Reduction of computing time for seismic applications based on the Helmholtz equation by Graphics Processing Units

The oil and gas industry makes use of computational intensive algorithms to provide an image of the subsurface. The image is obtained by sending wave energy into the subsurface and recording the signal required for a seismic wave to reflect back to the surface from the Earth interfaces that may have different physical properties. A seismic wave is usually generated by shots of known frequencies, placed close to the surface on land or close to the water surface in the sea. Returning waves are usually recorded in time by hydrophones in a marine environment or by geophones during land acquisition. The goal of seismic imaging is to transform the seismograms to a spatial image of the subsurface. Migration algorithms produce an image of the subsurface given the seismic data measured at the surface.
In this thesis we focus on solving the Helmholtz equation which represents the wave propagation in the frequency domain. We can easily convert fromthe timedomain to the frequency domain and vice-versa using the Fourier transformation. A discretizationwith second-order finite differences gives a sparse linear system of equations that needs to be solved for each frequency. Two- as well as three-dimensional problems are considered. Krylov subspace methods such as Bi-CGSTAB and IDR(s) have been chosen as solvers. Since the convergence of the Krylov subspace solvers deteriorates with an increasing wave number, a shifted Laplacian multigrid preconditioner is used to improve the convergence. Here, we extend the matrix-dependent multigrid method to solve complexvaluedmatrices in three dimensions. As the smoother,we have considered parallelizable methods such as weighted Jacobi (!-Jacobi), multi- colored Gauss-Seidel and damped multi-colored Gauss-Seidel (!-GS).