A monotone iterative reconstruction method for an inverse drift problem in a two-dimensional parabolic equation

TL;DR

This paper introduces a monotone iterative method for reconstructing the drift coefficient in a 2D parabolic inverse problem, demonstrating convergence and robustness to noise through numerical experiments.

Contribution

It develops a novel monotone operator-based iterative scheme that guarantees uniqueness and provides a constructive approach for inverse drift reconstruction.

Findings

The method converges monotonically to the true drift coefficient.

Numerical experiments confirm the effectiveness and stability of the approach.

The approach remains effective even with noisy data using denoising strategies.

Abstract

We study an inverse drift problem for a two-dimensional parabolic equation on the unit square with mixed boundary conditions, where the drift coefficient is recovered from terminal observation data $g=u(\cdot,T)$. A monotone operator is constructed whose fixed point coincides with the unknown drift, yielding uniqueness in an admissible class and a constructive iterative reconstruction scheme. Numerical experiments illustrate the monotone convergence and the effectiveness of the proposed method, and show that it remains effective for noisy terminal data under the denoising strategy.