Conductivity imaging by the method of characteristics in the 1-Laplacian

Abstract

We consider the problem of reconstruction of a sufficiently smooth planar conductivity from the knowledge of the magnitude |J| of one current density field inside the domain, and the corresponding voltage and current on a part of the boundary. Mathematically, we are lead to the Cauchy problem for the the 1-Laplacian with partial data. Different from existing works, we show that the equipotential lines are characteristics in a first order quasilinear partial differential equation. The conductivity can be recovered in the region flown by the characteristics originating at parts of the boundary where the data is available. Numerical experiments show the feasibility of this alternative method.

Stable reconstruction of regular 1-Harmonic maps with a given trace at the boundary

“…We prove below the finite length property for solutions of (6). The proof makes essential use of the curvature bound given by the third equation in (6), in conjunction with the fact that Hypothesis 1.1 implies that level sets of u describe global coordinates in . The following lemma is key to capturing the effect of the Euclidean curvature of the characteristics on their length.…”

“…where (x(τ (λ, β), λ, β), y(τ (λ, β), λ, β) is the corresponding solution of (6), and λ is as in (5). Then β → F λ (β) is a continuous at β λ .…”

“…The algorithm in Section 2 finds the root β λ of the function F λ (β) in (12), which together with (x (0, λ, β λ ), y(0, λ, β λ )) give the initial condition of a level curve of u which solves (6). To find β λ numerically, we use the boundary points which are joined by the level curve of u corresponding to λ, say (x 1 , y 1 ) and (x 2 , y 2 ).…”

“…Neumann problems associated with the 1-Laplacian were first considered in [5], and Cauchy problems in [1,6]. The Dirichlet problem for the 1-Laplacian (1) was first considered in [7], where level sets of the 1-harmonic maps are shown to be geodesics in a conformal metric, and a locally convergent algorithm (in the sense that a first guess is sufficiently close to the sought solution) was proposed.…”

Imaging from coupled physics

“…Mathematically, several of the ICP problems can be analyzed in terms of the 'p-Laplacian' which raises interesting research questions of non-linear partial differential equations. One approach for analyzing and for the 0266-5611/12/080201+02$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USAsolution of the CDI problem, using characteristics of the 1-Laplacian, is discussed by Tamasan and Veras [4]. Moreover, Moradifam et al [5] present a novel iterative algorithm based on Bregman splitting for solving the CDI problem.…”