This paper is concerned with the inverse problem of retrieving the initial value of a time-fractional fourth order parabolic equation from source and final time observation. The considered problem is an {\it ill-posed problem.} We obtain regularized approximations for the sought initial value by employing the quasi-boundary value method, its modified version and by Fourier truncation method(FTM). We provide both the apriori and aposteriori parameter choice strategies and derive the error estimates for all these methods under some {\it source conditions} involving some Sobolev smoothness. As an important implication of the obtained rates, we observe that for both the apriori and aposteriori cases, the rates obtained by all these three methods are same for some source sets. Moreover, we observe that in both the apriori and aposteriori cases, the FTM is free from the so-called {\it saturation effect}, whereas both the quasi-boundary value method and its generalizations possesses the saturation effect for both the cases. Further, we observe that the rates obtained by the FTM is always order optimal for all the considered source sets.

翻译：本文研究从源项和最终时刻观测数据反演时间分数阶四阶抛物型方程初值的不适定问题。我们采用拟边界值方法及其修正形式，以及傅里叶截断方法（FTM）来获得所求初值的正则化逼近。我们提供了先验和后验参数选取策略，并在包含某些Sobolev光滑性的源条件下推导了所有方法的误差估计。基于所获得的收敛速率，一个重要发现是：对于某些源集，无论先验还是后验情形，三种方法得到的收敛速率相同。此外，我们观察到在先验和后验两种情形下，FTM方法均不存在所谓的饱和效应，而拟边界值方法及其推广形式在两种情形下均存在饱和效应。进一步，我们发现对于所有考虑的源集，FTM方法获得的收敛速率始终是最优阶的。