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| '''Multigrid (MG) methods''' in [[numerical analysis]] are a group of [[algorithm]]s for solving [[differential equations]] using a [[hierarchy]] of [[discretization]]s. They are an example of a class of techniques called [[Multiresolution analysis|multiresolution methods]], very useful in (but not limited to) problems exhibiting [[Multiscale modeling|multiple scales]] of behavior. For example, many basic [[relaxation method]]s 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.<ref>{{cite book |title=Practical Fourier analysis for multigrid methods |author1=Roman Wienands |author2=Wolfgang Joppich |page=17 |url=http://books.google.com/books?id=IOSux5GxacsC&pg=PA17 |isbn=1-58488-492-4 |publisher=CRC Press |year=2005}}</ref> MG methods can be used as solvers as well as [[preconditioner]]s.
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| The main idea of multigrid is to accelerate the convergence of a basic iterative method by ''global'' correction from time to time, accomplished by solving a [[coarse problem]]. This principle is similar to [[interpolation]] between coarser and finer grids. The typical application for multigrid is in the numerical solution of [[elliptic partial differential equation]]s in two or more dimensions.<ref>{{cite book |title=Multigrid |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |publisher=Academic Press |year=2001 |isbn=0-12-701070-X |url=http://books.google.com/books?id=-og1wD-Nx_wC&printsec=frontcover&dq=isbn:012701070X#v=onepage&q=elliptic&f=false}}</ref>
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| 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.<ref>{{cite book |title=Multigrid finite element methods for electromagnetic field modeling |author1=Yu Zhu |author2=Andreas C. Cangellaris |url=http://books.google.com/books?id=amq9j71_nqAC&pg=PA132 |page=132 ''ff'' |isbn=0-471-74110-8 |year=2006 |publisher=Wiley}}</ref> 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 condition]]s. They do not depend on the [[Separable partial differential equation|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 [[Lamé system]] of [[Elasticity (physics)|elasticity]] or the [[Navier-Stokes equations]].<ref>{{cite thesis |last=Shah |first=Tasneem Mohammad |title=Analysis of the multigrid method |url=http://adsabs.harvard.edu/abs/1989STIN...9123418S |publisher=Oxford University |year=1989 |accessdate=8 January 2013}}</ref>
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| == Algorithm ==
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| There are many variations of multigrid algorithms, but the common features are that a hierarchy of discretizations (grids) is considered. The important steps are:<ref>{{cite book |title=Scientific Computing: An Introductory Survey |url=http://books.google.com/books?id=DPkYAQAAIAAJ&dq=isbn:007112229X |author=M. T. Heath |page=478 ''ff'' |chapter=Section 11.5.7 Multigrid Methods |year=2002 |publisher=McGraw-Hill Higher Education |isbn=0-07-112229-X }}</ref><ref>{{cite book |title=An Introduction to Multigrid Methods |author=P. Wesseling |year=1992 |publisher=Wiley |isbn=0-471-93083-0 |url=http://books.google.com/books?id=ywOzQgAACAAJ&dq=isbn:0471930830&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0}}</ref> | |
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| * '''[[Smoothing]]''' – reducing high frequency errors, for example using a few iterations of the [[Gauss–Seidel method]].
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| * '''Restriction''' – downsampling the [[residual (numerical analysis)|residual]] error to a coarser grid.
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| * '''[[Interpolation]]''' or '''prolongation''' – interpolating a correction computed on a coarser grid into a finer grid.
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| == Computational cost ==
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| This approach has the advantage over other methods that it often scales linearly with the number of discrete nodes used. That is: It can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns.
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| Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid <math>i</math> with a given grid point
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| density <math>N_i</math>. Assume furthermore that a solution on any grid <math>N_i</math> may be obtained with a given
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| effort <math>W_i = \rho K N_i</math> from a solution on a coarser grid <math>i+1</math>. Here, <math>\rho = N_{j+1} / N_j < 1</math> is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and <math>K</math> is some constant modeling the effort of computing the result for one grid point.
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| The following recurrence relation is then obtained for the effort of obtaining the solution on grid <math>k</math>: | |
| :<math>W_k = W_{k+1} + \rho K N_k</math>
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| And in particular, we find for the finest grid <math>N_1</math> that
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| :<math>W_1 = W_2 + \rho K N_1</math>
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| Combining these two expressions (and using <math>N_{k} = \rho^{k-1} N_1</math>) gives
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| :<math>W_1 = K N_1 \sum_{p=0}^n \rho^p </math>
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| Using the [[geometric series]], we then find (for finite <math>n</math>)
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| :<math>W_1 < K N_1 \frac{1}{1 - \rho}</math>
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| that is, a solution may be obtained in <math>O(N)</math> time.
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| ==Multigrid preconditioning==
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| A multigrid method with an intentionally reduced tolerance can be used as an efficient [[preconditioning|preconditioner]] for an external iterative solver. The solution may still be obtained in <math>O(N)</math> time as well as in the case where the multigrid method is used as a solver. Multigrid preconditioning is used in practice even for linear systems. Its main advantage versus a purely multigrid solver is particularly clear for nonlinear problems, e.g., [[eigenvalue]] problems.
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| ==Generalized multigrid methods==
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| Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of [[parabolic partial differential equation]]s, or they can be applied directly to time-dependent [[partial differential equation]]s.<ref>{{cite book |url=http://books.google.com/books?id=GKDQUXzLTkIC&pg=PA165 |editors=Are Magnus Bruaset, Aslak Tveito |title=Numerical solution of partial differential equations on parallel computers |page=165 |chapter=Parallel geometric multigrid |author1=F. Hülsemann |author2=M. Kowarschik |author3=M. Mohr |author4=U. Rüde |publisher=Birkhäuser |year=2006 |isbn=3-540-29076-1}}</ref> Research on multilevel techniques for [[hyperbolic partial differential equation]]s is underway.<ref>For example, {{cite book |title=Computational fluid dynamics: principles and applications |page=305 |url=http://books.google.com/books?id=asWGy362QFIC&pg=PA305&lpg=PA305&dq=%22The+goal+of+the+current+research+is+the+significant+improvement+of+the+efficiency+of+multigrid+for+hyperbolic+flow+problems%22&source=bl&ots=TCSODVg_KV&sig=ZM1V6j3z4MXY6u-r0HmKF_fpjIc&hl=en&ei=cosaS8GEEIqIswPQsP2HBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAgQ6AEwAA#v=onepage&q=%22The%20goal%20of%20the%20current%20research%20is%20the%20significant%20improvement%20of%20the%20efficiency%20of%20multigrid%20for%20hyperbolic%20flow%20problems%22&f=false |author= J. Blaz̆ek |year=2001 |isbn=0-08-043009-0 |publisher=Elsevier}} and {{cite book |url=http://books.google.com/books?id=TapltAX3ry8C&pg=PA369 |author=Achi Brandt and Rima Gandlin |chapter=Multigrid for Atmospheric Data Assimilation: Analysis |page=369 |editors=Thomas Y. Hou, Eitan Tadmor |title=Hyperbolic problems: theory, numerics, applications: proceedings of the Ninth International Conference on Hyperbolic Problems of 2002 |year=2003 |isbn=3-540-44333-9 |publisher=Springer}}</ref> Multigrid methods can also be applied to [[integral equation]]s, or for problems in [[statistical physics]].<ref>{{cite book |title=Multiscale and multiresolution methods: theory and applications |author=Achi Brandt |url=http://books.google.com/books?id=mtsy6Ci2TRoC&pg=PA53 |editors=Timothy J. Barth, Tony Chan, Robert Haimes |page=53 |chapter=Multiscale scientific computation: review |isbn=3-540-42420-2 |year=2002 |publisher=Springer}}</ref>
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| Other extensions of multigrid methods include techniques where no partial differential equation nor geometrical problem background is used to construct the multilevel hierarchy.<ref>{{cite book |title=Matrix-based multigrid: theory and applications |author=Yair Shapira |url=http://books.google.com/books?id=lCDGhpDDk5IC&pg=PA66 |chapter=Algebraic multigrid |page=66 |isbn=1-4020-7485-9 |publisher=Springer |year=2003}}</ref> Such '''algebraic multigrid methods''' (AMG) construct their hierarchy of operators directly from the system matrix, and the levels of the hierarchy are simply subsets of unknowns without any geometric interpretation. Thus, AMG methods become true black-box solvers for [[sparse matrices]]. However, AMG is regarded as advantageous mainly where geometric multigrid is too difficult to apply.<ref>{{cite book |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |title=''op. cit.'' |url=http://books.google.com/books?id=-og1wD-Nx_wC&pg=PA417 |page=417 |isbn=0-12-701070-X}}</ref>
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| Another set of multiresolution methods is based upon [[wavelets]]. These wavelet methods can be combined with multigrid methods.<ref>{{cite book |url=http://books.google.com/books?id=mtsy6Ci2TRoC&pg=PA140 |author1=Björn Engquist |author2=Olof Runborg |editors=Timothy J. Barth, Tony Chan, Robert Haimes |chapter=Wavelet-based numerical homogenization with applications |title=Multiscale and Multiresolution Methods |isbn=3-540-42420-2 |volume=Vol. 20 of Lecture notes in computational science and engineering |publisher=Springer |year=2002 |page=140 ''ff''}}</ref><ref>{{cite book |url=http://books.google.com/books?id=-og1wD-Nx_wC&dq=wavelet+multigrid&printsec=frontcover&source=in&hl=en&ei=bx8ZS_v1KIaQsgO-5pn3Bw&sa=X&oi=book_result&ct=result&resnum=12&ved=0CD0Q6AEwCw#v=snippet&q=wavelet%20&f=false |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |title=''op. cit.'' |isbn=0-12-701070-X}}</ref> For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method.<ref>{{cite book |title=Numerical Analysis of Wavelet Methods |author=Albert Cohen |url=http://books.google.com/books?id=Dz9RnDItrAYC&pg=PA44 |page=44 |publisher=Elsevier |year=2003 |isbn=0-444-51124-5}}</ref>
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| '''Adaptive multigrid''' exhibits [[adaptive mesh refinement]], that is, it adjusts the grid as the computation proceeds, in a manner dependent upon the computation itself.<ref>{{cite book |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |title=''op. cit.'' |chapter=Chapter 9: Adaptive Multigrid |url=http://books.google.com/books?id=-og1wD-Nx_wC&pg=PA356 |page=356 |isbn=0-12-701070-X}}</ref> The idea is to increase resolution of the grid only in regions of the solution where it is needed.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * G. P. Astrachancev (1971), An iterative method of solving elliptic net problems. USSR Comp. Math. Math. Phys. 11, 171–182.
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| * N. S. [[Bakhvalov]] (1966), On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comp. Math. Math. Phys. 6, 101–13.
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| * [[Achi Brandt]] (April 1977), "[http://www.jstor.org/stable/2006422 Multi-Level Adaptive Solutions to Boundary-Value Problems]", ''Mathematics of Computation'', '''31''': 333–90.
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| * William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000), ''[http://www.llnl.gov/casc/people/henson/mgtut/welcome.html A Multigrid Tutorial]'' (2nd ed.), Philadelphia: [[Society for Industrial and Applied Mathematics]], ISBN 0-89871-462-1.
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| * R. P. Fedorenko (1961), A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, p. 1092.
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| * R. P. Fedorenko (1964), The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, p. 227.
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| * {{cite book | last1=Press | first1=W. H. | last2=Teukolsky | first2=S. A. | last3=Vetterling | first3=W. T. | last4=Flannery | first4=B. P. | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 20.6. Multigrid Methods for Boundary Value Problems | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1066}}
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| == External links ==
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| *[http://www.mgnet.org/ Repository for multigrid, multilevel, multiscale, aggregation, defect correction, and domain decomposition methods]
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| *[http://www.mgnet.org/mgnet/tutorials/xwb/xwb.html Multigrid tutorial]
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| *[https://computation.llnl.gov/casc/linear_solvers/talks/AMG_TUT_PPT/sld001.htm Algebraic multigrid tutorial]
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| *[https://computation.llnl.gov/casc/linear_solvers/present.html Links to AMG presentations]
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Multigrid Method}}
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| [[Category:Numerical analysis]]
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| [[Category:Partial differential equations]]
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| [[Category:Wavelets]]
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