SERIES REVERSION IN CALDERON'S PROBLEM

This work derives explicit series reversions for the solution of Calderon's problem. The governing elliptic partial differential equation is del . (A del u) = 0 in a bounded Lipschitz domain and with a matrix-valued coefficient. The corresponding forward map sends A to a projected version of a local Neumann-to-Dirichlet operator, allowing for the use of partial boundary data and finitely many measurements. It is first shown that the forward map is analytic, and subsequently reversions of its Taylor series up to specified orders lead to a family of numerical methods for solving the inverse problem with increasing accuracy. The convergence of these methods is shown under conditions that ensure the invertibility of the Frechet derivative of the forward map. The introduced numerical methods are of the same computational complexity as solving the linearised inverse problem. The analogous results are also presented for the smoothened complete electrode model.

Automated summary

This study presents explicit series reversions for the solution of Calderon's problem and derives a family of numerical methods to improve the accuracy of the solution by reversing the Taylor series of the forward map. The convergence of these numerical methods is shown under conditions that ensure the invertibility of the Frechet derivative of the forward map. The introduced numerical methods have the same computational complexity as solving the linearized inverse problem.