Series reversion in Calder\'on's problem

TL;DR

This paper develops explicit series reversion techniques for Calderón's inverse problem, enabling numerical methods that improve accuracy using partial boundary data and analyzing their convergence and computational complexity.

Contribution

It introduces explicit series reversion formulas for the forward map in Calderón's problem, leading to new numerical methods with proven convergence and similar complexity to linearized solutions.

Findings

Series reversion formulas for Calderón's problem derived

Numerical methods with increasing accuracy developed

Convergence proven under invertibility conditions

Abstract

This work derives explicit series reversions for the solution of Calder\'on's problem. The governing elliptic partial differential equation is $\nabla\cdot(A\nabla u)=0$ 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 Fr\'echet 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.