A Spring-Damping Regularization and a Novel Lie-Group Integration Method for Nonlinear Inverse Cauchy Problems

In this paper, the solutions of inverse Cauchy problems for quasi-linear elliptic equations are resorted to an unusual mixed group-preserving scheme (MGPS). The bottom of a finite rectangle is imposed by overspecified boundary data, and we seek unknown data on the top side. The spring-damping regularization method (SDRM) is introduced by converting the governing equation into a new one, which includes a spring term and a damping term. The SDRM can further stabilize the inverse Cauchy problems, such that we can apply a direct numerical integration method to solve them by using the MGPS. Several numerical examples are examined to show that the SDRM+MGPS can overcome the ill-posed behavior of the inverse Cauchy problem. The present algorithm has good efficiency and stability against the disturbance from random noise, even with an intensity being large up to 10%, and the computational time is very saving.