An efficient iteration method to reconstruct the drift term from the final measurement
Dakang Cen, Wenlong Zhang, Zhidong Zhang
TL;DR
The paper tackles the problem of recovering a drift term $q$ in a 1D parabolic equation from final-time data by building a monotone-operator framework. A carefully crafted operator $K$ has fixed points that coincide with the inverse problem’s solution, and a monotonicity-based uniqueness proof is obtained via an iterative scheme that converges from an explicit upper bound. To address ill-posedness, the authors employ mollification of the final data and implement a backward-Euler finite-difference numerical scheme to perform reconstructions, demonstrating accurate results for diverse drift profiles, including non-smooth cases and noisy data. The work provides a solid foundation for extending the method to higher dimensions, where the drift becomes a vector field and poses additional construction and stability challenges.
Abstract
This work investigates the inverse drift problem in the one-dimensional parabolic equation with the final time data. The authors construct an operator first, whose fixed points are the unknown drift, and then apply it to prove the uniqueness. The proof of uniqueness contains an iteration converging to the drift, which inspires the numerical algorithm. To handle the ill-posedness of the inverse problem, the authors add the mollification on the data first in the iterative algorithm, and then provide some numerical results.