A two-step approach for recovering both time and space dependent Robin coefficients in parabolic systems

This study addresses an inverse Robin problem for parabolic partial differential equations, aiming to identify an unknown space-time dependent Robin coefficient on an inaccessible boundary from Cauchy data measured on an accessible boundary. Such a problem arises in many applications but is inherently ill-posed due to instability and nonlinear dependence of the solution on the data. To overcome these difficulties, a two-step solution strategy is proposed that transforms the nonlinear inverse problem into a sequence of well-posed problems. In the first step, the problem is reduced to a linear parabolic Cauchy problem and is solved using a coupled complex boundary method with a complex Robin condition, allowing Dirichlet and Neumann data to be incorporated simultaneously and stably. Space-time Tikhonov regularization is employed to ensure well-posedness. In the second step, the recovered boundary data are used to reconstruct the Robin coefficient by applying Newton's method combined with Lavrentiev regularization. The proposed approach is supported by theoretical results, including convergence, stability, error estimates, and uniform boundedness. Numerical experiments demonstrate accurate reconstruction of smooth, discontinuous, and time-dependent Robin coefficients under various noise levels ranging from 0.1% to 5%. The results validate the theoretical convergence rates and demonstrate the method's robustness in handling complex three-dimensional geometries. Overall, the proposed framework provides a stable and robust tool for inverse Robin problems arising in engineering, physics, and environmental applications.