Abstract
In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.
Highlights
Diffusion phenomenon is ubiquitous in our physical world
We focus on the backward problem for equation (2)
By using operator semigroup and fractional operator semigroup theory, we prove the backward problems for an abstract fractional differential equation satisfying the conditions in our general theory
Summary
Diffusion phenomenon is ubiquitous in our physical world. From the point of view of probability theory, we can derive diffusion equations by applying. For our model (15), because the edges will change dramatically during the evolution process, we must iterate the value of p(x) during our computation Theories about existence, uniqueness and stability will be proved in a very general setting, restricted to backward problem for equation (2) we propose a modified Bregman iterative algorithm to solve problem (15). In order to prove existence, uniqueness and stability of problem (15), we first generalize the theory constructed in [1] to our variable total variation regularization model, during the proof we propose a concept named CBV-coercive.
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