Variational Regularization of the Inverse Problem of a Class of Nonlinear Time-Fractional Diffusion Equations
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摘要:
考虑了一类二维非线性时间分数阶扩散方程，并从最终位置获取的测量数据来反演物质在u(0, y, t)处的物理信息。这个问题是严重不适定的，即问题的解并不连续依赖于测量数据，因此提出了变分型正则化方法来稳定求解该问题。给出了精确解与正则近似解之间的误差估计，数值算例验证了该方法的有效性。
Abstract:The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state
\begin{document}$ u(0,y,t) $\end{document} of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.
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图 1 在
$ {x}=0 $ 时的精确解和不同误差水平下的正则近似解$ {{u}}^{{\alpha },{\varepsilon }} $ Figure 1. Exact solution
$ u $ at$ x=0 $ and regularized approximate solution$ {u}^{\alpha ,\varepsilon } $ curves corresponding to different error levels
图 2 正则化解和精确解在不同x取值下的比较(y=0)
Figure 2. The exact solution and its regularized solution curves corresponding to different x values
( y =0) 表 1 不同误差水平下的相对误差
$ {E}_{{\rm{r}}} $ Table 1. Relative errors corresponding to different error levels
$ {x} $ $ {{E}}_{\rm{r}} $ $ {\varepsilon }=1{\rm{E}}-1 $ $ {\varepsilon }=1{\rm{E}}-2 $ $ {\varepsilon }=1{\rm{E}}-3 $ 0 2.749E−1 2.685E−1 2.682E−1 0.1 2.639E−1 2.522E−1 2.520E−1 0.2 2.409E−1 2.367E−1 2.348E−1 0.3 2.251E−1 2.219E−1 2.227E−1 0.4 2.047E−1 2.017E−1 2.026E−1 0.5 1.848E−1 1.709E−1 1.713E−1 0.6 1.551E−1 1.366E−1 1.345E−1 0.7 1.213E−1 1.030E−1 1.026E−1 0.8 1.003E−1 8.010E−2 7.600E−2 0.9 9.800E−2 9.750E−2 9.470E−2 
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