This paper is concerned with the mathematical analysis of the inverse random source problem for the time fractional diffusion equation, where the source is assumed to be driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag-Leffler function and the stochastic integrals associated with the fractional Brownian motion.2010 Mathematics Subject Classification. 35R30, 35R60, 65M32.

[1] In this paper, the essential technique of Grad-Shafranov (GS) reconstruction is reformulated into an inverse boundary value problems (IBVPs) for Laplace's equation on a circle by introducing a Hilbert transform between the normal and tangent component of the boundary gradients. It is proved that the specified IBVPs have unique solution, given the known Dirichlet and Neumann conditions on certain arc. Even when the arc is reduced to only one point on the circle, it can be inferred logically that the unique solution still exists there on the remaining circle. New solution approach for the specified IBVP is suggested with the help of the introduced Hilbert transform. An iterated Tikhonov regularization scheme is applied to deal with the ill-posed linear operators appearing in the discretization of the new approach. Numerical experiments highlight the efficiency and robustness of the proposed method. According to linearity of the elliptic operator in GS equation, its solution can be divided into two parts. One is solved from a semilinear elliptic equation with an homogeneous Dirichlet boundary condition. The other is solved from the IBVP of Laplace's equation. It is concluded that there exists a unique solution for the so-called elliptic Cauchy problem for the essential technique of GS reconstruction.

A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested.