热传导方程辐射系数及初始条件反问题的数值求解

Inverse Problem of Reconstructing the Heat-conduction Equation's Initial Value and the Radiative Coefficient

讨论用某一时刻的温度测量值及某一子区域中各时刻的温度测量值同时重构热传导方程的辐射系数和初始条件这一反问题的数值求解方法,用最小二乘法,将此反问题化为一个变分问题,且将此变分问题离散化为一个非线性规划问题,其目标函数值依赖于热传导方程正问题的数值解。同时用差分法和径向基函数(RBF)方法求正问题的数值解并导出相应目标函数的梯度公式,在此基础上用拟牛顿方法实现一般情形下的数值重构。数值实验表明,这一方法是可行的。

A numerical method of reconstructing the radiative coefficient and initial condition simultaneously by measuring the domain temperature at a fixed time and the temperature of a subdomain all the time is studied.By the least-square technique,this inverse problem can be transformed into a variational problem and discretized into a nonlinear programming problem with the cost function depending on the numerical solution of the corresponding direct problem of heat equation.The numerical solution of the direct problem is obtained by the finite difference method and the radial basis function (RBF) method respectively and the gradient formula for cost function is derived.Then the numerical reconstruction is realized by the quasi-Newton technique.Numerical results show that this method is available.