Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate
For researchers working on inverse problems and numerical methods for parabolic equations, this provides a more general theoretical foundation and convergent numerical schemes for reversed-time problems.
This paper develops a new Carleman estimate for parabolic equations with reversed time that works on arbitrary time intervals, unlike previous estimates limited to small intervals. It proves stability, proposes a quasi-reversibility method with convergence rate for linear cases, and establishes global convergence of a gradient projection method for quasilinear equations.
The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability estimate is proven. Next, the quasi-reversibility numerical method is proposed for an arbitrary time interval for the linear case. This is unlike a sufficiently small time interval in the previous work. The convergence rate for the quasi-reversibility method is established. Finally, the quasilinear parabolic equation with reversed time is considered. A weighted globally strictly convex Tikhonov-like functional is constructed. The weight is the Carleman Weight Function which is involved in that Carleman estimate. The global convergence of the gradient projection method to the exact solution is proved for this functional.