The CUDA implementation of the method of lines for the curvature dependent flows

Tomáš Oberhuber; Atsushi Suzuki; Vítězslav Žabka

Kybernetika (2011)

  • Volume: 47, Issue: 2, page 251-272
  • ISSN: 0023-5954

Abstract

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We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the integration time step. We implement this solver on CPU but also on GPU using the CUDA toolkit. We demonstrate that the mean-curvature flow can be successfully approximated in single precision arithmetic with the speed-up almost 17 on the Nvidia GeForce GTX 280 card compared to Intel Core 2 Quad CPU. On the same card, we obtain the speed-up 7 in double precision arithmetic which is necessary for the fourth order problem - the Willmore flow of graphs. Both speed-ups were achieved without affecting the accuracy of the approximation. The article is structured in such way that the reader interested only in the implementation of the Runge-Kutta-Merson solver on the GPU can skip the sections containing the mathematical formulation of the problems.

How to cite

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Oberhuber, Tomáš, Suzuki, Atsushi, and Žabka, Vítězslav. "The CUDA implementation of the method of lines for the curvature dependent flows." Kybernetika 47.2 (2011): 251-272. <http://eudml.org/doc/197096>.

@article{Oberhuber2011,
abstract = {We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the integration time step. We implement this solver on CPU but also on GPU using the CUDA toolkit. We demonstrate that the mean-curvature flow can be successfully approximated in single precision arithmetic with the speed-up almost 17 on the Nvidia GeForce GTX 280 card compared to Intel Core 2 Quad CPU. On the same card, we obtain the speed-up 7 in double precision arithmetic which is necessary for the fourth order problem - the Willmore flow of graphs. Both speed-ups were achieved without affecting the accuracy of the approximation. The article is structured in such way that the reader interested only in the implementation of the Runge-Kutta-Merson solver on the GPU can skip the sections containing the mathematical formulation of the problems.},
author = {Oberhuber, Tomáš, Suzuki, Atsushi, Žabka, Vítězslav},
journal = {Kybernetika},
keywords = {GPGPU; CUDA; parallel algorithms; high performance computing; differential geometry; mean-curvature flow; Willmore flow; Runge--Kutta method; method of lines; explicit scheme; complementary finite volume method; GPGPU; CUDA; parallel algorithms; high performance computing; differential geometry; mean-curvature flow; Willmore flow; Runge-Kutta method; method of lines; explicit scheme; complementary finite volume method; computer graphics; image processing; graphics processing unit; compute unified device architecture},
language = {eng},
number = {2},
pages = {251-272},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The CUDA implementation of the method of lines for the curvature dependent flows},
url = {http://eudml.org/doc/197096},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Oberhuber, Tomáš
AU - Suzuki, Atsushi
AU - Žabka, Vítězslav
TI - The CUDA implementation of the method of lines for the curvature dependent flows
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 2
SP - 251
EP - 272
AB - We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the integration time step. We implement this solver on CPU but also on GPU using the CUDA toolkit. We demonstrate that the mean-curvature flow can be successfully approximated in single precision arithmetic with the speed-up almost 17 on the Nvidia GeForce GTX 280 card compared to Intel Core 2 Quad CPU. On the same card, we obtain the speed-up 7 in double precision arithmetic which is necessary for the fourth order problem - the Willmore flow of graphs. Both speed-ups were achieved without affecting the accuracy of the approximation. The article is structured in such way that the reader interested only in the implementation of the Runge-Kutta-Merson solver on the GPU can skip the sections containing the mathematical formulation of the problems.
LA - eng
KW - GPGPU; CUDA; parallel algorithms; high performance computing; differential geometry; mean-curvature flow; Willmore flow; Runge--Kutta method; method of lines; explicit scheme; complementary finite volume method; GPGPU; CUDA; parallel algorithms; high performance computing; differential geometry; mean-curvature flow; Willmore flow; Runge-Kutta method; method of lines; explicit scheme; complementary finite volume method; computer graphics; image processing; graphics processing unit; compute unified device architecture
UR - http://eudml.org/doc/197096
ER -

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