Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs

Journal of Computational Physics

Volumn 231, Issue 21, 2012, Pages 7133-7151

  Bollig, Evan F ^a   Flyer, Natasha ^b   Erlebacher, Gordon ^a  

^a FLORIDA STATE UNIVERSITY   (United States)

^b NATIONAL CENTER FOR ATMOSPHERIC RESEARCH   (United States)

Author keywords

High order finite differencing; Multi GPU computing; OpenCL; Parallel computing; Radial basis functions; RBF FD

Indexed keywords

GRAPHICS PROCESSING UNIT; KNOWLEDGE MANAGEMENT; MESSAGE PASSING; PROGRAM PROCESSORS; RADIAL BASIS FUNCTION NETWORKS;

BASE FUNCTION; FINITE DIFFERENCING; GPU COMPUTING; HIGH-ORDER; HIGH-ORDER FINITE DIFFERENCING; HIGHER-ORDER; MULTI-GPU COMPUTING; OPENCL; PARALLEL COM- PUTING; RADIAL BASE FUNCTION; RADIAL BASE FUNCTION FINITE-DIFFERENCE; RADIAL BASIS;

FINITE DIFFERENCE METHOD;

EID: 84865400698     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2012.06.030     Document Type: Article

References (59)

1. BOOST C++ Libraries. http://www.boost.org.
2. AccelerEyes. Jacket User Guide - The GPU Engine for MATLAB, 1.2.1ed., November 2009.
3. N. Bell M. Garland, Implementing sparse matrix-vector multiplication on throughput-oriented processors, in: Proceedings of the Conference on High Performance Computing Networking Storage and Analysis - SC '09, vol. 1, 2009, p. 1.
4. Bollig E.F. Multi-GPU solutions of geophysical PDEs with radial basis function-generated finite differences. PhD thesis Florida State University 2012.
5. Brandstetter A., Artusi A. Radial basis function networks GPU based implementation. IEEE Transaction on Neural Network 2008, 19(12):2150-2161.
6. Buhmann M.D. Radial Basis Functions: Theory and Implementations 2003, Cambridge University Press.
7. Carr J.C., Beatson R.K., McCallum B.C., Fright W.R., McLennan T.J., Mitchell T.J. Smooth Surface Reconstruction from Noisy Range Data. GRAPHITE '03: Proceedings of the First International Conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia 2003, 119-126. ACM, New York, NY, USA.
8. Cecil T., Qian J., Osher S. Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. Journal of Computational Physics 2004, 196:327-347.
9. Chandhini G., Sanyasiraju Y. Local RBF-FD solutions for steady convection-diffusion problems. International Journal for Numerical Methods in Engineering 2007, 72(3).
10. P.P. Chinchapatnam, K. Djidjeli, P.B. Nair, M. Tan, A compact RBF-FD based meshless method for the incompressible Navier-Stokes equations, in: Proceedings of the Institution of Mechanical Engineers Part M: Journal of Engineering for the Maritime Environment, 223 (3) (2009) pp. 275-290.
11. Connor M., Kumar P. Fast construction of k-nearest neighbor graphs for point clouds. IEEE Transactions on Visualization and Computer Graphics 2010, 16:599-608.
12. Corrigan A., Dinh H. Computing and rendering implicit surfaces composed of radial basis functions on the GPU. International Workshop on Volume Graphics 2005.
13. N. Cuntz, M. Leidl, T. Darmstadt, G. Kolb, C. Salama, M. Böttinger, D. Klimarechenzentrum, G. Hamburg, GPU-based Dynamic Flow Visualization for Climate Research Applications, in: Proceedings of SimVis, 2007, pp. 371-384.
14. Divo E., Kassab A. An efficient localized radial basis function meshless method for fluid flow and conjugate heat transfer. Journal of Heat Transfer 2007, 129:124.
15. Flyer N., Fornberg B. Radial basis functions: developments and applications to planetary scale flows. Computers & Fluids 2011, 46(1):23-32.
16. Flyer N., Lehto E. Rotational transport on a sphere: local node refinement with radial basis functions. Journal of Computational Physics 2010, 229(6):1954-1969.
17. Flyer N., Lehto E., Blaise S., Wright G.B., St-Cyr A. Rbf-generated finite differences for nonlinear transport on a sphere: shallow water simulations. Journal of Computational Physics 2012, 231(11):4078-4095.
18. Flyer N., Wright G.B. Transport schemes on a sphere using radial basis functions. Journal of Computational Physics 2007, 226 1:1059-1084.
19. Flyer N., Wright G.B. A radial basis function method for the shallow water equations on a sphere. Proceedings of the Royal Society A 2009, 465(2106):1949-1976.
20. Fornberg B., Driscoll T., Wright G., Charles R. Observations on the behavior of radial basis function approximations near boundaries. Computers & Mathematics with Applications 2002, 43(3-5):473-490.
21. Fornberg B., Flyer N. Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids. Advances in Computational Mathematics 2005, 23(1-2):5-20.
22. Fornberg B., Larsson E., Flyer N. Stable computations with gaussian radial basis functions. SIAM Journal on Scientific Computing 2011, 33(2):869-892.
23. Fornberg B., Lehto E. Stabilization of RBF-generated finite difference methods for convective PDEs. Journal of Computational Physics 2011, 230(6):2270-2285.
24. Fornberg B., Piret C. A stable algorithm for flat radial basis functions on a sphere. SIAM Journal on Scientific Computing 2007, 30 1:60-80.
25. Fornberg B., Piret C. On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere. Journal of Computational Physics 2008, 227 5:2758-2780.
26. Fornberg B., Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Computers & Mathematics with Applications 2004, 48(5-6):853-867.
27. E. Gabriel, G.E. Fagg, G. Bosilca, T. Angskun, J.J. Dongarra, J.M. Squyres, V. Sahay, P. Kambadur, B. Barrett, A. Lumsdaine, R.H. Castain, D.J. Daniel, R.L. Graham, T.S. Woodall, OpenMPI: Goals concept and design of a next generationMPI implementation, in: Proceedings 11th European PVM/MPI Users' Group Meeting, Budapest, Hungary, September 2004, pp. 97-104.
28. P. Goswami, P. Schlegel, B. Solenthaler, R. Pajarola, Interactive sph simulation and rendering on the gpu, in: Proceedings of the 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Aire-la-Ville Switzerland Switzerland 2010) SCA '10 Eurographics Association, pp. 55-64.
29. G. Guennebaud, B. Jacob et al., Eigen v3. 2010. http://eigen.tuxfamily.org.
30. Hardy R. Multiquadratic equations of topography and other irregular surfaces. Journal of Geophysical Research 1971, 76:1-905.
31. L. Ivan, H.D. Sterck, S.A. Northrup, C.P.T. Groth, Three-dimensional MHD on cubed-sphere grids: parallel solution-adaptive simulation framework, in: 20th AIAA CFD Conference, vol. 3382, 2011, pp. 1325-1342.
32. Jakob-Chien R., Hack J., Williamson D. Spectral transform solutions to the shallow water test set. Journal of Computational Physics 1995, 119 1:164-187.
33. Kansa E.J. Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics. I. Surface approximations and partial derivative estimates. Computers & Mathematics with Applications 1990, 19:127-145.
34. Karypis G., Kumar V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing 1999, 20 1:359-392.
35. Khronos OpenCL Working Group, The OpenCL Specification (Version: 1.0.48), October 2009.
36. Kosec G., Šarler B. Solution of thermo-fluid problems by collocation with local pressure correction. International Journal of Numerical Methods for Heat & Fluid Flow 2008, 18.
37. Larsson E., Fornberg B. A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers & Mathematics with Applications 2003, 46:891-902.
38. Nair R.D., Thomas S., Loft R. A discontinuous Galerkin transport scheme on the cubed sphere. Monthly Weather Review 2005, 133(4):814-828.
39. Nair R.D., Jablonowski C. Moving vortices on the sphere: a test case for horizontal advection problems. Monthly Weather Review 2008, 136(2):699-711.
40. NVidia, NVIDIA CUDA - NVIDIA CUDA C - Programming Guide version 4.0, March 2011.
41. J. Pan, D. Manocha, Fast GPU-based locality sensitive hashing for K-nearest neighbor computation, in: Proceedings of the 19th ACM SIGSPATIAL GIS '11, 2011.
42. Randall D., Ringler T., Heikes R. Climate modeling with spherical geodesic grids. Computing in Science & Engineering 2002, 32-41.
43. C. Sanderson, Armadillo: An open source c++ linear algebra library for fast prototyping and computationally intensive experiments, Tech. rep. NICTA, 2010.
44. chaback R. Multivariate interpolation and approximation by translates of a basis function. Approximaton Theory VIII-Vol. 1: Approximation and Interpolation 1995, 491-514. World Scientific Publishing Co. Inc. C. Chui, L. Schumaker (Eds.).
45. Schmidt J., Piret C., Zhang N., Kadlec B., Yuen D., Liu Y., Wright G., Sevre E. Modeling of Tsunami waves and atmospheric swirling flows with graphics processing unit (GPU) and radial basis functions (RBF). Concurrency and Computation: Practice and Experience 2009.
46. Shu C., Ding H., Yeo K.S. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 2003, 192(7-8):941-954.
47. Sloan I.H., Womersley R.S. Extremal systems of points and numerical integration on the sphere. Advance in Computational Mathmatics 2003, 21:107-125.
48. Stevens D., Power H., Lees M., Morvan H. The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems. Journal of Computational Physics 2009.
49. A.I. Tolstykh, On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations, in: Proceedings of the 16 IMACS World Congress Lausanne, 2000, 1-6.
50. Tolstykh A.I., Shirobokov D.A. On using radial basis functions in a "finite difference mode" with applications to elasticity problems. Computational Mechanics 2003, 33(1).
51. Vetter J., Glassbrook R., Dongarra J., Schwan K., Loftis B., McNally S., Meredith J., Rogers J., Roth P., Spafford K., Yalamanchili S. Keeneland: Bringing heterogeneous GPU computing to the computational science community. IEEE Computing in Science and Engineering 2011, 13(5):90-95.
52. R. Vuduc, J.W. Demmel, K.A. Yelick, Oski: A library of automatically tuned sparse matrix kernels, in: Institute of Physics Publishing, 2005.
53. Weiler M., Botchen R., Stegmaier S., Ertl T., Huang J., Jang Y., Ebert D., Gaither K. Hardware-assisted feature analysis and visualization of procedurally encoded multifield volumetric data. IEEE Computer Graphics and Applications 2005, 25 5:72-81.
54. Wendland H., polynomial Piecewise positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 1995, 4:389-396. 10.1007/BF02123482.
55. Wendland H. Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics 2005, Cambridge University Press.
56. G.B. Wright, Radial basis function interpolation: numerical and analytical developments, PhD thesis University of Colorado, 2003.
57. Wright G.B., Flyer N., Yuen D.A. A hybrid radial basis function-pseudospectral method for thermal convection in a 3-D spherical shell. Geochemistry, Geophysics, Geosystems 2010, 11:Q07003.
58. Wright G.B., Fornberg B. Scattered node compact finite difference-type formulas generated from radial basis functions. Journal of Computational Physics 2006, 212(1):99-123.
59. Yokota R., Barba L., Knepley M.G. PetRBF - a parallel O(N) algorithm for radial basis function interpolation with Gaussians. Computer Methods in Applied Mechanics and Engineering 2010, 199(25-28):1793-1804.