On the coefficient inverse problem of heat conduction in a degenerating domain

ABSTRACT

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In conclusion, statements of possible generalizations and the results of numerical calculations are given.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

A comparison of some inverse methods for estimating the initial condition of the heat equation

In this work we analyze two explicit methods for the solution of an inverse heat conduction problem and we confront them with the least-squares method, using for the solution of the associated direct problem a classical finite difference method and a method based on an integral formulation. Finally, the Tikhonov regularization connected to the least-squares criterion is examined. We show that the explicit approaches to this inverse heat conduction problem will present disastrous results unless some kind of regularization is used.     

On Determining Sources with Compact Supports in a Bounded Plane Domain for the Heat Equation

The inverse problem of determining the source for the heat equation in a bounded domain on the plane is studied. The trace of the solution of the direct problem on two straight line segments inside the domain is given as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions whose derivatives satisfy the Hölder condition.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.     

Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

An Inverse Heat Conduction Problem

In this paper, an inverse heat conduction problem will be considered. By reducing this inverse problem and using an overspecified condition, it is shown that the solution to the problem exists, and this solution is unique.AMS Subject Classification (2000): 45D05, 34A55     

Uniqueness of the finite-difference solution of the inverse problem of heat conduction

The article examines the problem of determination of the coefficients of heat conduction and heat capacity from a system of difference equations for the equation of heat conduction with additional information about the solution of the difference problem. Uniqueness of the solution of the inverse problem is proved.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 22–29, 1993.     

Removal of numerical instability in the solution of an inverse heat conduction problem

In this paper, we consider an inverse heat conduction problem (IHCP). A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a transformation, the ill-posed IHCP becomes a Cauchy problem. Since the solution of Cauchy problem, exists and is unique but not always stable, the ill-posed problem is closely approximated by a well-posed problem. For this new well-posed problem, the existence, uniqueness, and stability of the solution are proved.     

Two regularization methods to identify time-dependent heat source through an internal measurement of temperature

This paper deals with an inverse problem for identifying an unknown time-dependent heat source in a one-dimensional heat equation, with the aid of an extra measurement of temperature at an internal point. Since this problem is ill-posed, two regularization solutions are obtained by employing a Fourier truncation regularization and a Quasi-reversibility regularization. Furthermore, the Hölder type stability estimate between the regularization solutions and the exact solution, are obtained, respectively. Numerical examples show that these regularization methods are effective and stable.     

FOURIER REGULARIZATION FOR DETERMINING SURFACE HEAT FLUX FROM INTERIOR OBSERVATION BASED ON A SIDEWAYS PARABOLIC EQUATION

In this paper we consider a non-standard inverse heat conduction problem for determining surface heat flux from an interior observation which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier method is applied to formulate a regularized approximation solution, and some sharp error estimates are also given.     

On regularization and discretization control for the numerical solution of inverse problems in parabolic equations

In this paper, the influence of modelling, a priori information, discretization and measurement error to the numerical solution of inverse problems is investigated. Given an a priori approximation of the unknown parameter function in a parabolic problem, we propose a strategy for the regularized determination of a skeleton solution to the inverse problem. This strategy is based on a discretization control of the forward problem in order to find a trade-off between accuracy and computational efficiency. Numerical results with regard to a nonlinear inverse heat conduction problem illustrate the study.     

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.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

Numerical solution of the problem of optimal control of the heating of a thermoelastic plate by internal heat sources

We give the statement and numerical solution of the problem of optimal control (in the sense of rapidity of response) of the heating of an unbounded plate by internal heat sources in the presence of restrictions on the maximal (in absolute value) thermal stresses. In constructing the solution of this nonlinear optimal control problem we use the method of the inverse heat conduction problem.     

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.     

Identification of the thermal growth characteristics of coagulated tumor tissue in laser‐induced thermotherapy

We consider an inverse problem arising in laser‐induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues are destroyed by coagulation. For the dosage planning, numerical simulations play an important role. To this end, a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a nonconvex cost functional in this paper. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient. Numerical implementations are presented to show the validity of the optimization schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

An Inverse Problem for the Heat Equation II

Completeness of the set of products of the derivatives of the solutions to the equation ( av ')' m u v = 0, v (0, u ) = 0 is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $ \dot u = (a(x)u')' $ , u ( x , 0) = 0, u (0, t ) = 0, u (1, t ) = f ( t ) known for all t > 0, from the heat flux a (1) u '(1, t ) = g ( t ). Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point x = 0 in our case) does not have a unique solution, in general.