User:Daviddaved/Layered discretization

< User:Daviddaved

The continued fraction approach to the inverse problems can be applied to domains w/inhomogenuous isotropic conductivity in 2D and higher dimensions using layered discretization. The resulting Dirichlet-to-Neumann operator can be written as a ratio of two high order differential operators that satisfy three-term recurrence (similar to the numerators and denominators of functional continued fraction). The fundumental solutions of the differential operators can be directly read from the kernel of the Dirichlet-to-Neumann operator. The conductivity can be then found by a Eucledian type algorithm, reversing the three-term recurrence.

More formally: Let f be the potential and g be the current on a layered domain. Then, on the k'th layer:

Therefore, for ordinary differential operators F and G,

since, the Dirichlet-to-Neumann operator is self-adjoint. And

Exercise (**). Prove that the eigenvalues of the monodromy matrices of the operators F and G are simple, positive and interlace.

Since, the operator G is differential, the support of the function Gf belongs to the closure of the support of the potential f, which allows one to read the solutions of the equation F*g=0 from the Dirichlet-to-Neumann operator and G*f=0 from the dual problem. (Need to find a family of fundamental solutions). The coefficients of the differential operators F, G, F* and G* can be then obtained from the Wronskians of the fundamental solutions.

One also gets from the systems above that,

where and are conductivities of the layers. Therefore, one can find the conductivity of the outmost layer as a ratio of the leading coefficients of the corresponding operators F an G, which allows one to reverse the three-term recurrence and find conductivities of all layers by induction.