Geometry, Imaging and Computing

Volume 1 (2014)

Number 2

High performance computing for spherical conformal and Riemann mappings

Pages: 223 – 258

DOI: http://dx.doi.org/10.4310/GIC.2014.v1.n2.a2

Authors

Wei-Qiang Huang (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Xianfeng David Gu (Department of Computer Science, Stony Brook University, Stony Brook, New York, U.S.A.)

Tsung-Ming Huang (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)

Song-Sun Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Wen-Wei Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

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.

Keywords

spherical conformal mapping, Riemann mapping, nonlinear heat diffusion equation, quasi-implicit Euler method, adaptive controlling time step, two-phase

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