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Calibri 83ffff̙̙3f3fff3f3f33333f33333.9TU Delft Repositoryg @uuidrepository linktitleauthorcontributorpublication yearabstract
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departmentresearch group programmeprojectcoordinates)uuid:b2da5584c4064eec9c03a8b10f9d3d1eDhttp://resolver.tudelft.nl/uuid:b2da5584c4064eec9c03a8b10f9d3d1e@On a class of preconditioners for solving the Helmholtz equation)Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W.5Helmholtz equation; preconditioners; GMRES; BiCGSTABenreportDelft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics8Electrical Engineering, Mathematics and Computer Science)uuid:04de891244334222a0da14ee2a6301c9Dhttp://resolver.tudelft.nl/uuid:04de891244334222a0da14ee2a6301c9KA novel multigrid based preconditioner for heterogeneous Helmholtz problems)Erlangga, Y.A.; Oosterlee, C.W.; Vuik, C.xAn iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is made with the help of Fourier analysis. Multigrid analysis results are verifed by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.cHelmholtz equation; nonconstant high wavenumber; complex multigrid preconditioner; Fourier analysis)uuid:0d5a64ef90b245e695e5fb6c75b4a58bDhttp://resolver.tudelft.nl/uuid:0d5a64ef90b245e695e5fb6c75b4a58b\Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian*van Gijzen, M.B.; Erlangga, Y.A.; Vuik, C.jShifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRESresidual norm we are able to provide an optimal value for the shift, and to explain the meshdependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.kHelmholtz equation; shifted Laplace preconditioner; iterative solution methods; GMRES; convergence analysis)uuid:32be8cc98e3444f4896f5fad3e1546afDhttp://resolver.tudelft.nl/uuid:32be8cc98e3444f4896f5fad3e1546aftReduction of computing time for leastsquares migration based on the Helmholtz equation by graphics processing units%Knibbe, H.; Vuik, C.; Oosterlee, C.W.(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 highperformance 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 BiCGSTAB preconditioned with the shifted Laplace matrixdependent 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 f<
actor but also increases the efficiency of the matrixvector 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 higher speedups and allows us to solve larger problem sizes. Summarizing the effects of each improvement, the resulting speedup can be at least an order of magnitude compared to the original LSM method.leastsquares migration; Helmholtz equation; wave equation; frequency domain; multigrid method; GPU acceleration; matrix storage format; frequency decimationjournal articleSpringer&Delft Institute of Applied Mathematics)uuid:0a2bcdfd6b4d433e8aaed876234b9703Dhttp://resolver.tudelft.nl/uuid:0a2bcdfd6b4d433e8aaed876234b9703Knibbe, H.P. (TU Delft Numerical Analysis); Vuik, C. (TU Delft Numerical Analysis); Oosterlee, C.W. (TU Delft Numerical Analysis; Center for Mathematics and Computer Science)s Leastsquares migration; Helmholtz equation; Wave equation; Frequency domain; Multigrid method; GPU acceleration; Matrix storage format; Frequency decimation)uuid:6c51bab413f3436f9216f286e6394333Dhttp://resolver.tudelft.nl/uuid:6c51bab413f3436f9216f286e6394333MConvergence analysis of multilevel sequentially semiseparable preconditionersGQiu, Y.; van Gijzen, M.B.; van Wingerden, J.W.; Verhaegen, M.; Vuik, C.Multilevel sequentially semiseparable (MSSS) matrices form a class of structured matrices that have lowrank offdiagonal structure, which allows the matrixmatrix operations to be performed in linear computational complexity. MSSS preconditioners are computed by replacing the Schur complements in the block LU factorization of the global linear system by MSSS matrix approximations with low offdiagonal rank. In this manuscript, we analyze the convergence properties of such preconditioners. We show that the spectrum of the preconditioned system is contained in a circle centered at (1, 0) and give an analytic bound of the radius of this circle. This radius can be made arbitrarily small by properly setting a parameter in the MSSS preconditioner. Our results apply to a wide class of linear systems. The system matrix can be either symmetric or unsymmetric, definite or indefinite. We demonstrate our analysis by numerical experiments.tmultilevel sequentially semiseparable preconditioners; convergence analysis; saddlepoint systems; Helmholtz equation
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