[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/multigrid-method-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/multigrid-method-wikipedia\/","headline":"Multigrid method &#8211; Wikipedia","name":"Multigrid method &#8211; Wikipedia","description":"In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations.","datePublished":"2017-09-11","dateModified":"2017-09-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8c\/Multigrid_Visualization.png\/456px-Multigrid_Visualization.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8c\/Multigrid_Visualization.png\/456px-Multigrid_Visualization.png","height":"282","width":"456"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/multigrid-method-wikipedia\/","about":["Wiki"],"wordCount":11874,"articleBody":"In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid.[1] MG methods can be used as solvers as well as preconditioners.The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a global correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors. It can also be solved by a combination of relaxation and appeal to still coarser grids. This recursive process is repeated until a grid is reached where the cost of direct solution there is negligible compared to the cost of one relaxation sweep on the fine grid. This multigrid cycle typically reduces all error components by a fixed amount bounded well below one, independent of the fine grid mesh size. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.[2]Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite element method may be recast as a multigrid method.[3] In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation.  They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like the Lam\u00e9 equations of elasticity or the Navier-Stokes equations.[4]Table of ContentsAlgorithm[edit]Multigrid preconditioning[edit]Bramble\u2013Pasciak\u2013Xu preconditioner[edit]Generalized multigrid methods[edit]Algebraic multigrid (AMG)[edit]Multigrid in time methods[edit]Multigrid for nearly singular problems[edit]References[edit]External links[edit]Algorithm[edit]  Visualization of iterative Multigrid algorithm for fast O(n) convergence.There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretizations (grids) is considered. The important steps are:[5][6]Smoothing \u2013 reducing high frequency errors, for example using a few iterations of the Gauss\u2013Seidel method.Residual Computation \u2013 computing residual error after the smoothing operation(s).Restriction \u2013 downsampling the residual error to a coarser grid.Interpolation or prolongation \u2013 interpolating a correction computed on a coarser grid into a finer grid.Correction \u2013 Adding prolongated coarser grid solution onto the finer grid.There are many choices of multigrid methods with varying trade-offs between speed of solving a single iteration and the rate of convergence with said iteration. The 3 main types are V-Cycle, F-Cycle, and W-Cycle. For a discrete 2D problem, F-Cycle takes 83% more time to compute than a V-Cycle iteration while a W-Cycle iteration takes 125% more. If the problem is set up in a 3D domain, then a F-Cycle iteration and a W-Cycle iteration take about 64% and 75% more time respectively than a V-Cycle iteration ignoring overheads. Typically, W-Cycle produces similar convergence to F-Cycle. However, in cases of convection-diffusion problems with high P\u00e9clet numbers, W-Cycle can show superiority in its rate of convergence per iteration over F-Cycle. The choice of smoothing operators are extremely diverse as they include Krylov subspace methods and can be preconditioned.Any geometric multigrid cycle iteration is performed on a hierarchy of grids and hence it can be coded using recursion. Since the function calls itself with smaller sized (coarser) parameters, the coarsest grid is where the recursion stops. In cases where the system has a high condition number, the correction procedure is modified such that only a fraction of the prolongated coarser grid solution is added onto the finer grid.These steps can be used as shown in the MATLAB style pseudo code for 1 iteration of V-Cycle Multigrid:function phi = V_Cycle(phi,f,h)    % Recursive V-Cycle Multigrid for solving the Poisson equation (nabla^2 phi = f) on a uniform grid of spacing h    % Pre-Smoothing    phi = smoothing(phi,f,h);    % Compute Residual Errors    r = residual(phi,f,h);    % Restriction    rhs = restriction(r);    eps = zeros(size(rhs));    % stop recursion at smallest grid size, otherwise continue recursion    if smallest_grid_size_is_achieved        eps = smoothing(eps,rhs,2*h);    else        eps = V_Cycle(eps,rhs,2*h);    end    % Prolongation and Correction    phi = phi + prolongation(eps);    % Post-Smoothing    phi = smoothing(phi,f,h);endThe following represents F-cycle multigrid. This multigrid cycle is slower than V-Cycle per iteration but does result in faster convergence.function phi = F_Cycle(phi,f,h)    % Recursive F-cycle multigrid for solving the Poisson equation (nabla^2 phi = f) on a uniform grid of spacing h    % Pre-smoothing    phi = smoothing(phi,f,h);    % Compute Residual Errors    r = residual(phi,f,h);    % Restriction    rhs = restriction(r);    eps = zeros(size(rhs));    % stop recursion at smallest grid size, otherwise continue recursion    if smallest_grid_size_is_achieved        eps = smoothing(eps,rhs,2*h);    else        eps = F_Cycle(eps,rhs,2*h);    end    % Prolongation and Correction    phi = phi + prolongation(eps);    % Re-smoothing    phi = smoothing(phi,f,h);    % Compute residual errors    r = residual(phi,f,h);    % Restriction    rhs = restriction(r);    % stop recursion at smallest grid size, otherwise continue recursion    if smallest_grid_size_is_achieved        eps = smoothing(eps,rhs,2*h);    else        eps = V_Cycle(eps,rhs,2*h);    end    % Prolongation and Correction    phi = phi + prolongation(eps);    % Post-smoothing    phi = smoothing(phi,f,h);endSimilarly the procedures can modified as shown in the MATLAB style pseudo code for 1 iteration of W-cycle multigrid for an even superior rate of convergence in certain cases:function phi = W_cycle(phi,f,h)    % Recursive W-cycle multigrid for solving the Poisson equation (nabla^2 phi = f) on a uniform grid of spacing h    % Pre-smoothing    phi = smoothing(phi,f,h);    % Compute Residual Errors    r = residual(phi,f,h);    % Restriction    rhs = restriction(r);    eps = zeros(size(rhs));    % stop recursion at smallest grid size, otherwise continue recursion    if smallest_grid_size_is_achieved        eps = smoothing(eps,rhs,2*h);    else        eps = W_cycle(eps,rhs,2*h);    end    % Prolongation and correction    phi = phi + prolongation(eps);    % Re-smoothing    phi = smoothing(phi,f,h);    % Compute residual errors    r = residual(phi,f,h);    % Restriction    rhs = restriction(r);    % stop recursion at smallest grid size, otherwise continue recursion    if smallest_grid_size_is_achieved        eps = smoothing(eps,rhs,2*h);    else        eps = W_cycle(eps,rhs,2*h);    end    % Prolongation and correction    phi = phi + prolongation(eps);    % Post-smoothing    phi = smoothing(phi,f,h);end  Assuming a 2-dimensional problem setup, the computation moves across grid hierarchy differently for various multigrid cycles.This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used. In other words, it can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns.Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid i{displaystyle i} with a given grid pointdensity Ni{displaystyle N_{i}}. Assume furthermore that a solution on any grid Ni{displaystyle N_{i}} may be obtained with a giveneffort Wi=\u03c1KNi{displaystyle W_{i}=rho KN_{i}} from a solution on a coarser grid i+1{displaystyle i+1}. Here, \u03c1=Ni+1\/NiKN1{displaystyle W_{1}=W_{2}+rho KN_{1}}Combining these two expressions (and using Nk=\u03c1k\u22121N1{displaystyle N_{k}=rho ^{k-1}N_{1}}) givesW1=KN1\u2211p=0n\u03c1p{displaystyle W_{1}=KN_{1}sum _{p=0}^{n}rho ^{p}}Using the geometric series, we then find (for finite n{displaystyle n})W1{displaystyle W_{1}{displaystyle varepsilon }. Here A{displaystyle A} is symmetric semidefinite operator with large null space, while M{displaystyle M} is a symmetric positive definite operator. There were many works to attempt to design a robust and fast multigrid method for such nearly singular problems. A general guide has been provided as a design principle to achieve parameters (e.g., mesh size and physical parameters such as Poisson&#8217;s ratio that appear in the nearly singular operator) independent convergence rate of the multigrid method applied to such nearly singular systems,[24] i.e., in each grid, a space decomposition based on which the smoothing is applied, has to be constructed so that the null space of the singular part of the nearly singular operator has to be included in the sum of the local null spaces, the intersection of the null space and the local spaces resulting from the space decompositions.^ Roman Wienands; Wolfgang Joppich (2005). Practical Fourier analysis for multigrid methods. CRC Press. p.\u00a017. ISBN\u00a0978-1-58488-492-7.^ U. Trottenberg; C. W. Oosterlee; A. Sch\u00fcller (2001). Multigrid. Academic Press. ISBN\u00a0978-0-12-701070-0.^ Yu Zhu; Andreas C. Cangellaris (2006). Multigrid finite element methods for electromagnetic field modeling. Wiley. p.\u00a0132 ff. ISBN\u00a0978-0-471-74110-7.^ Shah, Tasneem Mohammad (1989). Analysis of the multigrid method (Thesis). Oxford University. Bibcode:1989STIN&#8230;9123418S.^ M. T. Heath (2002). &#8220;Section 11.5.7 Multigrid Methods&#8221;. Scientific Computing: An Introductory Survey. McGraw-Hill Higher Education. p.\u00a0478 ff. ISBN\u00a0978-0-07-112229-0.^ P. Wesseling (1992). An Introduction to Multigrid Methods. Wiley. ISBN\u00a0978-0-471-93083-9.^ Andrew V Knyazev, Klaus Neymeyr. Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method. Electronic Transactions on Numerical Analysis, 15, 38\u201355, 2003.^ Bouwmeester, Henricus; Dougherty, Andrew; Knyazev, Andrew V. (2015). &#8220;Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods 1&#8221;. Procedia Computer Science. 51: 276\u2013285. doi:10.1016\/j.procs.2015.05.241. S2CID\u00a051978658.^ Xu, Jinchao. Theory of multilevel methods. Vol. 8924558. Ithaca, NY: Cornell University, 1989.^ Bramble, James H., Joseph E. Pasciak, and Jinchao Xu. &#8220;Parallel multilevel preconditioners.&#8221; Mathematics of Computation 55, no. 191 (1990): 1\u201322.^ Xu, Jinchao. &#8220;Iterative methods by space decomposition and subspace correction.&#8221; SIAM review 34, no. 4 (1992): 581-613.^ F. H\u00fclsemann; M. Kowarschik; M. Mohr; U. R\u00fcde (2006). &#8220;Parallel geometric multigrid&#8221;.  In Are Magnus Bruaset; Aslak Tveito (eds.). Numerical solution of partial differential equations on parallel computers. Birkh\u00e4user. p.\u00a0165. ISBN\u00a0978-3-540-29076-6.^ For example, J. Blaz\u0306ek (2001). Computational fluid dynamics: principles and applications. Elsevier. p.\u00a0305. ISBN\u00a0978-0-08-043009-6. and Achi Brandt and Rima Gandlin (2003). &#8220;Multigrid for Atmospheric Data Assimilation: Analysis&#8221;.  In Thomas Y. Hou; Eitan Tadmor (eds.). Hyperbolic problems: theory, numerics, applications: proceedings of the Ninth International Conference on Hyperbolic Problems of 2002. Springer. p.\u00a0369. ISBN\u00a0978-3-540-44333-9.^ Achi Brandt (2002). &#8220;Multiscale scientific computation: review&#8221;.  In Timothy J. Barth; Tony Chan; Robert Haimes (eds.). Multiscale and multiresolution methods: theory and applications. Springer. p.\u00a053. ISBN\u00a0978-3-540-42420-8.^ Bj\u00f6rn Engquist; Olof Runborg (2002). &#8220;Wavelet-based numerical homogenization with applications&#8221;.  In Timothy J. Barth; Tony Chan; Robert Haimes (eds.). Multiscale and Multiresolution Methods. Vol.\u00a020 of Lecture Notes in Computational Science and Engineering. Springer. p.\u00a0140 ff. ISBN\u00a0978-3-540-42420-8.^ U. Trottenberg; C. W. Oosterlee; A. Sch\u00fcller (2001). Multigrid. ISBN\u00a0978-0-12-701070-0.^ Albert Cohen (2003). Numerical Analysis of Wavelet Methods. Elsevier. p.\u00a044. ISBN\u00a0978-0-444-51124-9.^ U. Trottenberg; C. W. Oosterlee; A. Sch\u00fcller (2001). &#8220;Chapter 9: Adaptive Multigrid&#8221;. Multigrid. p.\u00a0356. ISBN\u00a0978-0-12-701070-0.^ Yair Shapira (2003). &#8220;Algebraic multigrid&#8221;. Matrix-based multigrid: theory and applications. Springer. p.\u00a066. ISBN\u00a0978-1-4020-7485-1.^ U. Trottenberg; C. W. Oosterlee; A. Sch\u00fcller (2001). Multigrid. p.\u00a0417. ISBN\u00a0978-0-12-701070-0.^ Xu, J. and Zikatanov, L., 2017. Algebraic multigrid methods. Acta Numerica, 26, pp.591-721.^ Hackbusch, Wolfgang (1985). &#8220;Parabolic multi-grid methods&#8221;. Computing Methods in Applied Sciences and Engineering, VI: 189\u2013197. ISBN\u00a09780444875976. Retrieved 1 August 2015.^ Horton, Graham (1992). &#8220;The time-parallel multigrid method&#8221;. Communications in Applied Numerical Methods. 8 (9): 585\u2013595. doi:10.1002\/cnm.1630080906.^ Young-Ju Lee, Jinbiao Wu, Jinchao Xu and Ludmil Zikatanov, Robust Subspace Correction Methods for Nearly Singular Systems, Mathematical Models and Methods in Applied Sciences, Vol. 17, No 11, pp. 1937-1963 (2007)References[edit]G. P. Astrachancev (1971), An iterative method of solving elliptic net problems. USSR Comp. Math. Math. Phys. 11, 171\u2013182.N. S. Bakhvalov (1966), On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comp. Math. Math. Phys. 6, 101\u201313.Achi Brandt (April 1977), &#8220;Multi-Level Adaptive Solutions to Boundary-Value Problems&#8220;, Mathematics of Computation, 31: 333\u201390.William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000), A Multigrid Tutorial (2nd ed.), Philadelphia: Society for Industrial and Applied Mathematics, ISBN\u00a00-89871-462-1.R. P. Fedorenko (1961), A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, p.\u00a01092.R. P. Fedorenko (1964), The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, p.\u00a0227.Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). &#8220;Section 20.6. Multigrid Methods for Boundary Value Problems&#8221;. Numerical Recipes: The Art of Scientific Computing (3rd\u00a0ed.). New York: Cambridge University Press. ISBN\u00a0978-0-521-88068-8.External links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/multigrid-method-wikipedia\/#breadcrumbitem","name":"Multigrid method &#8211; Wikipedia"}}]}]