A gradient descent method for solving an inverse coefficient heat conduction problem

An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient. These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration convergence rates are compared for different descent parameters.     

Wavelet regularization for an inverse heat conduction problem

In this paper we consider an inverse heat conduction problem 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. The Meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero.     

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.     

Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods

A method for constructing numerical schemes for an inverse coefficient heat conduction problem with boundary measurement data and piecewise-constant coefficients is considered. Some numerical schemes for a gradient optimization algorithm to solve the inverse problem are presented. The method is based on locally-adjoint problems in combination with approximation methods in Hilbert spaces.     

On a radially symmetric inverse heat conduction problem

In this paper, we treat an inverse problem for a radially symmetric heat equation, which arises from non-destructive evaluation by thermal imaging. The problem can also be considered as an inverse heat conduction problem. Based on a weighted energy method, we give a conditional stability estimate. A feasible regularization method is provided for numerical simulation. The reconstruction experiment is done for verifying the efficiency of the regularization method.     

A modified method for a non-standard inverse heat conduction problem

A non-standard inverse heat conduction problem is considered. Data are given along the line x = 1 and the solution at x = 0 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 solve the problem numerically it is necessary to employ some regularization method. In this paper, we study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the heat equation. A numerical implementation is considered and a simple example is given. Some numerical results show the usefulness of the modified method.     

A numerical approximation to the solution of an inverse heat conduction problem

The aim of this article is to study the parabolic inverse problem of determination of the leading coefficient in the heat equation with an extra condition at the terminal. After introducing a new variable, we reformulate the problem as a nonclassical parabolic equation along with the initial and boundary conditions. The uniqueness and continuous dependence of the solution upon the data are demonstrated, and then finite difference methods, backward Euler and Crank–Nicolson schemes are studied. The results of some numerical examples are presented to demonstrate the efficiency and the rapid convergence of the methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

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 for a spherically symmetric inverse heat conduction problem

In this paper, we consider a spherically symmetric inverse heat conduction problem of determining the internal surface temperature of a hollow sphere from the measured data at a fixed location inside it. This is an ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Tikhonov type’s regularization method and a Fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates.     

多层介质中逆热传导问题的傅里叶正则化方法

逆热传导问题是数学物理反问题中的热点和前沿课题之一,在钢铁生产等领域中具有重要的应用背景.讨论一个多层介质中的逆热传导问题,它是一个极度不适定问题.通过傅里叶截断方法构造正则化近似解,并给出相应的稳定性估计.     

On an initial inverse problem in nonlinear heat equation associated with time-dependent coefficient

In this paper, a nonlinear backward heat problem with time-dependent coefficient in the unbounded domain is investigated. A modified regularization method is established to solve it. New error estimates for the regularized solution are given under some assumptions on the exact solution.     

Uniqueness of the density in an inverse problem for isotropic elastodynamics

We consider the unique determination of the density of a nonhomogeneous, isotropic elastic object from measurements made at the surface. We model the behavior of the bounded, 3-dimensional object by the linear, hyperbolic system of operators for isotropic elastodynamics. The material properties of the object (its density and elastic properties) correspond to the smooth coefficients of these differential operators. The data for this inverse problem, in the form of the correspondence between applied surface tractions and resulting surface displacements, is modeled by the dynamic Dirichlet-to-Neumann map on a finite time interval. In an earlier paper we show that the speeds of (compressional and sheer) wave propagation through the object are uniquely determined by the Dirichlet-to-Neumann map. Here we extend that result by showing that the density is also determined in the interior by the Dirichlet-to-Neumann map in the case, for example, that at only isolated points in the object. We use techniques from microlocal analysis and integral geometry to solve this fully three-dimensional problem.

     

An inverse heat conduction problem with a nonlinear source term

In this paper, the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered. The surface heat flux history of a heated conducting body will be identified. The approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data. Some numerical examples will also be presented.     

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.     

Determination of the coefficient of heat output in the inverse problem for the heat equation

To determine the variable coefficient of heat output in a boundary condition of third kind we propose applying a representation of the solution of the one-sided heat conduction problem obtained by the symbolic operator method. We consider the cases of the one- and two-dimensional inverse problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 9–13.     

On the unsolvability of the boundary inverse problem of heat conduction

It is shown that the boundary inverse nonstationary problem of heat conduction with given heat flux and temperature on one of the boundary surfaces is essentially ill-posed, since it has no solution in the class of continuous functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 114–118.     

The inverse coefficient problem for unions of homogeneous bodies

We propose a model that makes it possible to determine the complex of thermophysical characteristics of a union of homogeneous bodies based on the solution of the nonstationary two-dimensional problem of thermal conductivity for a semibounded body with a foreign platelike inclusion perpendicular to the boundary surface; the body is subject to rapid heating by heat fluxes along thin strips.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 63–66.     

Uniqueness and stability of the minimizer for a binary functional arising in an inverse heat conduction problem

The local well-posedness of the minimizer of an optimal control problem is studied in this paper. The optimization problem concerns an inverse problem of simultaneously reconstructing the initial temperature and heat radiative coefficient in a heat conduction equation. Being different from other ordinary optimization problems, the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters. Particularly, since the status of the two unknown coefficients in the cost functional are different, the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem. The necessary condition which must be satisfied by the minimizer is deduced. By assuming the terminal time T is relatively small, an L² estimate regarding the minimizer is obtained, from which the uniqueness and stability of the minimizer can be deduced immediately.     

On the inverse problem of finding the coefficient of heat transfer

We establish conditions for the existence and uniqueness of a solution to the problem of finding the coefficient of heat transfer in a boundary condition of third kind. We give an explicit formula for computing this coefficient. Translated fromMatermaticheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 45–50.