Regularization of the backward stochastic heat conduction problem

In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rates are established under different a priori assumptions on the sought solution.

On a backward problem for nonlinear time fractional wave equations

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity

This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

On the inverse problem for nonlinear strongly damped wave equations with discrete random noise

In this paper, we consider the existence of a solution u(x, t) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q (q > 0) norms. Moreover, a numerical example is given to illustrate our results.