Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

<abstract><p>The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.</p></abstract>

Regularization of backward time-fractional parabolic equations by Sobolev-type equations

The problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

On an initial inverse problem for a diffusion equation with a conformable derivative

In this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

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.

Using Landweber iteration to quantify source conditions – a numerical study

Source conditions of the type {x^{\dagger}\in\mathcal{R}((A^{\ast}A)^{\mu})} are a standard assumption in the theory of inverse problems to show convergence rates of regularized solutions as the noise in the data goes to zero. Unfortunately, it is rarely possible to verify these conditions in practice, rendering data-independent parameter choice rules unfeasible. In this paper we show that such a source condition implies a Kurdyka–Łojasiewicz inequality with certain parameters depending on μ. While the converse implication is unclear from a theoretical point of view, we propose an algorithm which represents a first attempt that allows to approximate the value of μ numerically. It is based on combining the Landweber iteration with the Kurdyka–Łojasiewicz inequality. We conduct several numerical experiments to demonstrate the potential and limitations of the current method. We also show that the source condition implies a lower bound on the convergence rate which is of optimal order and observable without the knowledge of μ.