Parallel resolution of the 3D Helmholtz equation based on multi-graphics processing unit clusters

Concurrency and Computation: Practice and Experience

Volumn 27, Issue 13, 2015, Pages 3205-3219

  Ortega, Gloria ^a   Lobera, Julia ^b   García, Inmaculada ^c   Pilar Arroyo, M ^d   Garzõn, Ester M ^a  

^a UNIVERSITY OF ALMERÍA   (Spain)

^b CENTRO UNIVERSITARIO DE LA DEFENSA   (Spain)

^c UNIVERSITY OF MÁLAGA   (Spain)

^d UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

biconjugate gradient method; Helmholtz equation; MPI paradigm; multi GPU

Indexed keywords

COMPUTER GRAPHICS EQUIPMENT; GRADIENT METHODS; HELMHOLTZ EQUATION; MESSAGE PASSING; PROGRAM PROCESSORS; THREE DIMENSIONAL COMPUTER GRAPHICS;

BI-CONJUGATE GRADIENT METHODS; HIGH-PERFORMANCE COMPUTING RESOURCES; LARGE SPARSE MATRIX; MESSAGE PASSING INTERFACE; MPI PARADIGM; MULTI-GPU; SPARSE MATRIX VECTOR PRODUCTS; TECHNOLOGICAL APPLICATIONS;

GRAPHICS PROCESSING UNIT;

EID: 84939529782     PISSN: 15320626     EISSN: 15320634     Source Type: Journal    
DOI: 10.1002/cpe.3212     Document Type: Article

References (43)

1. Dautray R, Lions JL,. Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag: Berlin, 1990.
2. Junger MC, Feit D,. Sound, Structures, and Their Interaction (Second Edition). The MIT Press: Cambridge, Massachusetts, 1986.
3. Harris M,. Fast fluid dynamics simulation on the GPU, ACM SIGGRAPH 2005 Courses, New York, NY, USA, 2005; 637-665.
4. Sadiku MNO,. Numerical Techniques in Electromagnetics (Second Edition). CRC Press: Boca Raton, 2001.
5. Nail A, Gumerov RD,. Fast Multipole Methods for the Helmholtz Equation in Three Dimensions. Elsevier Science: Amsterdam, London, 2004.
6. Ihlenburg F, Babuska I,. Solution of Helmholtz problems by knowledge-based FEM. CAMES 1997; 4: 397-415.
7. Babuska IM, Sauter SA,. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Review 2000; 42 (3): 451-484. DOI: 10.1137/S0036142994269186.
8. Bao G, Wei GW, Zhao S,. Numerical solution of the Helmholtz equation with high wave numbers. International Journal for Numerical Methods in Engineering 2004; 59: 389-408.
9. Lobera J, Coupland JM,. Optical diffraction tomography in fluid velocimetry: The use of a priori information. Measurement Science and Technology 2008; 19 (7): 074013.
10. TOP 500 supercomputing site. Available form: http://www.top500.org/ [Accessed on 21 january 2014].
11. Jacobsen DA, Thibault JC, Senocak I,. An MPI-CUDA implementation for massively parallel incompressible flow computations on multi-GPU clusters, 2010. Available from: http://scholarworks.boisestate.edu/cgi/viewcontent.cgi?article=1004&context=mecheng-facpubs [Accessed on 21 january 2014].
12. Ortega G, Lobera J, Arroyo MP, García I, Garzõn EM,. High performance computing for optical diffraction Tomography. Proceedings of The 2012 International Conference on High Performance Computing & Simulation (HPCS 2012), 2012; 195-201.
13. Saad Y,. Iterative Methods for Sparse Linear Systems (Second Edition). SIAM: Philadelphia, 2003.
14. Barrett R, Berry M, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, der Vorst HV,. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (2nd Edition). SIAM: Philadelphia, PA, 1994.
15. Van der Vorst HA,. Iterative Krylov Methods for Large Linear Systems, Cambridge monographs on applied and computational mathematics, Cambridge University Press: Cambridge, UK, New York, 2003. Available from: http://opac.inria.fr/record=b1100090 [Accessed on 21 january 2014].
16. Greenbaum A,. Iterative Methods for Solving Linear Systems. Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1997.
17. Axelsson O, Kucherov A,. Real valued iterative methods for solving complex symmetric linear systems. Numerical Linear Algebra with Applications 2000; 7 (4): 197-218, DOI: 10.1002/1099-1506(200005)7:4197::AID-NLA1943.0.CO;2-S.
18. Van der Vorst HA, Melissen JBM,. A Petrov-Galerkin type method for solving Axk=b, where A is symmetric complex. IEEE Transactions on Magnetics 1990; 26 (2): 706-708, DOI: 10.1109/20.106415.
19. Balay S, et al,. PETSc Users Manual. Revision 3.3. Available from: http://www.mcs.anl.gov/petsc/petsc-current/docs/manual.pdf [Accessed on 21 january 2014].
20. Li L, Xue W, Ranjan R, Jin Z,. A scalable Helmholtz solver in GRAPES over large-scale multicore cluster. Concurrency and Computation: Practice and Experience 2013; 25 (12): 1722-1737. DOI: 10.1002/cpe.2979.
21. Knibbe H, Oosterlee CW, Vuik C,. 3D Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method on multi-GPUs. In Numerical Mathematics and Advanced Applications 2011, Cangiani A, Davidchack R, Georgoulis E, Gorban A, Levesley J, Tretyakov M, (eds). Springer: Berlin Heidelberg, 2013; 653-661.
22. Bordage C,. Parallelization on heterogeneous multicore and multi-GPU systems of the fast multipole method for the Helmholtz equation using a runtime system. ADVCIMP12, Barcelone, Espagne, September 2012; 90-95. Available from: http://hal.inria.fr/hal-00773114 [Accessed on 21 january 2014].
23. Kononov AV, Riyanti CD, de Leeuw SW, Oosterlee CW, Vuik C,. Numerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation. Computing and Visualization in Science April 2008; 11 (3): 139-146. DOI: 10.1007/s00791-007-0069-6.
24. Williams S, Oliker L, Vuduc R, Shalf J, Yelick K, Demmel J,. Optimization of sparse matrix-vector multiplication on emerging multicore platforms. Parallel Computing 2009; 35 (3): 178-194. DOI: http://dx.doi.org/ 10.1016/j.parco.2008.12.006.
25. INTEL. Math kernel library, 2013. Available from: http://software.intel.com/en-us/articles/intel-math-kernel-library-documentation [Accessed on 21 january 2014].
26. Kurzak J, Alvaro W, Dongarra J,. Optimizing matrix multiplication for a short-vector SIMD architecture-CELL processor. Parallel Computing 2009; 35 (3): 138-150. DOI: http://dx.doi.org/ 10.1016/j.parco.2008.12.010.
27. Bell N, Garland M,. Implementing sparse matrix-vector multiplication on throughput-oriented processors. Sc '09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis, New York, NY, USA, 2009; 1-11.
28. Buatois L, Caumon G, Lévy B,. Concurrent number cruncher-a GPU implementation of a general sparse linear solver. International Journal of Parallel Emergent and Distributed Systems 2009; 24 (3): 205-223.
29. Monakov A, Lokhmotov A, Avetisyan A,. Automatically tuning sparse matrix-vector multiplication for GPU architectures. Proceedings of HiPEAC 2010, LNCS 5952, Pisa, Italy, 2010; 111-125.
30. Vázquez F, Fernández JJ, Garzõn EM,. A new approach for sparse matrix vector product on NVIDIA GPUs. Concurrency and Computation: Practice And Experience 2011; 23 (8): 815-826. DOI: 10.1002/cpe.1658.
31. Vázquez F, Ortega G, Fernández JJ, Garzõn EM,. Improving the performance of the sparse matrix vector product with GPUs. 10th IEEE International Conference on Computer and Information Technology. CIT 2010, 2010; 1146-1151.
32. NVIDIA. Cusparse library, V5.5, 2013. Available from: http://docs.nvidia.com/cuda/cusparse/ [Accessed on 21 january 2014].
33. Golub GH, van Van Loan CF,. Matrix Computations (Johns Hopkins Studies in Mathematical Sciences) (3rd Edition). The Johns Hopkins University Press, 1996.
34. Vázquez F, Garzõn EM, Fernández JJ,. A matrix approach to tomographic reconstruction and its implementation on GPUs. Journal of Structural Biology 2010; 170 (1): 146-151.
35. Ihlenburg F, Babuska I,. Finite element solution of the Helmholtz equation with high wave number part I: The h-version of the FEM. Computers & Mathematics with Applications 1995; 30 (9): 9-37. DOI: 10.1016/0898-1221(95)00144-N.
36. Babuska I, Banerjee U, Osborn JE,. Generalized finite element methods- main ideas, results and perspective. International Journal of Computational Methods 2004; 1: 153-156.
37. Basermann A, Reichel B, Schelthoff C,. Preconditioned CG methods for sparse matrices on massively parallel machines. Parallel Computing 1997; 23 (3): 381-398.
38. Bisseling RH,. Parallel Scientific Computation. Oxford University Press: New York, USA, 2004.
39. Ortega G, Garzõn EM, Vázquez F, García I,. The biconjugate gradient method on GPUs. The Journal of Supercomputing 2013; 64: 49-58. DOI: 10.1007/s11227-012-0761-2.
40. Patterson DA, Hennessy JL,. Computer Organization and Design-The Hardware / Software Interface (Revised 4th Edition), The Morgan Kaufmann Series in Computer Architecture and Design, Academic Press: Amsterdam, 2012.
41. Snir M, Otto S, Huss-Lederman S, Walker D, Dongarra J,. MPI-The Complete Reference, Volume 1: The MPI Core (2nd. (Revised)). MIT Press: Cambridge, MA, USA, 1998.
42. NVIDIA Corporation 2701 San Tomas Expressway. Santa Clara 95050, USA. CUDA C Best Practices Guide., 2013. Available from: http://docs.nvidia.com/cuda/cuda-c-best-practices-guide/index.html [Accessed on 21 january 2014].
43. Anzt H, Castillo M, Fernández JC, Heuveline V, Igual FD, Mayo R, Quintana-Ortí ES,. Optimization of power consumption in the iterative solution of sparse linear systems on graphics processors. Computer Science-R&D 2012; 27 (4): 299-307.