On Numerical Algorithms for the Solution of a Beltrami Equation

Abstract

Abstract. The paper concerns numerical algorithms for solving the Beltrami equation fz(z) = µ(z)fz (z) for a compactly supported µ.First, we study an efficient algorithm that has been proposed in the literature, and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of a greater simplicity of implementation.

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

“…Regarding different ways to discretize (1.5), fast numerical solution of the C-linear Beltrami equation was discussed in [6][7][8] with differing conditions in place of the asymptotics (1.2) leading to approaches not involving (1.5). However, the uniform polar coordinate grid discretizations for the operators S and C developed by these authors can be applied in a straightforward manner to (1.5) enabling numerical comparison of the polar coordinate case with the uniform square grid discretizations.…”

Complex geometrical optics solutions for anisotropic equations and applications

“…Based on Daripa's algorithm, a slightly modified version was given in [6]. Here we shall follow the description of Daripa's algorithm given in [6]. Since it is often desirable to refine the grid in the part of boundary where the measurements are taken, we shall replace FFT in Darpia's algorithm by NUFFT described in [7].…”

“…Based on Daripa's algorithm, a slightly modified version was given in [6]. Here we shall follow the description of Daripa's algorithm given in [6].…”

“…(see [3], [6]). Armed with recurrence formulae (4.7), (4.8), we can evaluate the Fourier coefficients t k (r) and p k (r) more efficiently.…”

“…In Section 4, we will describe a numerical scheme to solve the Beltrami equation. This algorithm is a combination of the fast algorithm developed by Daripa et al [3], [4], [5] (also see [6]) and the NUFFT (nonuniform fast Fourier transform) [7]. The rest of the paper is organized as follows.…”