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.     

A simplified Tikhonov regularization method for determining the heat source

This paper is to discuss the inverse problem of determining a spacewise dependent heat source in one-dimensional heat equation in a bounded domain where data is given at some fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The regularization solution is given by a simplified Tikhonov regularization. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable.     

The revised generalized Tikhonov regularization for the inverse time-dependent heat source problem

This paper investigates the inverse problem of finding a time-dependent heat source in a parabolic equation where the data is given at a fixed location. A conditional stability result is given, and a revised generalized Tikhonov regularization method with error estimate is also provided. Numerical examples show that the regularization method is effective and stable.     

A quasi‐boundary value regularization method for determining the heat source

In this paper, we consider the inverse problem of determining the heat source, which depends only on spatial variable in one‐dimensional heat equation in a bounded domain where data is given at some fixed time. A conditional stability result is given, and a quasi‐boundary value regularization method is also provided. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable. Copyright © 2013 John Wiley & Sons, Ltd.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.     

The modified regularization method for identifying the unknown source on Poisson equation

This paper discusses the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation from one supplementary temperature measurement at an internal point. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the modified regularization method. For the regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is given. Numerical results are presented to illustrate the accuracy and efficiency of this method.     

Simultaneous identification of diffusion coefficient,spacewise dependent source and initial value for one‐dimensional heat equation

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one‐dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross‐validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

A numerical method for identifying heat transfer coefficient

In this paper, we consider an inverse problem of heat equation with Robin boundary condition for identifying heat transfer coefficient. Combining the method of fundamental solutions with discrepancy principle for the choice of the locations for source points, we give a method for solving the reconstruction problem. Since the resultant matrix is severe ill-conditioning, Tikhonov regularization with L-curve method is employed. Some numerical examples are given for verifying the efficiency and accuracy of the presented method.     

The truncation method for identifying an unknown source in the Poisson equation

This paper deals with the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation, with the aid of an extra measurement at an internal point. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. We obtain the regularization solution by the truncation method. For the regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is given. Numerical results are presented to illustrate the accuracy and efficiency of this method.     

Two Tikhonov‐type regularization methods for inverse source problem on the Poisson equation

In this paper, we investigate a problem of the identification of an unknown source on Poisson equation from some fixed location. A conditional stability estimate for an inverse heat source problem is proved. We show that such a problem is mildly ill‐posed and further present two Tikhonov‐type regularization methods (a generalized Tikhonov regularization method and a simplified generalized Tikhonov regularization method) to deal with this problem. Convergence estimates are presented under the a priori choice of the regularization parameter. Numerical results are presented to illustrate the accuracy and efficiency of our methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

In this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation. Under an a priori condition we answer the question concerning the best possible accuracy for the problem. The Fourier regularization method is utilized for solving the problem, and its convergent rate is analyzed. Numerical results are presented to illustrate the accuracy and efficiency of the method.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

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.     

Numerical solution of forward and backward problem for 2-D heat conduction equation

For a two-dimensional heat conduction problem, we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements. The conditional stability for this inverse problem and the error analysis for the Tikhonov regularization are presented. An implicit inversion method, which is based on the regularization technique and the successive over-relaxation (SOR) iteration process, is established. Due to the explicit difference scheme for a direct heat problem developed in this paper, the inversion process is very efficient, while the application of SOR technique makes our inversion convergent rapidly. Numerical results illustrating our method are also given.     

A regularization method for solving the radially symmetric backward heat conduction problem

This work is devoted to solving the radially symmetric backward heat conduction problem, starting from the final temperature distribution. The problem is ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified Tikhonov regularization method is proposed for solving this inverse problem. A quite sharp estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter. A numerical example is presented to verify the efficiency and accuracy of the method.     

A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem

In this paper, we consider a radially symmetric inverse heat conduction problem of determining the surface heat flux distribution from a fixed location inside a cylinder. This problem is ill-posed in the Hadamard sense and a conditional stability estimate is given for it. A modified quasi-boundary value regularization method is applied to formulate a regularized solution, and a sharp error estimate between the approximate solution and the exact solution is established by choosing a suitable regularization parameter. A numerical example is presented to verify the efficiency of the regularization method.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Fourier regularization method of a sideways heat equation for determining surface heat flux

We consider a non-standard inverse heat conduction problem in a quarter plane which appears in some applied subjects. We want to know the surface heat flux in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier regularization method together with order optimal logarithmic stability estimates is given. A numerical example shows that the theoretical results are valid.     

求解热源识别问题的修正吉洪诺夫正则化方法

该文研究了一个热源识别问题,通过引入修正吉洪诺夫方法来处理问题的不适定性,在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步验证了方法的有效性和稳定性.     

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.