A multiscale hierarchical method of fundamental solutions for severely ill-posed elliptic Cauchy problems

We propose a multiscale hierarchical extension of the method of fundamental solutions (MH-MFS) for inverse Cauchy problems. The problems considered in this paper are governed by homogeneous elliptic partial differential equations and characterized by severe ill-posedness. The method hierarchically constructs the MFS solution by approximating the boundary data from coarse to fine scales, through progressive refinement of the Tikhonov regularization parameter at each level. Numerical experiments indicate that the MH-MFS, combined with adaptive stopping criteria, can recover the solution on the inaccessible boundary in severely ill-posed scenarios while improving robustness and computational efficiency compared with the standard MFS with L-curve parameter selection. • Proposes MH-MFS for severely ill-posed elliptic Cauchy problems. • Recovers coarse-to-fine boundary features via progressive Tikhonov regularization. • Incorporates adaptive stopping criteria to enhance stability and robustness. • Improves robustness and efficiency over MFS with Tikhonov-L-curve.