A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains

In this paper, the inverse Cauchy problems for elliptic equations, including the Laplace equation, the Poisson equation, and the Helmholtz equation, defined in annular domains are investigated. When the outer boundary of an annulus is imposed by overspecified boundary data, we seek unknown data in the inner boundary through a combination of the spring-damping regularization method (SDRM) and the mixed group-preserving scheme (MGPS). Several numerical examples are examined to show that the MGPS plus the SDRM can overcome the ill-posed behavior of this highly ill-conditioned inverse Cauchy problem. The presently proposed novel algorithm has good efficiency and stability against the disturbance from large random noise even up to 50%, and the computational cost of MGPS is very time saving.