In this paper, we first establish a weak unique continuation property for timefractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.
We shall study the convergence rates of the Tikhonov regularizations for the identification of the diffusivity q(x) in a parabolic–elliptic system. The H1 regularization and a mixed Lp–H1 regularization are considered. For the H1 regularization, we present a simple and easily interpretable source condition, under which the regularized solutions will be shown to converge at the standard rate in terms of the noise level of the data. The convergence is analyzed in three different approaches, which result in the same convergence rate but require quite different conditions on the measurement time and the identifying parameters. For the mixed Lp–H1 regularization, we will achieve some desired convergence rate by using the Bregman distance and some new source condition and nonlinearity condition.
We shall study in this paper the convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions in general dimensional spaces. The conditional stability estimates are first derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the conditional stability estimates. We shall finally derive the reasonable convergence rates, which explicitly reveals the relation between the regularity of the radiativities and the convergence rates results.
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