Simultaneous reconstruction of boundary coefficients in a parabolic system via regularized Levenberg-Marquardt optimization

Abstract

This work addresses the problem of simultaneously recovering two spatially varying boundary coefficients in a parabolic system. To handle the ill-posedness, we use Tikhonov regularization. The forward model is approximated using finite elements for spatial discretization together with an implicit Euler scheme for the temporal direction. The resulting nonlinear least-squares problem is solved by the Levenberg--Marquardt algorithm, improved with a surrogate functional approach that gives explicit update formulas and speeds up convergence. Numerical tests with noisy data show that our method is accurate, stable, and efficient.