# Numerical investigation of an inverse problem based on regularization method

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## Abstract

In this study, a numerical approach of the spectral collocation method coupled with a regularization technique is applied for solving an inverse parabolic problem of the heat equation in a quarter plane. The problem includes the estimation of an unknown boundary condition from an overspecified condition. The stable solution of the problem exists and is proved by Tikhonov regularization technique. The algorithm works without any mesh points or elements, and accurate results can be obtained efficiently. By employing the numerical algorithm on the problem, the resultant matrix equation is ill-condition. To regularize this matrix equation, we apply regularization technique, with the L-curve and general cross-validation criteria for choosing the regularization parameter. For demonstrating the performance and ability of the proposed algorithm, a test example is presented. The numerical results showed that the solution obtained with the algorithm designed in this paper is stable with the noisy data and the unknown boundary condition was recovered very well.

## Keywords

Inverse heat conduction problem Collocation method Ill-conditioned system Regularization method## Introduction

Inverse problems of partial differential equations have many applications in different sciences, e.g., physics, engineering, geoscience, finance, etc. In the last decades, the scientific research has begun to focus on inverse problems of heat conduction equation, e.g., for more details, see [1, 2, 3, 4, 5, 6, 7].

The heat conduction problems include two categories: direct and inverse problems. The direct problems are related to the determination of unknown temperature distribution at the domain in the presence of the known initial and boundary conditions and other physical properties. The inverse problems consist of not only determination of temperature distribution, but also an estimation of the diffusion coefficient or temperature/heat flux on the surface or other unknown parameters from the extra condition which is given usually by temperature measurements taken within the body.

The major difficulty of IHCPs is their instability, that is, the small error in input data causes the large change in output data according to Hadamard definition [1, 2, 3].

Due to the instability of the inverse problem, the computational treatment of such problems needs applying more accurate numerical methods to obtain a stable approximate solution. In recent years, IHCPs have been investigated from the analytical and numerical point of view, in the literature, e.g., [3, 4, 5, 6, 8, 9, 10].

Some more relevant works are published regarding the estimation of boundary condition and heat flux in [9, 11, 12, 13, 14].

We present in this paper a numerical algorithm based on collocation method for an IHCP, which does not require domain discretization. The solution is constructed from the linear combination of basis functions which are chosen in an appropriate way.

By using the solution of the direct problem, a numerical approach based on the collocation method is introduced. An ill-conditioned system is derived from discretization. The condition number of the resultant matrix increases with respect to the number of collocation points, and so by using standard algorithms, we cannot achieve good accuracy. Some kind of regularization is necessary for finding a stable solution.

The outline of this paper is organized as follows:

In Sect. 2, a mathematical model of our interest problem is introduced. In Sect. 3, the Tikhonov regularization method is applied to construct a stable solution of the ill-posed integral equation. Section 4 contains a numerical procedure based on the collocation method for solving the inverse problem. In Sect. 5, the regularization method for discrete ill-posed problem is applied to find a stable solution and introduced GCV and L-curve criteria for determining regularization parameter. In Sect. 6, a test problem is considered to investigate the validity and efficiency of the method. Section 7 contains the conclusions.

## Problem formulation

*M*is a positive constant.

*x*= 1 and using (2.5), we obtain

The above first-kind integral Eq. (2.8) cannot be reduced into an integral equation of the second kind by differentiation, and so the problem is ill-posed. We can uniquely determine \(g\left( t \right)\) subject to \(G\left( t \right)\) belong to the range of \(A^{ - 1}\). Since the \(A^{ - 1}\) is unbounded, the noisy data of \(p\left( t \right)\) lead to large errors in \(g\left( t \right)\). In the next section, we show the existence stable solution of the operator integral Eq. (2.9) by Tikhonov regularization technique in continuous form.

## The regularization method for ill-posed integral equation

The regularized solution \(g_{\lambda }\) for Eq. (2.9) is constructed by using the following minimization problem, which is stated by the following theorem:

### **Theorem 1**

*For every function*\(G \in L_{2} \left[ {0,T} \right]\)

*and any positive*\(\lambda ,\)

*there exists an element*\(g_{\lambda } \in W_{2}^{1} \left[ {0,T} \right]\)

*such that the functional*\(M^{\lambda } \left[ {g,G} \right]\)

*attains its greatest lower bound*.

### *Proof*

Equation (3.2) is called the Euler–Lagrange equation, and the minimizer \(g_{\lambda }\) of Tikhonov functional is determined by the solution of this equation. The solution of Eq. (3.2) with consistency conditions (3.3) is unique based on classical theorems in the ODE.

Now, the regularized solution of the above minimization problem, i.e.,\(g_{\lambda } ,\) is assumed in the form \(g_{\lambda } = R\left( {G,\lambda } \right),\) where \(\lambda = \lambda \left( {\delta ,G_{\lambda \left( \delta \right)} } \right)\), \(R\left( {G,\lambda } \right)\) is a regularizing operator, and \(\delta\) is the error level of input data. The discrete form of Eq. (2.9) based on the proposed numerical algorithm is investigated in Sect. 4.

## The collocation method

*t*; by using the collocation method at discrete points \(t_{j} ,\) we have

## Regularization method for discrete ill-conditioned system

The ill-posedness of the operator integral Eq. (2.9) results in the ill-conditioned system (4.2) from the discretization by the proposed algorithm. Because of the ill-conditionary of matrix A in Eq. (4.2), some type of regularization must then be applied to get stable and accurate results [16, 17, 18]. For the linear system (4.2), we are faced with the same issues of Hadamard definition, and so we must investigate the sensitivity analysis of solution with respect to random noise in input data \(p\left( t \right)\). In our computation, the zero-order Tikhonov regularization technique is applied to solve the matrix Eq. (4.2).

The regularization method for the discrete ill-posed problem of the form (4.2) is an extension of the least square method (LSM) to find a stable solution for \(AC = G\). The definition of the regularization method is as the following:

- (I)
L-curve

The L-curve criterion is defined by the log–log plot regularized solution versus the norm of residual and is sketched in the following:The graph of the curve is as the L-shape and so is called L-curve, and the corner of curve is selected as a suitable regularization parameter \(\lambda\) [18].$$L = \left\{ {\left( {\log \left(||{C_{\lambda }||^{2} } \right),\quad \log \left( ||{AC_{\lambda } - G||^{2} } \right)} \right), \quad \lambda > 0} \right\}$$ - (II)
GCV

The GCV criterion is an approach for choosing the regularization parameter [18, 19]. The GCV criterion leads to minimization of the following GCV function with respect to \(\lambda\):where \(A_{\lambda } = \left( {A^{\text{tr}} A + \lambda I} \right)^{ - 1} \cdot A^{\text{tr}}\) is a matrix which produces the regularized solution \(C_{\lambda }^{\delta }\) of Eq. (6.1) from the normal equation$$G\left( \lambda \right) = \frac{||{AC_{\lambda }^{\delta } - G^{\delta}||^2 }}{{\left( {{\text{trace}}\left( {I - AA_{\lambda } } \right)} \right)^{2} }} ,$$$$\left( {A^{\text{tr}} A + \lambda I} \right)C_{\lambda }^{\delta } = A^{\text{tr}} G^{\delta } .$$

## Numerical results and discussion

### Test problem

The obtained numerical results obtained for both exact and noisy data are reported.

*g*(

*t*) without regularization is plotted in Fig. 3.

Condition number of matrix A corresponding to the collocation points *n*

| Cond (A) | RMSE—Tikhonov | RMSE—unregularization |
---|---|---|---|

5 | 10.7130 | 1.6739 | 1.8374 |

10 | 9.4538 × 10 | 0.8106 | 2.4763 × 10 |

15 | 8.6240 × 10 | 0.6122 | 5.2312 × 10 |

20 | 3.4086 × 10 | 0.4656 | 4.3863 × 10 |

35 | 9.5507 × 10 | 0.3390 | 2.1258 × 10 |

50 | 6.4591 × 10 | 0.2617 | 6.5804 × 10 |

Condition number of matrix A in Eq. (4.2) corresponding to the collocation points *n*

| Cond (A) | RMSE—Tikhonov | RMSE—unregularization |
---|---|---|---|

5 | 10.7130 | 0.5799 | 0.5696 |

10 | 9.4538 × 10 | 0.5146 | 0.6383 |

15 | 8.6240 × 10 | 0.2807 | 3.6452 |

20 | 3.4086 × 10 | 0.1734 | 2.8813 × 10 |

35 | 9.5507 × 10 | 0.0607 | 3.9181 × 10 |

50 | 6.4591 × 10 | 0.0366 | 9.2144 × 10 |

RMSE values and regularization parameters for different criteria (with *n *= 50)

Method | Reg parameter | RMSE |
---|---|---|

L-curve | 5.1300 × 10 | 0.036638219244414 |

GCV | 2.1090 × 10 | 0.036638228291466 |

Representation of RMSE values for different values of \(\delta\) without regularization and in the presence of regularization method

RMSE | \(\delta\) | ||
---|---|---|---|

\(\delta = .01\) | \(\delta = .001\) | \(\delta = .005\) | |

RMSE—unregularization | 1.3899 × 10 | 2.255 × 10 | 1.1002 × 10 |

RMSE—Tikhonov | 0.0617 | 0.0399 | 0.0463 |

## Conclusions

In this paper, the spectral collocation method along with Tikhonov regularization technique was successfully applied to the solution of IHCP to determine the unknown heat temperature on the boundary by employing temperature reading at a fixed point. A test example involving different collocation point and measurement errors with noise was considered.

The results show that the approximate solution obtained by the proposed algorithm and using regularization techniques remain stable as the collocation points are increased even when noise is included in the input data.

## Notes

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