A Parallel Preconditioned Conjugate Gradient Solver for the Poisson Problem on a Multi-GPU Platform

Ament, Marco; Knittel, Günter; Weiskopf, Daniel; Straßer, Wolfgang

doi:10.1109/pdp.2010.51

Cited by 75 publications

(49 citation statements)

References 17 publications

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“…To compute execution time, we used system-specific high-resolution timers: mach absolute time() on Mac OS X* and clock gettime() on Linux*. We further refer the reader to [1] for GPU parallel implementations of the incomplete Poisson preconditioner. Figure 2 shows that PIPP scales very well with the number of available cores.…”

Section: Parallelization Resultsmentioning

confidence: 99%

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Parallelized Incomplete Poisson Preconditioner in Cloth Simulation

2011

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Efficient cloth simulation is an important problem for interactive applications that involve virtual humans, such as computer games. A common aspect of many methods that have been developed to simulate cloth is a linear system of equations, which is commonly solved using conjugate gradient or multi-grid approaches. In this paper, we introduce to the computer gaming community a recently proposed preconditioner, the incomplete Poisson preconditioner, for conjugate gradient solvers. We show that the parallelized incomplete Poisson preconditioner (PIPP) performs as well as the current state-of-the-art preconditioners, while being much more amenable to standard thread-level parallelism. We demonstrate our results on an 8-core Apple* Mac* Pro and a 32-core code name Emerald Ridge system. Efficient cloth simulation is an important problem for interactive applications that involve virtual humans, such as computer games. A common aspect of many methods that have been developed to simulate cloth is a linear system of equations, which is commonly solved using conjugate gradient or multi-grid approaches. In this paper, we introduce to the computer gaming community a recently proposed preconditioner, the incomplete Poisson preconditioner, for conjugate gradient solvers. We show that the parallelized incomplete Poisson preconditioner (PIPP) performs as well as the current state-of-the-art preconditioners, while being much more amenable to standard threadlevel parallelism. We demonstrate our results on an 8-core Apple* Mac* Pro and a 32-core code name Emerald Ridge system.

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Section: Parallelization Resultsmentioning

confidence: 99%

“…We also examine a new preconditioning scheme, the incomplete Poisson preconditioner, proposed by Ament et al [1] for the Poisson problem. Their motivation was to find an easily parallelizable preconditioner for simulations on multi-GPU systems.…”

Section: Preconditioning Methodsmentioning

confidence: 99%

Parallelized Incomplete Poisson Preconditioner in Cloth Simulation

2011

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“…In the case of very large sparse matrices, problems concerning the speed-up are considerably smaller [18]. Resigning from the standard libraries of universal functions [5] and introducing programs directly adapted to the analysis of the given problem [19][20][21][22][23] can additionally increase the efficiency of parallel computations which use a GPU.…”

Section: Discussionmentioning

confidence: 99%

Parallel computations of the step response of a floor heater with the use of a graphics processing unit. Part 2: results and their evaluation

Gołębiowski¹,

Forenc²

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. Using models and algorithms presented in the first part of the article, a spatio-temporal distribution of the step response of a floor heater was determined. The results have been presented in the form of heating curves and temperature profiles of the heater in the selected time moments. The computations results were verified through comparing them with the solution obtained with the use of a commercial program -NISA. Additionally, the distribution of the average time constant of thermal processes occurring in the heater was determined. The analysis of the use of a graphics processing unit in numerical computations based on the conjugate gradient method was done. It was proved that the use of a graphics processing unit is profitable in the case of solving linear systems of equations with dense coefficient matrices. In the case of a sparse matrix, the speed-up depends on the number of its non-zero elements.Key words: air floor heating, step response computation, parallel computations, GPGPU. Computational program parameters, material properties of the model, computing platformThe construction of the analyzed model of the heater was described in the first part of this paper ( Fig. 2 in [1], where L = 0.15 m). In order to conduct the discrete approximation of the model, an appropriate fragment of the floor slab was covered with a finite difference mesh of an identical step along x, y axes which equal ∆x = ∆y = 2.34375 · 10 −3 m. The mesh has 65×97 nodes. However, the number of nodes taken into consideration (and thus the number of unknown quantities of the obtained system of equations) is smaller and equals to 4289. As 2016 nodes, which are placed in the crosssection of the duct of the given temperature T H , should be omitted. The step response was determined with the assumed time step ∆t = 2 s and the N = 27000 number of steps. As a result, the considered time interval equals to 54 000 s (15 hours from letting the hot air into the ducts).Denotations of material properties were explained in the first part of the article (chapter 2 in [1]). The following set of data was assumed:Computations were done with the use of a personal computer equipped with an Intel Core 2 Quad CPU Q9650 3.00 GHz processor, RAM memory of 4 GB and a 64-bit Windows 7 Professional operating system. A Gigabyte Nvidia GeForce GTX480 graphics card equipped with 480 streaming processors, and 1536 MB of GDDR5 memory were installed in the computer. It also contained the CUDA parallel computing environment [2,3] Step responses and their computational verificationThe computations of the step response were done according to the algorithm described in the first part of the article [1]. The obtained curves can be presented in a graph for selected points of the analyzed model of a floor heater. Characteristic points (A-J) were marked in Fig. 1, and the obtained graphs of the step response are presented in Fig. 2. Fig. 1. Analyzed fragment of the heater with the characteristic points and adiabats *

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“…The preconditioner in [2] offers as much parallelism as the number of degrees of freedom, N (or the number of unknowns). However, our experiments [1] show its use is limited for two-phase (high condition number, (κ)) flow problems.…”

Section: Related Workmentioning

confidence: 99%

“…However, our experiments [1] show its use is limited for two-phase (high condition number, (κ)) flow problems. An extension to [2] is provided in [8] wherein a relaxation factor is utilized. In [9] an incomplete LU decomposition based preconditioner with fill-in is used combined with reordering using multi-coloring.…”

Section: Related Workmentioning

confidence: 99%

Efficient Two-Level Preconditioned Conjugate Gradient Method on the GPU

Gupta

Gijzen

Vuik

2013

Lecture Notes in Computer Science

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Abstract. We present an implementation of a Two-Level Preconditioned Conjugate Gradient Method for the GPU. We investigate a Truncated Neumann Series based preconditioner in combination with deflation. This combination exhibits fine-grain parallelism and hence we gain considerably in execution time when compared with a similar implementation on the CPU. Its numerical performance is comparable to the Block Incomplete Cholesky approach. Our method provides a speedup of up to 16 for a system of one million unknowns when compared to an optimized implementation on one core of the CPU.

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A Parallel Preconditioned Conjugate Gradient Solver for the Poisson Problem on a Multi-GPU Platform

Cited by 75 publications

References 17 publications

Parallelized Incomplete Poisson Preconditioner in Cloth Simulation

Parallelized Incomplete Poisson Preconditioner in Cloth Simulation

Parallel computations of the step response of a floor heater with the use of a graphics processing unit. Part 2: results and their evaluation

Efficient Two-Level Preconditioned Conjugate Gradient Method on the GPU

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