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GMRES-based multigrid for the complex scaled preconditoner for the indefinite Helmholtz equation [article]

Bram Reps, Wim Vanroose, Hisham bin Zubair
2013 arXiv   pre-print
Multigrid preconditioners and solvers for the indefinite Helmholtz equation suffer from non-stability of the stationary smoothers due to the indefinite spectrum of the operator.  ...  In this paper we explore GMRES as a replacement for the stationary smoothers of the standard multigrid method.  ...  While this strategy leads to a stable smoother for the homogeneous Helmholtz problem, a more robust smoother is preferred for heterogeneous problems, where the spectrum of the discretization matrix could  ... 
arXiv:1012.5379v2 fatcat:j3hn6hmmkbda3iijkggowptxqy

A new level-dependent coarse grid correction scheme for indefinite Helmholtz problems

Siegfried Cools, Bram Reps, Wim Vanroose
2013 Numerical Linear Algebra with Applications  
In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems.  ...  This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme  ...  Two-level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics.  ... 
doi:10.1002/nla.1895 fatcat:dmoxbxi2grgrhfcxls2vneumwm

Scaling to the stars -- a linearly scaling elliptic solver for p-multigrid [article]

Immo Huismann, Jörg Stiller, Jochen Fröhlich
2018 arXiv   pre-print
The resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n_DOF for polynomial degrees at least up to  ...  It scales with p^3 per element, i.e. n_DOF globally, and allows for a linearly scaling additive Schwarz smoother, yielding a p-multigrid cycle with an operation count of O(n_DOF).  ...  The authors would like to thank their colleagues in the Orchestration path of cfaed for stimulating discussions and ZIH, Dresden, for the provided computational resources and M.  ... 
arXiv:1808.03595v1 fatcat:6jji3n7auze35a5uxhog5zjgui

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Tuomas Airaksinen, Sanna Mönkölä
2010 Journal of Computational and Applied Mathematics  
We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well.  ...  The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as  ...  multigrid code.  ... 
doi:10.1016/ fatcat:wrjqqspmvje5jcq7pglao664hm

On the optimality of shifted Laplacian in the class of expansion preconditioners for the Helmholtz equation [article]

Siegfried Cools, Wim Vanroose
2016 arXiv   pre-print
The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for the Helmholtz problem for growing values of the number of series terms m.  ...  This paper introduces and explores the class of expansion preconditioners EX(m) that forms a direct generalization to the classic complex shifted Laplace (CSL) preconditioner for Helmholtz problems.  ...  Acknowledgments This research in this chapter was funded by the Research Council of the University of Antwerp under BOF grant number 41-FA070300-FFB5342.  ... 
arXiv:1501.04445v3 fatcat:gejztazca5dnpofzldixc7roz4

Extending the applicability of multigrid methods

J Brannick, M Brezina, R Falgout, T Manteuffel, S McCormick, J Ruge, B Sheehan, J Xu, L Zikatanov
2006 Journal of Physics, Conference Series  
Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements  ...  In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods.  ...  One must decide which iterative method to use as a smoother, how to coarsen the problem, and how to transfer information between the grids.  ... 
doi:10.1088/1742-6596/46/1/061 fatcat:ydn3cm5zazct5c67wydrbijmdy

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations [article]

Vasseur Xavier
2016 Zenodo  
This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations.  ...  of the Habilitation manuscript: Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications  ...  This research benefited from a large number of people whom I wish to thank.  ... 
doi:10.5281/zenodo.5813538 fatcat:jhsbgwinxngcpe53ewid6ygo7i

A Multilevel Approach for Trace System in HDG Discretizations [article]

Sriramkrishnan Muralikrishnan, Tan Bui-Thanh, John N. Shadid
2019 arXiv   pre-print
This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h-coarser edges/faces.  ...  Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends only on the smoothness of the solution and is independent of the nature of the PDE under consideration  ...  The first author would like to thank Stephen Shannon for his generous help regarding the derivation of projections used in this paper and also Shinhoo Kang for various fruitful discussions on this topic  ... 
arXiv:1903.11045v1 fatcat:34rjwd4s7zclng7upt6v4fub2y

Coupling vs decoupling approaches for PDE/ODE systems modeling intercellular signaling

Thomas Carraro, Elfriede Friedmann, Daniel Gerecht
2016 Journal of Computational Physics  
Based on a sensitivity analysis, we present a systematic comparison between coupling and decoupling approaches for this class of problems and show numerical results.  ...  For biologically relevant configurations of the model, our quantitative study shows that a coupling approach performs much better than a decoupling one.  ...  Acknowledgements The authors thank Prof. Rolf Rannacher for the numerous fruitful discussions on numerical aspects of this work. Furthermore, the authors acknowledge Prof.  ... 
doi:10.1016/ fatcat:se5f6laurrbdxa7splxx74y2j4

On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains

A. Naumovich, F. J. Gaspar
2007 Computing and Visualization in Science  
A set of numerical experiments shows necessity of use of the operator-dependent restriction and prolongation in the multigrid solver for the considered class of problems.  ...  In this paper, we present problem-dependent prolongation and problem-dependent restriction for a multigrid solver for the three-dimensional Biot poroelasticity system, which is solved in a multilayered  ...  iterations (in brackets) for the multigrid solvers: 1.  ... 
doi:10.1007/s00791-007-0059-8 fatcat:tjazta2djfhjfdxdm42au7rbfu

Isogeometric simulation of acoustic radiation [article]

Victoria Hernández Mederos
2021 arXiv   pre-print
additionally smoother solutions which depend on less degrees of freedom.  ...  In this paper we discuss the numerical solution on a simple 2D domain of the Helmoltz equation with mixed boundary conditions.  ...  Our contribution The main contribution of this paper is the application of IgA to the solution of a radiation problem, mathematically modeled with a 2D Helmholtz equation with mixed boundary conditions  ... 
arXiv:2111.02546v1 fatcat:cvdd4pkqlvfldcnrbkctngfpbi

Preparing sparse solvers for exascale computing

Hartwig Anzt, Erik Boman, Rob Falgout, Pieter Ghysels, Michael Heroux, Xiaoye Li, Lois Curfman McInnes, Richard Tran Mills, Sivasankaran Rajamanickam, Karl Rupp, Barry Smith, Ichitaro Yamazaki (+1 others)
2020 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
Sparse solvers provide essential functionality for a wide variety of scientific applications.  ...  This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'.  ...  While the default smoother is based on Gauss-Seidel requiring triangular solves, one can also use matvec-based smoothers, such as Jacobi or polynomial smoothers.  ... 
doi:10.1098/rsta.2019.0053 pmid:31955673 fatcat:bqw6xqixbrabddmxglmtcbw2wa

Multigrid - Adaptive Local Refinement Solver for Incompressible Flows [chapter]

Oleg Iliev, Dimitar Stoyanov
2001 Lecture Notes in Computer Science  
Vorwort Das Tätigkeitsfeld des Fraunhofer Instituts für Techno-und Wirt schafts ma the ma tik ITWM um fasst an wen dungs na he Grund la gen for schung, angewandte For schung so wie Be ra tung und kun den  ...  spe zi fi sche Lö sun gen auf allen Gebieten, die für Techno-und Wirt schafts ma the ma tik be deut sam sind.  ...  Nickel A Polynomial Case of the Batch Presorting Problem This paper presents new theoretical results for a special case of the batch presorting problem (BPSP).  ... 
doi:10.1007/3-540-45346-6_38 fatcat:7y2c3sgbnzgdrethfvyly3c43u

Pragmatic solvers for 3D Stokes and elasticity problems with heterogeneous coefficients: evaluating modern incomplete LDLT preconditioners

Patrick Sanan, Dave A. May, Matthias Bollhöfer, Olaf Schenk
2020 Solid Earth  
Modern direct solvers, which robustly produce solutions to almost any system, do so at the cost of rapidly growing time and memory requirements for large problems, especially in 3D.  ...  While ABF solvers have emerged as the state-of-the-art scalable option, they are invasive solvers requiring splitting of pressure and velocity degrees of freedom, a multigrid hierarchy with tuned transfer  ...  Patrick Sanan acknowledges Radim Janalik, who helped develop the PETSc interface to PARDISO.  ... 
doi:10.5194/se-11-2031-2020 fatcat:ejvxjzn5orevtnn3ujqjrjmbxq

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Oleg Iliev, Daniela Vassileva, Dimitar Stoyanov, Willy Dörfler
2008 Computing and Visualization in Science  
Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.  ...  A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach.  ...  Vassileva acknowledges the fellowship from European Research Consortium for Informatics and Mathematics (ERCIM) for her work at ITWM, Kaiserslautern.  ... 
doi:10.1007/s00791-008-0119-8 fatcat:5wjtllasmndfng4saoontq3csy
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