# User:DVD206/Layered discretization

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:

${\displaystyle {\begin{cases}g_{-1}=0,\\g_{0}=D\alpha _{0}Df_{0},\\f_{k+1}=f_{k}+g_{k}/\beta _{k},\\g_{k}=g_{k-1}+D\alpha _{k}Df_{k}.\end{cases}}{\mbox{ In the dual domain, }}{\begin{cases}f_{0}={\frac {\int \beta _{0}f_{1}}{\int \beta _{0}}},\\g_{0}=\beta _{0}(f_{1}-{\frac {\int \beta _{0}f_{1}}{\int \beta _{0}}}),\\f_{k+1}=f_{k}+g_{k}/\beta _{k},\\g_{k}=g_{k-1}+D\alpha _{k}Df_{k}.\end{cases}}}$

Therefore, for ordinary differential operators F and G,

${\displaystyle {\begin{cases}f_{k}=F_{k}f_{0},\\g_{k}=G_{k}f_{0},\\\Lambda _{k}=G_{k}F_{k}^{-1}=(F_{k}^{*})^{-1}G_{k}^{*},\end{cases}}}$

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

${\displaystyle {\begin{cases}F_{k}^{*}G_{k}=G_{k}^{*}F_{k},\\G_{k}^{*}f_{k}=F_{k}^{*}g_{k},\\G_{k}=\Lambda _{k}F_{k}.\end{cases}}}$

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,

${\displaystyle {\begin{cases}F_{0}=Id,\\G_{0}=\alpha _{0}D^{2}+(D\alpha _{0})D,\\F_{k}={\frac {\alpha _{0}\ldots ,\alpha _{k-1}}{\beta _{0}\ldots \beta _{k-1}}}D^{2k}+\ldots ,k>0{\mbox{ and }}\\G_{k}={\frac {\alpha _{0}\ldots ,\alpha _{k}}{\beta _{0}\ldots \beta {k-1}}}D^{2k+2}+\ldots ,\end{cases}}}$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ 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.