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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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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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