Backward Heat Conduction Problem Research Articles

A new shooting method for quasi‐boundary regularization of multi‐dimensional backward heat conduction problems

We employ a quasi‐boundary regularization to construct a two‐point boundary value problem for multi‐dimensional backward heat conduction equations. The multidimensional backward heat conduction problem (BHCP) is renowned as severely ill‐posed because the solution does not fullly depend on the data. In order to numerically tackle the multi‐dimensional BHCP, we propose a Lie‐group shooting method (LGSM) in the time direction to find the unknown initial conditions. The pivot point is based on the establishment of a one‐step Lie group element G(T) and the construction of a generalized mid‐point Lie group element G(r). Then, by imposing G(T) = G(r) we can search for the missing initial conditions through a minimum discrepancy to the real targets by the numerical ones, in terms of the weighting factor r ? (0, 1). When numerical examples are tested, we find that the LGSM is applicable to the BHCP. Even with noisy final data, the LGSM is also robust against disturbance.

Convergence analysis of gradient descend methods generated by two different functionals in a backward heat conduction problem

A backward heat conduction problem is reduced to an operator equation Aq = ƒ and is solved in two versions by minimization of functionals R(q) = 〈Aq – 2ƒ, q〉 and J(q) = 〈Aq – ƒ, Aq – ƒ〉. The performance of the versions is compared both theoretically and numerically. An a priori convergence rate estimate has been obtained in the case of exact data and an a posteriori two-side estimate of the solution error norm has been derived in the case of noisy data. The estimates are illustrated by a model image deblurring problem. Numerical experiments demonstrated that minimization of J(q) is more effective for noisy data while R(q) works better for exact data.

The backward group preserving scheme for 1D backward in time advection‐dispersion equation

AbstractIn this article, we propose a backward group preserving scheme (BGPS) on the advection‐dispersion equation (ADE) for tackling of the contamination problems. The BGPS has been successfully applied on the backward heat conduction problems as well as the backward in time Burgers equation, but it has never been applied to solve the ADE. The BGPS is able to recover the spatial distribution of groundwater contaminant concentration in this work. Several numerical examples are worked out, and we show, based on those numerical examples, that the BGPS is applicable to the ADE and the method can also handle the ADE with piecewise constant dispersion coefficients. When a steep gradient is appeared in the solution, several steps of the BGPS can be used to retrieve the desired initial data and its result is better than the marching‐jury backward beam equation (MJBBE) method as far as our examples are concerned. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution

Consider a two-dimensional backward heat conduction problem for a general domain with a boundary. Based on the fundamental solution to the heat equation, we propose to solve this problem by the boundary integral equation method, which generates a coupled ill-posed integral equation. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. Our regularizing scheme can be considered a quasi Tikhonov regularization, with the advantage of a relatively small amount of computation compared with the classical Tikhonov regularization. Numerical performances are given to show the validity of our inversion method.

ON THREE SPECTRAL REGULARIZATION METHODS FOR A BACKWARD HEAT CONDUCTION PROBLEM

We introduce three spectral regularization methods for solving a backward heat conduction problem (BHCP). For the three spectral regularization methods, we give the stability error estimates with optimal order under an a-priori and an a-posteriori regularization parameter choice rule. Numerical results show that our theoretical results are effective.

Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region

In this paper we consider a numerical approximation of solution of nonhomogeneous backward heat conduction problem (BHCP) in bounded region based on Tikhonov regularization method. Error estimate at t = 0 for this method is provided. According to the error estimate, a selection of regularization parameter is given. Meanwhile, a numerical implementation is described and the numerical results show that our algorithm is effective.

Some Numerical Experiments on Global Simulation of the Backward Heat Conduction Problem

Some Numerical Experiments on Global Simulation of the Backward Heat Conduction Problem

A new shooting method for quasi-boundary regularization of backward heat conduction problems

A quasi-boundary regularization leads to a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G( T) and the formation of a generalized mid-point Lie group element G( r). Then, by imposing G ( T ) = G ( r ) we can search for the missing initial conditions through a minimum discrepancy of the targets in terms of the weighting factor r ∈ ( 0 , 1 ) . Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the final temperature is almost undetectable and/or is disturbed by large noise, the Lie group shooting method is stable to recover the initial temperature very well.

A modified method for a backward heat conduction problem

We consider a backward heat conduction problem in a strip, where data is given at the final time t = T( T > 0) and a solution for 0 ⩽ t < T is sought. The problem is ill-posed in the sense that the solution(if it exists) does not depend continuously on the data. In order to numerically solve the problem, we study a modification of the equation, where a third-order mixed derivative term is added. Error estimates for this problem are given, which show that the modified problem is stable and its solution is an approximation of the backward heat conduction problem. Some numerical tests illustrate that the proposed method is feasible and effective.

Global Simulation of a Backward Heat Conduction Problem with a Variable Transform on Time

Global Simulation of a Backward Heat Conduction Problem with a Variable Transform on Time

Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems

In this paper we are concerned with the backward problems governed by differential equations. It is afirst timethatwe can constructabackward timedy- namics on the past cone, such that an augmented dynam- ical system of the Lie type ˙ X = B(X,t)X with t ∈ R − , X ∈ M n+1 lying on the past cone and B ∈ so(n,1) ,w as derived for the backward differential equations system ˙ x = f(x,t), t ∈ R − ,x ∈ R n . These two differential equa- tions systems are mathematically equivalent. Then we apply the backward group preserving scheme (BGPS), which is an explicit single-step algorithm formulated by an exponential mapping to preserve the group preperties of SOo(n,1), on the backward heat conduction problem (BHCP). It can retrieve all the initialdata withhigh order accuracy. Several numerical examples of the BHCP were work out,and weshow that theBGPS is applicableto the BHCP, even those of strongly ill-posed ones. Under the noisy final data the BGPS is also robust to against the disturbance. The one-step BGPS effectively reconstructs the initial data from a given final data, with a suitable grid length resulting into a high accuracy never seen be- fore. The results are very significant in the computations of BHCP. keyword: Past cone dynamics, Backward group pre- serving scheme, Backward heat conduction problem, Strongly ill-posed problem.

Two numerical methods for solving a backward heat conduction problem

We introduce a central difference method and a quasi-reversibility method for solving a backward heat conduction problem (BHCP) numerically. For these two numerical methods, we give the stability analysis. Meanwhile, we investigate the roles of regularization parameters in these two methods. Numerical results show that our algorithm is effective.

Error estimates of a difference approximationmethod for a backward heat conduction problem

We introduce a central difference method for a backward heat conduction problem (BHCP). Error estimates for this method are provided together with a selection rule for the regularization parameter (the space step length). A numerical experiment is presented in order to illustrate the role of the regularization parameter.

A numerical technique for backward inverse heat conduction problems in one-dimensional space

In this article, we consider an inverse heat conduction problem in one-dimensional space with two unknown terms. In this problem, the initial temperature and histories of temperature in the finite interval of time, may be determined. Then, by using a semi-implicit finite difference method, an estimation of these unknown terms will be found. Obviously, this problem is ill-posed. But by applying a suitable parameter, we can obtain the best of these estimations. Finally, some numerical experiments are given in the final section of this paper.

The method of fundamental solutions for the backward heat conduction problem

In this article a meshless numerical scheme for solving the backward heat conduction problem (BHCP) is proposed. The numerical solution is developed by using the fundamental solution of the heat equation as a basis function. The standard Tikhonov regularization technique and the L-curve method are adopted for solving the resultant ill-conditioned system of linear algebraic equations. The convergence and the stability of the method are investigated for a severe test example, hence revealing the computational performance and limitations of the method proposed. Numerical results are presented for bo th the one-dimensional and the two-dimensional backward heat conduction problems.

Numerical solution of backward heat conduction problems by a high order lattice‐free finite difference method

We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our method we show two tools using Fourier transform and Taylor expansion, respectively. On the other hand, the backward heat conduction problem is a typical example of ill‐posed problems in the sense that the solution is unstable for errors of data. Our aim is creation of a meshless method which can be applied to the ill‐posed problem. From numerical experiments we confirmed that our method is effective in solving two‐dimensional backward heat conduction equations subject to the Dirichlet boundary condition.

Lattice-free Finite Difference Method ForBackward Heat Conduction Problems

We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our method we show two tools using Fourier transform and Taylor expansion, respectively. On the other hand, the backward heat conduction problem is a typical example of ill-posed problems in the sense that the solution is unstable for errors in data. Our aim is to create a measles method which can be applied to the ill-posed problem. From numerical experiments we confirmed that our method is effective in order to solve the two-dimensional backward heat conduction equation subject to mixed boundary conditions.

Group preserving scheme for backward heat conduction problems

In this paper we numerically integrate the backward heat conduction equation ∂ u/∂ t= ν▵ u, in which the Dirichlet boundary conditions are specified at the boundary of a certain spatial domain and a final data is specified at time T>0. In order to treat this ill-posed problem we first convert it through the transformation s= T− t to an unstable initial-boundary-value problem: ∂ u/∂ s=− ν▵ u together with the same boundary conditions and the same data at s=0. Then, we consider the contraction map of u to v=exp[− as] u by a suitable contraction factor a>0, which is analyzed by considering the stability of the semi-discretization numerical schemes. The resulting ordinary differential equations at the interior grid points are then numerically integrated by the group preserving scheme, proposed by Liu [Int. J. Non-Linear Mech. 36 (2001) 1047], and the stable range of the index r= νΔ t/(Δ x) 2 is derived. Numerical tests for both forward and backward heat conduction problems are performed to confirm the effectiveness of the new numerical methods.

Different approaches for the solution of a backward heat conduction problem

This work presents a comparison of three different techniques to solve the inverse heat conduction problem involving the estimation of the unknown initial condition for a one-dimensional slab, whose solution is obtained through minimization of a known functional form. The following techniques are employed to solve the inverse problem: the conjugate gradient method with the adjoint equation, regularized solution using a quasi-Newton method, and regularized solution via genetic algorithm (GA) method. For the first one, a general form to compute the gradient of the functional form (considering the time and space domains) is presented, and for the GA method a new genetic operator named epidemical is applied.

Solution of inverse diffusion problems by operator-splitting methods

In this paper the inverse solution of the general (non-linear) diffusion problem or backward heat conduction problem. It is assumed that the direct solution can be satisfactorily modelled, for example by the finite difference method. The nature of the problem and typical approaches to its solution are briefly reviewed. An operator-splitting method is introduced as a means of solving the inverse diffusion problem. An error analysis of the method is given, particularly for the application of the method to the simple diffusion equation. The method is applied to a range of test problems to illustrate the points of the analysis and to demonstrate the properties and performance of the method.