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We are concerned with an inverse problem of simultaneously recovering the initial value and source strength in a transmission problem for a parabolic equation from extra measurements. First, we establish a conditional stability of the inverse problem by combining the Carleman estimate with the logarithmic convexity theory. Second, we construct a regularized minimizing functional and transform the inverse problem into an optimization problem. The existence and stability of solutions to the optimization problem are analyzed rigorously. Then, we introduce a surrogate functional and propose an iterative thresholding algorithm for solving the optimization problem. The algorithm is cheap and easy to be implemented numerically. Finally, we present several numerical examples for the two-dimensional (2D) and three-dimensional (3D) cases to show the validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem on determining a time dependent potential in a diffusion equation with temporal fractional derivative of variable order from a distributed observation. We shall study the existence, uniqueness and regularity estimates of the solution for the forward problem by utilizing the Freldhom alternative principle for compact operators. Based on a newly established coercivity for fractional derivatives of variable order, we prove a uniqueness result for the inverse potential problem. Numerically, we transform the inverse potential problem into an optimization problem with Tikhonov regularization. An iterative thresholding algorithm is proposed to find the minimizers by a newly constructed adjoint system, whose wellposedness is also verified. Several numerical experiments are presented to show the accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H1-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm. [ABSTRACT FROM AUTHOR]