GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A fast algorithm to solve nonhomogeneous Cauchy-Reimann equations in the complex plane
SIAM Journal on Scientific and Statistical Computing
A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings
Journal of Computational Physics
Electrical Impedance Tomography
SIAM Review
Global FOM and GMRES algorithms for matrix equations
Applied Numerical Mathematics
Direct Reconstructions of Conductivities from Boundary Measurements
SIAM Journal on Scientific Computing
Solution Methods for $\mathbb R$-Linear Problems in $\mathbb C^n $
SIAM Journal on Matrix Analysis and Applications
Numerical solution method for the dbar-equation in the plane
Journal of Computational Physics
On Numerical Algorithms for the Solution of a Beltrami Equation
SIAM Journal on Numerical Analysis
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This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation $\nabla\cdot\sigma\nabla u(\cdot,k)=0$ in $\mathbb{R}^2$ for piecewise smooth conductivities $\sigma$, where $k$ is a complex parameter. The key is to solve an $\mathbb R$-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anal., 29 (2010), pp. 2-17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the $\mathbb{R}$-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133-157] for singular integral transforms, we propose a simpler numerical method which solves the $\mathbb{R}$-linear integral equation in the unit disk directly. For the resulting $\mathbb{R}$-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method.