A non-typical Lie-group integrator to solve nonlinear inverse Cauchy problem in an arbitrary doubly-connected domain

The inverse Cauchy problem for a nonlinear elliptic equation defined in an arbitrary doubly-connected plane domain is solved numerically. When the overspecified boundary data are imposed on the outer boundary, we seek the unknown data on the inner boundary, by using a mixed group preserving scheme (MGPS) to integrate the nonlinear inverse Cauchy problem as an initial value problem. We also reverse the order of the above inverse Cauchy problem by giving overspecified boundary data on the inner boundary and seeking the unknown data on the outer boundary. Several numerical examples are examined to show that the MGPS can overcome the highly ill-posed behavior of nonlinear inverse Cauchy problem defined in arbitrary doubly-connected plane domain. The proposed algorithm is robust against large noise and very time saving without needing of any iteration.