解决一维线性扩散方程逆问题的一种迭代方法

On an Iterative Method for Solving Inverse Problem of 1-D Linear Diffusion Equation

从函数逼近论的观点出发，利用扰动法和正则化方法对扰动量进行优化，从而得到一种解决－维线性扩散方程逆问题近似数值解的迭代方法．数值计算表明：这种选代方法可行、收敛速度快．

Cotion et al in 1985 and 1987 applied nonlinear optimization method to solving inverse problcm of 1-D and 2-D acoustics[1,2]. Chen proposed GPST (General Pulse Spectrum Techniques) method to deal with inverse problem in 1985[3]. He and his associates applied GPST method to history matching of petroleum reservoirs, inverse scattering problem of electromagnetic field, determination of diffusion coefficient, etc, including many 2-D and 3-D inverse problems[4,5]. The two methods mentioned above both need to be improved, but of course the needed improvements arc different.In this paper, the author proposes a way to improve on Chen's and Cotion,s methods. For inverse problem of 1-D linear diffusion equation, the author sets up the optimization problem for perturbation,and obtains the lst order approximation of nonlinear functional of perturbation. thus forming a new iterative method. When a number of terms of a set of special functions is taken tO approximate the perturbation, 1-D GPST iterative algorithm can be derived. Numerical simulations confirms the feasibility of the new iterative method. Computation results show that the author's method is efficient and possesses good accuracy.Although the present paper dcals with only inverse problems of 1-D linear diffusion equation, it is easy to extend it to inverse problems of other types of equations, both 2-D and 3-D (in faCt the author's research after this paper is written shows that it can be successfully appled to determining the diffusion coefficient and to locating an air or water pollutant source).