GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method

Knibbe, H.; Oosterlee, Cornelis W.; Vuik, C.

doi:10.1016/j.cam.2011.07.021

Cited by 29 publications

(23 citation statements)

References 18 publications

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“…In this case, we choose α = 0.05. From our experiments, we have observed that this choice of α does not affect the quality of the image significantly within the LSM framework, however, it leads to faster computational times of the Helmholtz solver, see Knibbe et al [15].…”

Section: Helmholtz Solvermentioning

confidence: 75%

“…It has been shown in Knibbe et al [15] that the preconditioned Helmholtz solver is parallelizable on CPUs as well as on a single GPU and provides an interesting speedup on parallel architectures.…”

Section: Helmholtz Solvermentioning

confidence: 99%

“…In the first case, the data lives in GPU memory to avoid memory transfers between CPU and GPU memory. We have already investigated this approach for the Helmholtz equation in the frequency domain in Knibbe et al [15,16]. The advantage of the migration with a frequency domain solver is that it does not require large amounts of disk space to store the snapshots.…”

Section: Introductionmentioning

confidence: 99%

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Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

2015

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In geophysical applications, the interest in leastsquares migration (LSM) as an imaging algorithm is increasing due to the demand for more accurate solutions and the development of high-performance computing. The computational engine of LSM in this work is the numerical solution of the 3D Helmholtz equation in the frequency domain. The Helmholtz solver is Bi-CGSTAB preconditioned with the shifted Laplace matrix-dependent multigrid method. In this paper, an efficient LSM algorithm is presented using several enhancements. First of all, a frequency decimation approach is introduced that makes use of redundant information present in the data. It leads to a speedup of LSM, whereas the impact on accuracy is kept minimal. Secondly, a new matrix storage format Very Compressed Row Storage (VCRS) is presented. It not only reduces the size of the stored matrix by a certain factor but also increases the efficiency of the matrix-vector computations. The effects of lossless and lossy compression with a proper choice of the compression parameters are positive. Thirdly, we accelerate the LSM engine by graphics cards (GPUs). A GPU is used as an accelerator, where the data is partially transferred to a GPU to execute a set of operations or as a replacement, where the complete data is stored in the GPU memory. We demonstrate that using the GPU as a replacement leads to Summarizing the effects of each improvement, the resulting speedup can be at least an order of magnitude compared to the original LSM method.

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Section: Helmholtz Solvermentioning

confidence: 75%

Section: Helmholtz Solvermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

2015

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“…An iterative solver is an obvious alternative; for instance, the one with a preconditioner that uses a multigrid method to solve the same wave equation but with very strong damping Riyanti et al, 2006;Plessix, 2007;Knibbe et al, 2011). This method, however, needs a number of iterations that increases with frequency, causing the approach to be less efficient than a timedomain method.…”

Section: Introductionmentioning

confidence: 99%

“…To achieve the best performance, the data are kept on the GPU when possible. We have exploited this way of using a GPU for the Helmholtz equation earlier (Knibbe et al, 2011(Knibbe et al, , 2013.…”

Section: Introductionmentioning

confidence: 99%

Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3D migration on parallel architectures

et al. 2014

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Three-dimensional reverse-time migration with the constantdensity acoustic wave equation requires an efficient numerical scheme for the computation of wavefields. An explicit finitedifference scheme in the time domain is a common choice. However, it requires a significant amount of disk space for the imaging condition. The frequency-domain approach simplifies the correlation of the source and receiver wavefields, but requires the solution of a large sparse linear system of equations. For the latter, we use an iterative Krylov solver based on a shifted Laplace multigrid preconditioner with matrixdependent prolongation. The question is whether migration in the frequency domain can compete with a time-domain implementation when both are performed on a parallel architecture. Both methods are naturally parallel over shots, but the frequency-domain method is also parallel over frequencies. If we have a sufficiently large number of compute nodes, we can compute the result for each frequency in parallel and the required time is dominated by the number of iterations for the highest frequency. As a parallel architecture, we consider a commodity hardware cluster that consists of multicore central processing units (CPUs), each of them connected to two graphics processing units (GPUs). Here, GPUs are used as accelerators and not as an independent compute node. The parallel implementation of the 3D migration in frequency domain is compared to a time-domain implementation. We optimize the throughput of the latter with dynamic load balancing, asynchronous I/O, and compression of snapshots. Because the frequency-domain solver uses matrix-dependent prolongation, the coarse-grid operators require more storage than available on GPUs for problems of realistic size. Due to data transfer, there is no significant speedup using GPU-accelerators. Therefore, we consider an implementation on CPUs only. Nevertheless, with the parallelization over shots and frequencies, this approach could compete with the time-domain implementation on multiple GPUs.

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CUDA parallel programming for simulation of epidemiological models based on individuals

Filho

Paula

Coelho

et al. 2015

Math. Meth. Appl. Sci.

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Mathematical models are of great value in epidemiology to help understand the dynamics of the various infectious diseases, as well as in the conception of effective control strategies. The classical approach is to use differential equations to describe, in a quantitative manner, the spread of diseases within a particular population. An alternative approach is to represent each individual in the population as a string or vector of characteristic data and simulate the contagion and recovery processes by computational means. This type of model, referred in the literature as MBI (models based on individuals), has the advantage of being flexible as the characteristics of each individual can be quite complex, involving, for instance, age, sex, pre‐existing health conditions, environmental factors, social habits, etc. However, when it comes to simulations involving large populations, MBI may require a large computational effort in terms of memory storage and processing time. In order to cope with the problem of heavy computational effort, this paper proposes a parallel implementation of MBI using a graphics processor unit compatible with CUDA. It was found that, even in the case of a simple susceptible–infected–recovered model, the computational gains in terms of processing time are significant. Copyright © 2015 John Wiley & Sons, Ltd.

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GPU implementation of a Helmholtz Krylov solver preconditioned by a shifted Laplace multigrid method

Cited by 29 publications

References 18 publications

Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units

Closing the performance gap between an iterative frequency-domain solver and an explicit time-domain scheme for 3D migration on parallel architectures

CUDA parallel programming for simulation of epidemiological models based on individuals

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