1. Two Phase Flow using two levels of preconditioning on the GPU Prof. Kees Vuik and Rohit Gupta Delft Institute of Applied Mathematics

2. Problem Description Delft Institute of Applied Mathematics

3. Computational Model Boundary Conditions Delft Institute of Applied Mathematics

4. Graphical Processing Unit Delft Institute of Applied Mathematics SIMD based Architecture: Army of Smaller Simpler Processors Larger Memory Bandwidth Programmer Managed Caches

5. Preconditioning M -1 =(I-LD -1 )(I-D -1 L T ) Delft Institute of Applied Mathematics

6. Deflation Optimized Storage of AZ Stripe-Wise Domains Splitting Chosen X = ( I – P T ) x + P T x P=I-AQ Q=ZE -1 Z T E=Z T AZ 5758596061626364 4956 4148 3340 2532 1724 916 12345678 Delft Institute of Applied Mathematics

7. Factors Affecting Speed-Up Coalesced Memory Access More Deflation Vectors More preconditioning blocks

8. Results: Deflated Preconditioned CG HostDeviceHostDevice Number of Iterations34 37 Execution Time (seconds) 3.690.976.970.27 Relative Error Norm of the Solution 5.25e-03 4.59e-03 SpeedUp3.7926.14 Delft Institute of Applied Mathematics Poisson Type Matrix solved with Single Precision Math. ~1 Millions Unknowns (1024x1024). Precision Criteria 10e-04. Number of Blocks =512. Deflation Vectors=4096

9. Two Phase (Double Precision) Preliminary Results Deflated Preconditioned(IP) Conjugate Gradient Precision Criteria 10e-05. Deflation Vectors=4096 HostDevice Number of Iterations 394 Execution Time (seconds) 42.22.512 Relative Error Norm of the Solution 7.80e-022.98e-02 SpeedUp 16.8 Delft Institute of Applied Mathematics

10. Conclusion Deflation suits the many core platform Two Phase requires double precision Deflation with IP Preconditioning wins Delft Institute of Applied Mathematics

11. References 1.J. M. Tang and C. Vuik. Acceleration of preconditioned krylov solvers for bubbly ow problems. Lecture Notes in Computer Science, Parallel Processing and Applied Mathematics, 4967(1): 13231332, 2008. 2.S.P. Van der Pijl, A. Segal, C. Vuik, and P. Wesseling. A mass conserving level-set method for modelling of multi-phase ows. International Journal for Numerical Methods in Fluids, 47: 339361, 2005. 3.M. Ament, G. Knittel, D. Weiskopf, and W. Strbaer. A parallel preconditioned conjugate gradient solver for the poisson problem on a multi-GPU platform. http://www.vis.unistuttgart.de/ amentmo/docs/ament-pcgip-PDP-2010.pdf, 2010. 4.R. Gupta. Implementation of the Deated Preconditioned Conjugate Gradient Method for Bubbly Flow on the Graphical Processing Unit(GPU). Master's thesis, Delft University of Technology, Delft, 2010. http://ta.twi.tudelft.nl/nw/users/vuik/numanal/gupta_afst.pdf. Delft Institute of Applied Mathematics