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Multigrid methods for systems of linear equations Andrey Ponomarenko Sarntal St.Petersburg State University Faculty of Physics Earth physics department Ferienakademie 2008 26.09.2008 1

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OUTLINE The way to multigrid… Iterative methods for linear systems, relaxation schemes Coarse grid correction V-multigrid method Full-multigrid method 26.09.20082 Ferienakademie 2008

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The way to multigrid… Algorithm should not do a lot of empty work 1.Slow process – a lot of calculations 2.The convergence - it should be quite quick So, multigrid method – for high frequently components we use fine grids, for slow components we use coarse grids 26.09.20083 Ferienakademie 2008

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Iterative methods for linear systems Consider Let v be an approximation to u Residual equation Residual correction 1) At first, some basics: 26.09.20084 Ferienakademie 2008

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2) Iteration in the matrix form - the iteration error Let D – is diagonal matrix L and U – are lower and upper parts of A Then Now the iteration is 26.09.20085 Ferienakademie 2008

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Now consider the iteration properties on the next equation in 1D: One-dimensional boundary problem Introduce the grid: 26.09.20086 Ferienakademie 2008

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Then our problem can be presented in grid form: Then, with the help of Taylor series: 26.09.20087 Ferienakademie 2008

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Now we have The discrete model problem: So, we can used iteration scheme showed above Consider relaxation scheme: 26.09.20088 Ferienakademie 2008 symmetric, positive defined

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Relaxation schemes The equation: 1) Jacobi Method (simultaneous displacements): 2) Weighted Jacobi Method : D – is diagonal matrix L and U – are lower and upper parts of A Then as we have before 26.09.20089 Ferienakademie 2008

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Now we return to relaxation: λ – is eigenvalue of matrix B, w – its ascociated eigenvector Eigenvectors are linearly independent, they form the basis, and for any v it is possible to write (N×N matrix) Short observation about eigenvalues and eigenvectors: - for one iteration - we’ll have after n iterations - initial error 26.09.200810 Ferienakademie 2008

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Let’s make convergence analysis on weighted Jakobi on 1D model 26.09.200811 Ferienakademie 2008

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Let’s make convergence analysis on weighted Jakobi on 1D model Expand initial error in the terms of eigenvectors: So, after M iterations we have: (remember ) The k-th mode of the error is reduced by at each iteration 26.09.200812 Ferienakademie 2008

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Relaxation suppress eigenmodes unevenly If 0

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So, many relaxation schemes has smoothing property, but smooth modes of error are damped very slowly For instance (weighted Jakobi method): initial error: after 35 iterations: 26.09.200814 Ferienakademie 2008

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By using coarse grids we can use the smoothing property in good advantage! So why?? Relaxation on the coarse-grid is much cheaper (1/2 – 1D, ¼ - 2D e.t.c ) Relaxation on the coarse-grid has better convergence ( 1-O(4h 2 ) inst.of 1-O(h 2 ) ) After relaxing on the fine grid, the error will be smooth. On the coarse grid this error appears more oscillatory, and relaxation will be more effectively 26.09.200815 Ferienakademie 2008

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We have now the idea of The question appears: how to map and 26.09.200816 Ferienakademie 2008

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defined on Mapping from the coarse grid to the fine grid: 26.09.200817 Ferienakademie 2008

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defined on Mapping from the fine grid to the coarse grid: (there are others methods of mapping, we don’t consider them) 26.09.200818 Ferienakademie 2008

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The Coarse Grid Correction scheme: - is the “coarse-grid version” of operator A 26.09.200819 Ferienakademie 2008

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V-Multigrid method IDEA: it is unnecessary to It is enough to make several relaxation and to obtain approximate solution We effectively relax high-frequency components To make slow frequencies smooth we use the next, coarse grid So, we have recursive use of CGC sheme. GhGh G 2h G 4h G 8h 26.09.200820 Ferienakademie 2008

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Full-Multigrid method IDEA: the solution takes fewer iterations if the initial guess is good Interpolate coarse solution to the fine grid “Solve” the problem on the coarse grid first Use interpolated coarse solution as initial guess on fine grid How to get a good initial guess: Let’s use the V-multigrid cycle as the solver on each grid level. This defines the Full Multigrid (FMG) method 26.09.200821 Ferienakademie 2008

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Some properties of convergence: Some properties of FMG method: 1) Full Multigtid method computes solution to the error of truncation 2)The computational cost of FMG is O(N d ), where N – quantity of fine grid points, d – dimension of the problem 26.09.200822 Ferienakademie 2008

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So, MULTIGRID METHODS - increasingly the right tool: Multigrid methods are effective algorithms Multigrid method FMG is an optimal (O(N)) Multigrid algotithms can be parallelized effecintly – it is another real interesting and big topic to present! 26.09.200823 Ferienakademie 2008

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References Briggs W.M. “Multigrid tutorial”. Presentation by Henson V.E. Center for applied scientific computing, Lawrence livermore National Laboratory. Stankova E., Zatevahin M. “Multigrid methods. Introduction in the standard methods.” St.Petersburg. In PDF, in russian. Borzi A. “Introduction to multigrid methods.” In PDF. http://www.uni-graz.at/imawww/borzi/index.html 26.09.200824 Ferienakademie 2008

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Thank you for attention! 26.09.200825 Ferienakademie 2008

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