Backward Heat Conduction Problem Research Articles

A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique.

Homotopy analysis method for discrete quasi-reversibility mollification method of nonhomogeneous backward heat conduction problem

Abstract In this article, the inverse time problem is investigated. Regarding the ill-posed linear problem, utilize the quasi-reversibility method first. This problem has been regularized and after that provides an iterative regularizing strategy for noisy input data that are based on homotopy. For the regularizing solution, the error analysis is proved when we employ noisy measurement data as our initial guess. Finally, numerical implementations are presented.

A variational technique of mollification applied to backward heat conduction problems

A variational technique of mollification applied to backward heat conduction problems

A new regularization for time-fractional backward heat conduction problem

Abstract It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} at a time t ≥ 0 {t\geq 0} from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) {g:=u(\,\cdot\,,\tau)} for τ > t {\tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {t\in(0,\tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.

Application of compact local integrated RBF (CLI-RBF) for solving transient forward and backward heat conduction problems with continuous and discontinuous sources

It is well-known that the inverse heat problems (IHP) are ill-posed such that we encounter to some difficulties in the numerical techniques. In the current article, a numerical technique based on the compact local integrated radial basis function (CLI-RBF) method is developed for solving IHP with continuous/discontinuous heat source. For this aim, the derivative of space variable is discretized by the CLIRBF procedure that in this manner yields a system of ODEs related to the time variable. Then, the final system of ODEs is solved by adaptive fourth-order Runge–Kutta algorithm. Finally, several challenging examples are solved by the new numerical method. The computer implementations confirm that the present method has enough accuracy for solving IHP with continuous/discontinuous heat source in one- and two-dimensional cases.

Fast parallel-in-time quasi-boundary value methods for backward heat conduction problems

In this paper we propose two new quasi-boundary value methods for regularizing the ill-posed backward heat conduction problems. With a standard finite difference discretization in space and time, the obtained all-at-once nonsymmetric sparse linear systems have the desired block ω-circulant structure, which can be utilized to design an efficient parallel-in-time (PinT) direct solver that is built upon an explicit FFT-based diagonalization of the time discretization matrix. Convergence analysis is presented to justify the optimal choice of the regularization parameter under suitable assumptions. Numerical examples are reported to validate our analysis and illustrate the superior computational efficiency of our proposed PinT methods.

Regularization of the backward stochastic heat conduction problem

Abstract In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.

DeepBHCP: Deep neural network algorithm for solving backward heat conduction problems

This paper extends a deep neural network method, a semi-supervised one, to solve backward heat conduction problems which have been long-standing computational challenges due to being ill-posed. The effectiveness and robustness of the methodology are demonstrated through various problems including different types of boundary conditions, several types of time-dependent thermal diffusivity factors, and a variety of domains for two-dimensional tests. In spite of traditional methods, there is no need to apply any regularization technique. According to simulation results, this revolutionary strategy can efficiently and accurately extract the pattern of the solutions even when the noise corruption up to ten percent is imposed on input data. Moreover, when the final time is increased further, this approach is efficient in recovering the data at the initial time, which accentuates the method's robustness. To demonstrate the superiority of the semi-supervised neural network process, we make a comparison with the radial basis functions finite difference (RBF-FD) method which is a localized RBF method.

The Backward Heat Conduction Problem With Variable Coefficients in a Spherical Domain

The backward heat conduction problem with variable coefficients in a spherical domain was considered. This problem is ill-posed, i.e., the solution (if it exists) to this problem does not depend continuously on the measured data. A projected iteration regularization method was constructed to obtain the regularized approximate solution to this inverse problem, and the convergence error estimates between the exact solution and the corresponding regularized approximate solution were given under the a priori and a posteriori parameter choice rules. Numerical results verify the effectiveness of this method.

Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem

This paper proposes a new regularization technique using spectral graph wavelet for backward heat conduction problem (BHCP) on the graph. The method uses the fourth-order compact difference for an approximation of differential operators. Meanwhile, the error estimate between the exact and wavelet regularized solution is derived. Adaptive node arrangement is obtained using spectral graph wavelet. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator. The proposed method is applied to solve BHCP on graph arising from regular one-dimensional and two-dimensional mesh. Numerical results show that we can obtain a stable solution using spectral graph wavelet regularization. Moreover, spectral graph wavelet adaptivity and fourth-order compact difference scheme lead to a very efficient solution.

Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Different from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely differentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

Regularization Technique for an Inverse Space-Fractional Backward Heat Conduction Problem

This manuscript deals with a regularization technique for a generalized space-fractional backward heat conduction problem (BHCP) which is well-known to be extremely ill-posed. The presented technique is developed based on the Meyer wavelets in retrieving the solution of the presented space-fractional BHCP. Some sharp optimal estimates of the Holder-Logarithmic type are theoretically derived by imposing an a-priori bound assumption via the Sobolev scale. The existence, uniqueness and stability of the considered problem are rigorously investigated. The asymptotic error estimates for both linear and non-linear problems are all the same. Finally, the performance of the proposed technique is demonstrated through one- and two-dimensional prototype examples that validate our theoretical analysis. Furthermore, comparative results verify that the proposed method is more effective than the other existing methods in the literature.

Compact filtering as a regularization technique for a backward heat conduction problem

In this paper, the backward heat conduction problem is solved numerically using a fourth-order compact difference scheme. For the regularization of this ill-posed problem, fourth-order compact filtering which is a spatial filtering technique is proposed. The second derivative approximation in compact difference scheme has been compared with the fourth-order standard finite difference scheme via Fourier analysis. The comparison shows that the resolution ability of the compact difference scheme is better than the standard finite difference scheme. Meanwhile, an error estimate between the exact and regularized solution is derived. Numerical results are provided using the proposed method. The comparison of CPU time shows that the fourth-order compact difference scheme takes lesser CPU time than the fourth-order standard finite difference scheme.

Analytic series solutions of 2D forward and backward heat conduction problems in rectangles and a new regularization

In the paper, we solve a non-homogeneous heat conduction equation with non-homogeneous boundary conditions in a 2D rectangle. First, we derive the domain/boundary integral equations for both the forward and backward heat conduction problems. Then, by using the technique of homogenization, inserting the adjoint Trefftz test functions into the derived integral equations and expanding the solutions in terms of eigenfunctions, we can obtain the expansion coefficients in closed form. Hence, the analytic series solutions of forward heat conduction problems (FHCPs) and backward heat conduction problems (BHCPs) are available. For the FHCPs, only a few terms in the series render very high-order accurate solutions at any time, with errors of the order . For the BHCPs, we require to modify the closed-form series solutions via a new spring-damping regularization technique. Numerical tests for the BHCPs in a large space-time domain reveal that the present analytic series solution is very accurate to recover the initial temperature with an error of the order , although the measured final time temperature is very small when is large up to 100 and is even polluted by a large relative noise up to the level .

Nonhomogeneous backward heat conduction problem: Compact filter regularization and error estimates

In this paper, compact filter regularization method is introduced for the numerical solution of nonhomogeneous backward heat conduction problem. Compact filter regularization is a new, simple and convenient regularization method. The error estimate for this method is provided. Meanwhile, the numerical implementation is discussed. Numerical results conclude that the proposed algorithm is efficient and effective.

Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.

The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient

In the fields of continuous casting and the roll stepped cooling, the heat transfer coefficient is piecewise linear. However, few papers discuss the solution of the backward heat conduction problem in this situation. Therefore, the aim of this paper is to solve the backward heat conduction problem, which has the piecewise linear heat transfer coefficient. Firstly, the ill-posed of this problem is discussed and the truncated regularized optimization scheme is introduced to solve this problem. Secondly, because the regularization parameter is the key factor for the regularization method, this paper presents an improved method for choosing the regularization parameter to reduce the iterative number and proves the fourth-order convergence of this method. Furthermore, the numerical simulation experiments show that, compared with other methods, the improved method of fourth-order convergence effectively reduces the iterative number. Finally, the truncated regularized optimization scheme is used to estimate the initial temperature, and the results of numerical simulation experiments illustrate that the inverse values match the exact values very well.

Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting method

In this article, an explicit Lie-group shooting method (LGSM) is developed to solve the time-dependent heat source and the initial data for backward heat conduction problems. To recover both unknown data simultaneously, it is very difficult to obtain a stable solution by explicit or implicit schemes. To solve these problems by using conventional numerical schemes, numerical iterative regularization techniques and numerical integration techniques are necessary. To avoid these numerical techniques and to increase the computational efficiency, an explicit LGSM is developed. According to the solution of the quadratic equation of the LGSM, the initial condition can be directly obtained by using the final condition and boundary conditions at the initial time and final time. Using the reciprocal relationship of the solutions for the initial condition and the final condition, the proposed algorithm can avoid numerical integration and numerical iteration. Additionally, a closed-form formula from a two-point Lie-group equation can be directly used to calculate the heat source term. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and stable solution, even with noisy measurement data, by comparing the estimation results with the existing literature.

A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.