A novel Carleman estimation for high-dimensional parabolic equations and a class of inverse coefficient problems

This paper presents a novel construction method for the Carleman weight function based on variable separation for high-dimensional parabolic equations. By incorporating semi-classical parameters and leveraging principal symbolic analysis techniques, the global Carleman estimate was rigorously established. This enabled a strict proof of the existence, uniqueness, and stability of the equation’s solution. The study overcomes the limitations of traditional exponential weight functions and innovatively introduces the convexity adjustment technique, effectively addressing critical challenges in non-strongly convex estimation. Numerical experiments demonstrate that the algorithm maintains robust performance in the presence of noise ranging from 1% to 15%, and applies to practical problems, including heat conduction and pollutant diffusion.