"uuid","repository link","title","author","contributor","publication year","abstract","subject topic","language","publication type","publisher","isbn","issn","patent","patent status","bibliographic note","access restriction","embargo date","faculty","department","research group","programme","project","coordinates"
"uuid:b2da5584-c406-4eec-9c03-a8b10f9d3d1e","http://resolver.tudelft.nl/uuid:b2da5584-c406-4eec-9c03-a8b10f9d3d1e","On a class of preconditioners for solving the Helmholtz equation","Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W.","","2003","","Helmholtz equation; preconditioners; GMRES; Bi-CGSTAB","en","report","Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics","","","","","","","","Electrical Engineering, Mathematics and Computer Science","","","","",""
"uuid:04de8912-4433-4222-a0da-14ee2a6301c9","http://resolver.tudelft.nl/uuid:04de8912-4433-4222-a0da-14ee2a6301c9","A novel multigrid based preconditioner for heterogeneous Helmholtz problems","Erlangga, Y.A.; Oosterlee, C.W.; Vuik, C.","","2004","An 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.","Helmholtz equation; nonconstant high wavenumber; complex multigrid preconditioner; Fourier analysis","en","report","Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics","","","","","","","","Electrical Engineering, Mathematics and Computer Science","","","","",""
"uuid:0d5a64ef-90b2-45e6-95e5-fb6c75b4a58b","http://resolver.tudelft.nl/uuid:0d5a64ef-90b2-45e6-95e5-fb6c75b4a58b","Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian","van Gijzen, M.B.; Erlangga, Y.A.; Vuik, C.","","2006","Shifted 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 GMRES-residual 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.","Helmholtz equation; shifted Laplace preconditioner; iterative solution methods; GMRES; convergence analysis","en","report","Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics","","","","","","","","Electrical Engineering, Mathematics and Computer Science","","","","",""
"uuid:32be8cc9-8e34-44f4-896f-5fad3e1546af","http://resolver.tudelft.nl/uuid:32be8cc9-8e34-44f4-896f-5fad3e1546af","Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units","Knibbe, H.; Vuik, C.; Oosterlee, C.W.","","2015","In geophysical applications, the interest in least-squares 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 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.","least-squares migration; Helmholtz equation; wave equation; frequency domain; multigrid method; GPU acceleration; matrix storage format; frequency decimation","en","journal article","Springer","","","","","","","","Electrical Engineering, Mathematics and Computer Science","Delft Institute of Applied Mathematics","","","",""
"uuid:0a2bcdfd-6b4d-433e-8aae-d876234b9703","http://resolver.tudelft.nl/uuid:0a2bcdfd-6b4d-433e-8aae-d876234b9703","Reduction of computing time for least-squares migration based on the Helmholtz equation by graphics processing units","Knibbe, 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)","","2015","In geophysical applications, the interest in least-squares 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 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.","s Least-squares migration; Helmholtz equation; Wave equation; Frequency domain; Multigrid method; GPU acceleration; Matrix storage format; Frequency decimation","en","journal article","","","","","","","","","","","","","",""
"uuid:6c51bab4-13f3-436f-9216-f286e6394333","http://resolver.tudelft.nl/uuid:6c51bab4-13f3-436f-9216-f286e6394333","Convergence analysis of multilevel sequentially semiseparable preconditioners","Qiu, Y.; van Gijzen, M.B.; van Wingerden, J.W.; Verhaegen, M.; Vuik, C.","","2015","Multilevel sequentially semiseparable (MSSS) matrices form a class of structured matrices that have low-rank off-diagonal structure, which allows the matrix-matrix 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 off-diagonal 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.","multilevel sequentially semiseparable preconditioners; convergence analysis; saddlepoint systems; Helmholtz equation","en","report","Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics","","","","","","","","Electrical Engineering, Mathematics and Computer Science","Delft Institute of Applied Mathematics","","","",""