Abstract
In this paper, we study two inverse problems for stochastic parabolic equations with additive noise.
One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇.
For this inverse problem, we obtain a conditional stability result.
The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary.
The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.
The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.
A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.
Numerical solution of direct and inverse problems for degenerate parabolic equations with concentrated sources
