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In this paper, we consider a backward problem for a time-space fractionaldiffusion process. For this problem, we propose to construct the initial databy minimizing data residual error in fourier space domain and variable totalvariation (TV) regularizing term which can protect the edges as TV regularizingterm and reduce staircasing effect. The well-posedness of this optimizationproblem is studied under a very general setting. Actually, we write thetime-space fractional diffusion equation as an abstract fractional differentialequation and get our results by using fractional semigroup theory, so ourresults can be applied to other backward problems for more general fractionaldifferential equations. Then a modified Bregman iterative algorithm is proposedto approximate the minimizer. The new features of this algorithm is that theregularizing term changed in each step and we need not to solve the complexedEuler-Lagrange equations of variable TV regularizing term (just need to solve asimpler Euler-Lagrange equations). The convergence of this algorithm and thestrategy of choosing parameters are also obtained. Numerical implementationsare given to support our analysis to show the flexibility of our minimizationmodel.

Variable Total Variation Regularization for Backward Time-Space Fractional Diffusion Problem