An Iterative Regularization Method for Solving Backward Problems With 2 Perturbation Data
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摘要: 该文考虑了一类带有扰动扩散系数和扰动终值数据的空间分数阶扩散方程反向问题，从终值时刻的测量数据来反演初始时刻数据. 该问题是严重不适定的，因此该文提出了一种迭代正则化方法来处理该反向问题，并利用先验正则化参数选取规则得到了正则化解和精确解之间的误差估计，最后进行了一些数值模拟，验证了方法的有效性.Abstract: The backward problem of space-fractional diffusion equations with perturbed diffusion coefficients and perturbed final data was considered. The initial data were recovered from the measured data at the final time. Given the severe ill-posedness of this problem, an iterative regularization method was proposed to tackle it. The convergence error estimate between the exact and approximate solutions was obtained under the assumption of an a-priori bound on the exact solution. Finally, several numerical simulations were conducted to verify the effectiveness of this method.
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图 1 精确解和在不同误差水平下的正则化解
Figure 1. Exact solutions and regularized solutions with different noise levels
图 2 不同时刻下的精确解和正则化解
Figure 2. Comparison of exact solutions and regularized solutions at different moments
表 1 不同误差水平下u(x, t)的相对误差和绝对误差(α=0.6)
Table 1. Relative and absolute errors corresponding to different noise levels(α=0.6)
t ε1=10-1 ε2=0 ε1=0 ε2=10-1 ε1=10-1 ε2=10-1 ea(t) er(t) ea(t) er(t) ea(t) er(t) 0.9 0.017 0 0.032 5 0.023 8 0.045 4 0.024 8 0.047 4 0.5 0.076 9 0.062 8 0.074 3 0.068 8 0.093 5 0.076 4 0.3 0.114 0 0.087 3 0.109 8 0.084 6 0.137 3 0.096 0 t ε1=10-2 ε2=0 ε1=0 ε2=10-2 ε1=10-2 ε2=10-2 ea(t) er(t) ea(t) er(t) ea(t) er(t) 0.9 0.007 5 0.014 3 0.007 3 0.014 0 0.007 8 0.015 0 0.5 0.041 6 0.034 0 0.035 5 0.029 0 0.044 8 0.036 6 0.3 0.078 9 0.051 4 0.068 7 0.044 7 0.083 1 0.054 2 
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表 2 t=0.9时刻下β, ε与迭代步数k之间的关系
Table 2. Relationships between β, ε and iterative step number k at time t=0.9
β=0.1 β=0.5 β=1 k(ε=10-1) 68 15 7 k(ε=10-2) 418 126 18 k(ε=10-3) 549 193 56
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表 3 Tikhonov正则化方法(TRM)和迭代正则化方法(IRM)下u(x, t)的相对误差和绝对误差(α=0.6)
Table 3. Relative and absolute errors of u(x, t) under the TRM and IRM(α=0.6)
t=0.8 t=0.5 t=0.3 ε2=10-1 ε1=10-2 ε2=10-1 ε1=10-2 ε2=10-1 ε1=10-2 TRM ea(t) 0.079 6 0.018 5 0.097 2 0.028 8 0.260 9 0.048 0 er(t) 0.108 4 0.025 1 0.115 8 0.027 3 0.168 9 0.031 0 IRM ea(t) 0.033 0 0.003 4 0.060 9 0.011 6 0.137 3 0.060 4 er(t) 0.045 3 0.004 6 0.049 7 0.009 1 0.096 0 0.035 7
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