基于MFS的PDE-约束优化法求解热传导逆问题

The PDE-Constrained Optimization Method Based on MFS for Solving Inverse Heat Conduction Problems

中文关键词:  热传导问题  PDE-约束优化  基本解方法  与时间相关热源项  Tikhonov正则化方法

英文关键词:inverse heat conduction problem  PDE-constrained optimization  method of fundamental solutions  time-dependent heat source term  Tikhonov regularization method

中文摘要:

      在本文中,我们给出了一种有效的无网格方法来求解逆热传导问题,含有Neumann边界条件情形.所得到的PDE-约束优化法是一种在空间与时间域上的全局近似方法,其中将控制方程的基本解作为基函数.由于初始测量数据包含有噪声误差,则所得线性方程组的系数矩阵通常是病态的,文中利用广义交叉验证(GCV)的Tikhonov正则化方法来获得更加稳定的数值解.通过数值结果表明,本文给出的方法是精确、有效、鲁棒的.

英文摘要:

      In this paper, we present an effective meshless method for solving the inverse heat conduction problems, with the Neumann boundary condition. A PDE-constrained optimization method is developed to get a global approximation scheme in both spatial and temporal domains, by using the fundamental solution of the governing equation as the basis function. Since the initial measured data contain some noises, and the resulting systems of equations are usually ill-conditioned, the Tikhonov regularization technique with the generalized cross-validation criterion is applied to obtain more stable numerical solutions. It is shown that the proposed schemes are effective by some numerical tests.