Numerical solution of the $\mathbb R$-linear Beltrami equation

Abstract

Abstract. The R-linear Beltrami equation appears in applications, such as the inverse problem of recovering the electrical conductivity distribution in the plane. In this paper, a new way to discretize the R-linear Beltrami equation is considered. This gives rise to large and dense R-linear systems of equations with structure. For their iterative solution, norm minimizing Krylov subspace methods are devised. In the numerical experiments, these improvements combined are shown to lead to speed-ups of almost two orders of magnitude in the electrical conductivity problem.

Convergence of a numerical solver for an ℝ-linear Beltrami equation

“…While [10] benefitted from the weakly singular integral equation convergence theory [13], here the singularity in the Beurling transform renders this convergence theory inapplicable. Hence, while the numerical results in [3,9] were convincing, there was no guarantee of convergence to the correct solution. We now rectify the situation by providing L p -norm convergence results (Theorems 3.1 and 3.2).…”

“…In addition to being part of the solution algorithm to the inverse problem, this equation also emerges in the forward problem of computing complex geometric optics solutions [3]. In this paper we analyze the convergence rate of the numerical methods [3,9] to solve this equation. Furthermore, we compare their performance with the polar coordinate discretization suggested by Daripa [6].…”

“…In [9], (1.3) was solved by substituting ω = −Cu (1.4) leading to the R-linear singular integral equation…”

“…Moreover, this improved the solution accuracy by allowing to avoid the usage of truncated Neumann series at any stage. The discretizations in [3,9] were based on a collocation method on a uniform square grid containing generalizing the approach for the Cauchy transform in [10]. While [10] benefitted from the weakly singular integral equation convergence theory [13], here the singularity in the Beurling transform renders this convergence theory inapplicable.…”

“…We now rectify the situation by providing L p -norm convergence results (Theorems 3.1 and 3.2). This entails a modification in the matrix elements for the discretized operators of S and C over those considered in [9]. This causes a small precomputation penalty which is negligible in the EIT [4] application.…”