equation:

(2.5) K x u ( x , y , t ) = g ( y , t ) , x [ 0 , 1 ) ,

with a linear operator K x ( L 2 ( R 2 ) , L 2 ( R 2 ) ) . From (1.3) and (2.5), we have

(2.6) K x u ^ ( x , ξ , η ) = g ˆ ( ξ , η ) = u ˆ ( x , ξ , η ) [ cosh ( x ϑ ( ξ , η ) ) ] 1 , x [ 0 , 1 ) .

We apply Morozov’s discrepancy principle as a posteriori regularization parameter choice rule. Recalling the definition of Morozov’s discrepancy principle, the classical Morozov’s discrepancy principle chooses the regularization parameter α > 0 such that [43,48]

(2.7) K x u α , δ g δ = δ .

Scherzer [44] extended Morozov’s discrepancy principle and chose the regularization parameter α > 0 such that

(2.8) K x u α , δ g δ = τ δ ,

where τ > 1 is a constant. In this article, we select the regularization parameter α > 0 satisfied equation (2.8), because equation (2.7) will not fit in our framework.

In order to establish the existence and uniqueness of the solution of equation (2.8), the following lemma is needed.

## Lemma 2.1

Let d ( α ) = K x u α , δ g δ . If g δ > δ > 0 , then the following results exist:

1. d ( α ) is a continuous function;

2. lim α 0 d ( α ) = 0 ;

3. lim α + d ( α ) = g δ ;

4. d ( α ) is a strictly increasing function over ( 0 , ) .

## Proof

Due to the Parseval formula and (2.4), (2.6), we have

d ( α ) = R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) 2 g ˆ δ ( ξ , η ) 2 d ξ d η 1 2 .

From the above expression, the results of Lemma 2.1 are straightforward.□

## Lemma 2.2

For 0 < α < 1 , 0 < x < 1 , ϑ ( ξ , η ) = ξ 2 + i η , then there holds

(2.9) sup ( ξ , η ) R 2 cosh ( x ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) ( c α ) x .

## Proof

Using the inequality cosh ( z ) cosh ( z ) , z C , we have

(2.10) cosh ( x ϑ ( ξ , η ) ) cosh ( x ϑ ( ξ , η ) ) = e x ϑ ( ξ , η ) + e x ϑ ( ξ , η ) 2 e x ϑ ( ξ , η ) .

From Lemmas 2.1 and 2.2 in [49], we know there exists a positive constant c such that

sup ( ξ , η ) R 2 cosh ( x ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) = sup ( ξ , η ) R 2 cosh ( x ϑ ( ξ , η ) ) 1 + c α e ϑ ( ξ , η ) = sup ( ξ , η ) R 2 e x ϑ ( ξ , η ) 1 + c α e ϑ ( ξ , η ) = ( c α ) x .

## 3 Error estimate for the a posteriori quasi-boundary value method

In this section, we will give a Hölder-type error estimate between the exact solution of temperature and its regularized solution by using an a posteriori choice rule for the regularization parameter.

## Theorem 3.1

Suppose that the noise assumption (1.8) and the a priori condition (1.9) hold. If the regularization parameter α > 0 is chosen by Morozov discrepancy principle (2.8), then we have the following error estimate:

(3.1) u α , δ ( x , , ) u ( x , , ) ( ( τ + 1 ) 1 x + ( c ( τ 1 ) ) x ) E x δ 1 x .

## Proof

Using the Parseval formula and the triangle inequality, we have

(3.2) u α , δ ( x , , ) u ( x , , ) = u ˆ α , δ ( x , , ) u ˆ ( x , , ) u ˆ α , δ ( x , , ) u ˆ α , ( x , , ) + u ˆ α , ( x , , ) u ˆ ( x , , ) .

We first give an estimate for the second term.

From (1.3), (2.4), and (2.10), with the Hölder inequality, we obtain

u ˆ α , ( x , , ) u ˆ ( x , , ) 2 = cosh ( x ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ ( , ) cosh ( x ϑ ( , ) ) g ˆ ( , ) 2

= α cosh ( ϑ ( , ) ) cosh ( x ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ ( , ) 2 = R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) 2 cosh ( x ϑ ( ξ , η ) ) 2 g ˆ ( ξ , η ) 2 d ξ d η R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) 2 ( e x ϑ ( ξ , η ) ) 2 g ˆ ( ξ , η ) 2 d ξ d η = R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) 2 g ˆ ( ξ , η ) 2 ( 1 x ) e ϑ ( ξ , η ) g ˆ ( ξ , η ) 2 x d ξ d η R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) g ˆ ( ξ , η ) 2 ( 1 x ) e ϑ ( ξ , η ) g ˆ ( ξ , η ) 2 x d ξ d η R 2 α cosh ( ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) g ˆ ( ξ , η ) 2 d ξ d η ( 1 x ) R 2 e ϑ ( ξ , η ) g ˆ ( ξ , η ) 2 d ξ d η x .

From (2.4) and (2.8), we obtain

(3.3) τ δ = K x u α , δ g δ = K x u ^ α , δ g ˆ δ = α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ δ ( , ) ,

with the noise assumption (1.8) and the a priori condition (1.9), we have

(3.4) u ˆ α , ( x , , ) u ˆ ( x , , ) α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ ( , ) ( 1 x ) E x E x α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) ( g ˆ ( , ) g ˆ δ ( , ) ) + α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ δ ( , ) ( 1 x ) E x [ g ˆ ( , ) g ˆ δ ( , ) + τ δ ] ( 1 x ) E x [ ( τ + 1 ) δ ] ( 1 x ) .

Now we give the bound for the first term. Due to (2.4) and (2.9), we have

(3.5) u ˆ α , δ ( x , , ) u ˆ α , ( x , , ) = cosh ( x ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) ( g ˆ δ ( , ) g ˆ ( , ) ) sup ( ξ , η ) R 2 cosh ( x ϑ ( ξ , η ) ) 1 + α cosh ( ϑ ( ξ , η ) ) g ˆ δ ( , ) g ˆ ( , ) ( c α ) x δ .

From (3.3) and (1.9), we know

τ δ = α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ δ ( , ) α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) ( g ˆ δ ( , ) g ˆ ( , ) ) + α cosh ( ϑ ( , ) ) 1 + α cosh ( ϑ ( , ) ) g ˆ ( , ) g ˆ δ ( , ) g ˆ ( , ) + α 1 + α cosh ( ϑ ( , ) ) cosh ( ϑ ( , ) ) g ˆ ( , ) δ + α cosh ( ϑ ( , ) ) g ˆ ( , ) δ + α e ( ϑ ( , ) ) g ˆ ( , ) δ + α E .

This yields

(3.6) α δ E ( τ 1 ) .

Substituting (3.6) into (3.5), we have

(3.7) u ˆ α , δ ( x , , ) u ˆ α , ( x , , ) ( c ( τ 1 ) ) x E x δ 1 x .

Combining (3.2), (3.4), and (3.7), we obtain the error estimate (3.1).□

Estimate (3.1) is a Hölder-type stability estimate, and the error bounds in (3.1) are similar to error bounds in (4.14) of Theorem 4.2 in [46].

## 4 Conclusion

In this article, we investigate a two-dimensional IHCP, which determines the surface temperature distribution from measured data at the fixed location. We propose a quasi-boundary value method for obtaining a regularized solution. The Hölder-type error estimate between the approximate solution and its exact solution is obtained under Morozov’s discrepancy principle. Error analysis shows that our regularization method is effective.

## Acknowledgments

The authors would like to thanks the editor and the anonymous referees for their valuable comments and suggestions on this article.

1. Funding information: This work was supported by the National Natural Science Foundation of China (Grant Nos 11961044 and 11561045).

2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

3. Conflict of interest: The authors state no conflict of interest.

## References

[1] J. Hadamard, Lectures on the Cauchy Problems in Linear Partial Differential Equations, Yale University Press, New Haven, 1923. Search in Google Scholar

[2] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic, Boston, Mass, USA, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar

[3] J. V. Beck, B. Blackwell, and S. R. Clair, Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York, 1985. Search in Google Scholar

[4] L. Eldén, Approximations for a Cauchy problem for the heat equation, Inverse Problems 3 (1987), 263–273. 10.1088/0266-5611/3/2/009Search in Google Scholar

[5] Y. C. Wang, B. Wu, and Q. Chen, Numerical reconstruction of a non-smooth heat flux in the inverse radial heat conduction problem, Appl. Math. Lett. 111 (2021), 106658. 10.1016/j.aml.2020.106658Search in Google Scholar

[6] L. Eldén, F. Berntsson, and T. Regińska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), no. 6, 2187–2205. 10.1137/S1064827597331394Search in Google Scholar

[7] C. L. Fu, Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J. Comput. Appl. Math. 167 (2004), 449–463. 10.1016/j.cam.2003.10.011Search in Google Scholar

[8] A. Wróblewska, A. Frackowiak, and M. Cialkowski, Regularization of the inverse heat conduction problem by the discrete Fourier transform, Inverse Probl. Sci. Eng. 24 (2016), no. 2, 195–212. 10.1080/17415977.2015.1017480Search in Google Scholar

[9] A. Carasso, Determining surface temperature from interior observations, SIAM J. Appl. Math. 42 (1982), 558–574. 10.1137/0142040Search in Google Scholar

[10] W. Cheng and C. L. Fu, Two regularization methods for an axisymmetric inverse heat conduction problem, J. Inverse Ill-Posed Problems 17 (2009), 157–170. 10.1515/JIIP.2009.014Search in Google Scholar

[11] J. P. Ngendahayo, J. Niyobuhungiro, and F. Berntsson, Estimation of surface temperatures from interior measurements using Tikhonov regularization, Results Appl. Math. 9 (2021), 100140. 10.1016/j.rinam.2020.100140Search in Google Scholar

[12] B. T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the radially symmetric inverse heat conduction problem, Int. Commun. Heat Mass Transf. 39 (2012), 887–895. 10.1016/j.icheatmasstransfer.2012.05.011Search in Google Scholar

[13] Y. C. Hon and T. Wei, The method of fundamental solutions for solving multidimensional inverse heat conduction problems, CMES - Comput. Model. Eng. Sci. 7 (2005), no. 2, 119–132. Search in Google Scholar

[14] D. A. Murio, The Mollification Method and the Numerical Solution of Ill-posed Problem, John Wiley and Sons Inc, New York, 1993. 10.1002/9781118033210Search in Google Scholar

[15] M. Garshasbi and H. Dastour, Estimation of unknown boundary functions in an inverse heat conduction problem using a mollified marching scheme, Numer. Algorithms 68 (2015), no. 4, 769–790. 10.1007/s11075-014-9871-7Search in Google Scholar

[16] D. A. Murio, Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP, Inverse Probl. Sci. Eng. 17 (2009), no. 2, 229–243. 10.1080/17415970802082872Search in Google Scholar

[17] T. Regińska, and L. Eldén, Solving the sideways heat equation by a wavelet-Galerkin method, Inverse Problems 13 (1997), no. 4, 1093–1106. 10.1088/0266-5611/13/4/014Search in Google Scholar

[18] T. Regińska, and L. Eldén, Stability and convergence of wavelet-Galerkin method for the sideways heat equation, J. Inverse Ill-Posed Problems 8 (2000), 31–49. 10.1515/jiip.2000.8.1.31Search in Google Scholar

[19] T. Regińska, Application of wavelet shrinkage to solving the sideways heat equation, BIT 41 (2001), no. 5, 1101–1110. 10.1023/A:1021909816563Search in Google Scholar

[20] J. R. Wang, The multi-resolution method applied to the sideways heat equation, J. Math. Anal. Appl. 309 (2005), 661–673. 10.1016/j.jmaa.2004.11.025Search in Google Scholar

[21] C. L. Fu and C. Y. Qiu, Wavelet and error estimation of surface heat flux, J. Comput. Appl. Math. 150 (2003), 143–155. 10.1016/S0377-0427(02)00657-XSearch in Google Scholar

[22] W. Cheng, Y. Q. Zhang, and C. L. Fu, A wavelet regularization method for an inverse heat conduction problem with convection term, Electron. J. Differential Equations 2013 (2013), no. 122, 1–9. Search in Google Scholar

[23] D. N. Hào, A non-characteristic Cauchy problem for linear parabolic equations, II: A variational method, Numer. Funct. Anal. Optim. 13 (1992), 541–564. 10.1080/01630569208816498Search in Google Scholar

[24] J. G. Wang, Y. B. Zhou, and T. Wei, A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem, Appl. Math. Lett. 26 (2013), 741–747. 10.1016/j.aml.2013.02.006Search in Google Scholar

[25] F. Yang, F. Zhang, X. X. Li, and C. Y. Huang, The quasi-boundary value regularization method for identifying the initial value with discrete random noise, Bound. Value Probl. 2018 (2018), no. 108, 1–12. 10.1186/s13661-018-1030-ySearch in Google Scholar

[26] D. N. Hào, N. V. Duc, and D. Lesnic. Regularization of parabolic equations backward in time by a non-local boundary value problem method. IMA J. Appl. Math. 75 (2010), 291–315. 10.1093/imamat/hxp026Search in Google Scholar

[27] D. N. Hào, N. V. Duc, and H. Sahli, A non-local boundary value problem method for parabolic equations backward in time, J. Math. Anal. Appl. 345 (2008), 805–815. 10.1080/00036811.2014.970537Search in Google Scholar

[28] X. L. Feng and L. Eldn, Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method, Inverse Problems 30 (2014), no. 1, 15005–15021. 10.1088/0266-5611/30/1/015005Search in Google Scholar

[29] D. N. Hào, N. V. Duc, and D. Lesnic, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems 25 (2009), no. 25, 055002. 10.1088/0266-5611/25/5/055002Search in Google Scholar

[30] W. Cheng and Y. J. Ma, A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem, Appl. Anal. 96 (2017), no. 15, 2505–2515. 10.1080/00036811.2016.1227967Search in Google Scholar

[31] F. Yang, M. Zhang, and X. X. Li, A quasi-boundary value regularization method for identifying an unknown source in the Poisson equation, J. Inequal. Appl. 2014 (2014), 1–11. 10.1186/1029-242X-2014-117Search in Google Scholar

[32] T. Wei and J. G. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math. 78 (2014), 95–111. 10.1016/j.apnum.2013.12.002Search in Google Scholar

[33] K. Kurpisz and A. J. Nowak, BEM approach to inverse heat conduction problems, Eng. Anal. Bound. Elem. 10 (1992), 291–297. 10.1016/0955-7997(92)90142-TSearch in Google Scholar

[34] Z. Qian and C. L. Fu, Regularization strategies for a two-dimensional inverse heat conduction problem, Inverse Problems 23 (2007), no. 3, 1053–1068. 10.1088/0266-5611/23/3/013Search in Google Scholar

[35] Z. Qian and Q. Zhang, Differential-difference regularization for a 2D inverse heat conduction problem, Inverse Problems 26 (2010), no. 9, 095015. 10.1088/0266-5611/26/9/095015Search in Google Scholar

[36] G. F. Wei and H. F. Gao, Two-dimensional inverse heat conduction problem using a meshless manifold method, Phys. Procedia 25 (2012), no. 22, 421–426. 10.1016/j.phpro.2012.03.106Search in Google Scholar

[37] M. Bergagio, H. Li, and H. Anglart, An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems, Int. J. Heat Mass Transf. 126 (2018), 281–292. 10.1016/j.ijheatmasstransfer.2018.04.104Search in Google Scholar

[38] H. Engl and H. Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems, Appl. Numer. Math. 4 (1988), 395–417. 10.1016/0168-9274(88)90017-7Search in Google Scholar

[39] C. Shi, C. Wang, G. H. Zheng, and T. Wei, A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem, J. Comput. Appl. Math. 279 (2015), 233–248. 10.1016/j.cam.2014.11.013Search in Google Scholar

[40] J. H. Adler, X. Z. Hu, L. Mu, and X. Ye, An a posteriori error estimator for the weak Galerkin least-squares finite-element method, J. Math. Anal. Appl. 236 (2019), 383–399. 10.1016/j.cam.2018.09.049Search in Google Scholar

[41] D. D. Trong and D. N. D. Hac, Backward problem for time-space fractional diffusion equations in Hilbert scales, Comput. Math. Appl. 93 (2021), 253–264. 10.1016/j.camwa.2021.04.018Search in Google Scholar

[42] N. V. Duc, P. Q. Muoi, and N. T. V. Anh, Stability results for backward heat equations with time-dependent coefficient in the Banach space Lp(R), Appl. Numer. Math. 175 (2022), 40–55. 10.1016/j.apnum.2022.02.002Search in Google Scholar

[43] V. A. Morozov, On the solution of functional equations by the method of regularization, Dokl. Math. 7 (1966), 414–417. Search in Google Scholar

[44] O. Scherzer, The use of Morozov’s discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems, Computing 51 (1993), 45–60. 10.1007/BF02243828Search in Google Scholar

[45] T. Bonesky, Morozov’s discrepancy principle and Tikhonov-type functionals, Inverse Problems 25 (2009), 015015. 10.1088/0266-5611/25/1/015015Search in Google Scholar

[46] C. L. Fu, Y. J. Ma, Y. X. Zhang, and F. Yang, A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Model. 39 (2015), 4103–4120. 10.1016/j.apm.2014.12.030Search in Google Scholar

[47] X. L. Feng, M. X. Zhao, and Z. Qian, A Tikhonov regularization method for solving a backward time-space fractional diffusion problem, J. Comput. Appl. Math. 411 (2022), 1–20, 114236, https://doi.org/10.1016/j.cam.2022.114236. Search in Google Scholar

[48] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996. 10.1007/978-1-4612-5338-9Search in Google Scholar

[49] W. Cheng and Y. J. Ma, A modified regularization method for an inverse heat conduction problem with only boundary value, Bound. Value Probl. 2016 (2016), no. 100, 1–14. 10.1186/s13661-016-0606-7Search in Google Scholar