对流-扩散过程逆过程反问题的伴随同化研究

The numerical study of the inverse problem in reverse process of convection-diffusion equation with adjoint assimilation method

对流-扩散方程逆过程反问题是一个不适定问题。利用伴随同化方法及处理数学物理反问题的技巧对该问题进行了数值研究。为了克服反问题中不适定性带来的困难,利用反问题中正则化思想,这里在目标函数中引入了正则项,其目的是克服不适定和计算不稳定。数值模拟结果表明,与通常的伴随同化方法相比,该方法无论是目标函数的下降速度、解的精确度都有较明显改进。因而,利用此方法求解对流-扩散方程逆过程反问题具有稳定性好、精度高的特点。利用该方法反演对流-扩散方程逆过程反问题的初值是可行的。

The inverse problem in the reverse process of convection-diffusion equation is an ill-posed problem. Adjoint assimilation method is employed to solve the problem with regularization techniques in inverse problem. To overcome the difficulty of ill-posedness, a regularization term is introduced in the cost function as a stabilized functional. Numerical experiments show that this method has a fast decent speed and a high accuracy compared to the adjoint method without regularization term. So this method is characterized by good stability and high accuracy. Numerical results show that this method is feasible to inverse the initial condition of the inverse process of convectioniffusion equation.