Geometry, Imaging and Computing
Volume 1 (2014)
High performance computing for spherical conformal and Riemann mappings
Pages: 223 – 258
A classical way of finding the harmonic map is to minimize the harmonic energy by the time evolution of the solution of a nonlinear heat diffusion equation. To arrive at the desired harmonic map, which is a steady-state of this equation, we propose an efficient quasi-implicit Euler method and analyze its convergence under some simplifications. If the initial map is not close to the steady-state solution, it is difficult to find the stability region of the time steps. To remedy this drawback, we propose a two-phase approach for the quasi-implicit Euler method. In order to accelerate the convergence, a variant time step scheme and a heuristic method to determine an initial time step are developed. Numerical results clearly demonstrate that the proposed method achieves high performance for computing the spherical conformal and Riemann mappings.
spherical conformal mapping, Riemann mapping, nonlinear heat diffusion equation, quasi-implicit Euler method, adaptive controlling time step, two-phase