抛物方程的Landweber迭代正则化方法的后验误差估计

Posterior Error Estimation of Landweber Iterative Regularization Method for Parabolic Equations

考虑二维抛物方程Cauchy问题的反问题,该问题是严重不适定的.首先,用Landweber迭代正则化方法得到该问题的一个正则近似解,用Fourier变换求出该问题的精确解;其次,在后验正则化参数的选取规则下,给出精确解和正则解之间的H9lder型误差估计,并使用更强的先验条件给出端点x=1处的误差估计;最后,给出数值实例说明该方法的有效性.结果表明,该方法比已有方法收敛速度更快.

We considered the inverse problem of Cauchy problem of two dimensional parabolic equations,which was seriously ill-posed.Firstly,a regular approximate solution of the problem was obtained by using Landweber iterative regularization method,and Fourier transform was used to obtain the exact solution of the problem.Secondly,the H lder type error estimation between the exact solution and the regular solution was given under the selection rules of the posterior regularization parameters,and stronger prior conditions were used to give the error estimation at the end point x=1.Finally,numerical examples were given to demonstrate the effectiveness of the proposed method.The results show that the proposed method has a faster convergence rate than existing methods.