A Mollification Regularization Method for the Inverse Source Problem for a Time Fractional Diffusion Equation

We consider a time-fractional diffusion equation for an inverse problem to determine an unknown source term, whereby the input data is obtained at a certain time. In general, the inverse problems are ill-posed in the sense of Hadamard. Therefore, in this study, we propose a mollification regularization method to solve this problem. In the theoretical results, the error estimate between the exact and regularized solutions is given by a priori and a posteriori parameter choice rules. Besides, the proposed regularized methods have been verified by a numerical experiment.

Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method

In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them.

An a posteriori truncation regularization method for an ill-posed problem for the heat equation with an integral boundary condition

The aim of this paper is to investigate the problem of control by the initial conditions of the heat equation with an integral boundary condition. Using the truncation method with an a posteriori parameter choice rule, we give the error estimate between the exact and the regularized solutions. A numerical implementation shows the efficiency of the proposed method.

Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problems

We obtain rate of convergence estimates for approximations delivered by Tikhonov’s scheme of solving ill-posed nonconvex optimization problems in a Hilbert space. The problems under investigation involve minimization of twice Frechet differentiable functionals given with errors on a closed convex set having a nonempty interior and smooth boundary. Assuming that the desired solution satisfies an appropriate source condition which includes the second derivative of the cost functional and depends on the geometry of constraints near the solution, we establish accuracy estimates in terms of the error level. Both the a priori and a posteriori parameter choice rules are analyzed.