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CU-Boulder ASEN 5519 - Multigrid Tutorial

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AMultigrid TutorialByWilliam L. BriggsPresented byVan Emden HensonCenter for Applied Scientific ComputingLawrence Livermore National LaboratoryThis work was performed, in part, under the auspices of the United States Department of Energy by Universityof California Lawrence Livermore National Laboratory under contract number W-7405-Eng-48.2 of 119Outline•Model Problems• Basic Iterative Schemes– Convergence; experiments– Convergence; analysis• Development of Multigrid– Coarse-grid correction– Nested Iteration– Restriction and Interpolation– Standard cycles: MV, FMG•Performance– Implementation– storage and computation costs•Performance, (cont)–Convergence– Higher dimensions– Experiments• Some Theory– The spectral picture– The subspace picture–The whole picture!• Complications– Anisotropic operators and grids– Discontinuous or anisotropiccoefficients– Nonlinear Problems: FAS3 of 119Suggested Reading•Brandt, “Multi-level Adaptive Solutions to Boundary ValueProblems,” Math Comp., 31, 1977, pp 333-390.•Brandt, “1984 Guide to Multigrid Development, withapplications to computational fluid dynamics.”•Briggs, “A Multigrid Tutorial,” SIAM publications, 1987.• Briggs, Henson, and McCormick, “A Multigrid Tutorial, 2ndEdition,” SIAM publications, 2000.•Hackbusch, Multi-Grid Methods and Applications,” 1985.•Hackbusch and Trottenburg, “Multigrid Methods, Springer-Verlag, 1982”• Stüben and Trottenburg, “Multigrid Methods,” 1987.• Wesseling, “An Introduction to Multigrid Methods,” Wylie,19924 of 119Multilevel methods have beendeveloped for...• Elliptic PDEs• Purely algebraic problems, with no physical grid; forexample, network and geodetic survey problems.• Image reconstruction and tomography• Optimization (e.g., the travelling salesman and longtransportation problems)• Statistical mechanics, Ising spin models.• Quantum chromodynamics.• Quadrature and generalized FFTs.• Integral equations.5 of 119Model Problems• One-dimensional boundary value problem:•Grid:• Let and for>σ,<<)(=)(σ+)(″− xxfxuxu 010uu =)(=)( 010Nh =1xuvii)(≈xffii)(≈...,,=,=,Nihixi10Ni ...,,=10x0x1x2xixNx= 0x= 16 of 119We use Taylor Series to derivean approximation to u’’(x)• We approximate the second derivative usingTaylor series:• Summing and solving,hOxuhxuhxuhxuxu )(+)(′′′+)(″+)(′+)(=)(iiiii +4321!3!2hOxuhxuhxuhxuxu )(+)(′′′−)(″+)(′−)(=)(iiiii −4321!3!2hOhxuxuxuxu )(+)(+)(−)(=)(″iiii−+221127 of 119We approximate the equationwith a finite difference scheme• We approximate the BVP with the finite difference scheme:>σ,<<)(=)(σ+)(″− xxfxuxu 010uu =)(=)( 010−...,,==σ+−+−Nifvhvvviiiii +−2111212vv0==N08 of 119The discrete model problem• Letting and we obtain the matrix equation whereis (N-1) x (N-1), symmetric, positive definite, andAfv =v ),...,,(= vvv121TN −f),...,,(= fff121TN −AffffffvvvvvhhhhhhAö÷÷÷÷÷÷÷øæçççççççè=,ö÷÷÷÷÷÷÷øæçççççççè=,ö÷÷÷÷÷÷÷÷÷øσ+−−σ+−−σ+−−σ+−−σ+æçççççççççè=211211211211211232112321222222vNNNN−−−−9 of 119Solution Methods•Direct– Gaussian elimination– Factorization• Iterative–Jacobi–Gauss-Seidel–Conjugate Gradient, etc.• Note: This simple model problem can be solvedvery efficiently in several ways. Pretend it can’t,and that it is very hard, because it shares manycharacteristics with some very hard problems.10 of 119A two-dimensional boundary valueproblem• Consider the problem:• Where the grid is given:yyxxu>σ;=,=,=,=,= 010100NhMhyx,=,=11),(=),( hjhiyxyxji0 ≤≤Mi0≤≤Njxyz<<,<<,),(=σ+−− yxyxfuuuyyxx101011 of 119Discretizing the 2D problem• Let and . Again, using 2ndorder finite differences to approximate andwe obtain the approximate equation for theunknown , for i=1,2, … M-1 and j=1,2, …, N-1:• Ordering the unknowns (and also the vector f )lexicographically by y-lines:yxuvjiji),(≈yxffjiji),(≈uxxuyyyxu ),(ji=σ+−+−+−+−fvhvvvhvvvjijiyjijijixjijiji +,−,,+,−21121122MjjMiivji=,=,=,=,=000v ),...,,,...,...,,,,...,,(= vvvvvvvvv112111122212112111 −,−,−,−−,,,−,,,TNNNNNN12 of 119Yields the linear system• We obtain a block-tridiagonal system Av = f : where Iy is a diagonal matrix with on thediagonal andö÷÷÷÷÷÷÷øæçççççççè=ö÷÷÷÷÷øæçççççèö÷÷÷÷÷÷÷ø−−−−−−−−æçççççççèffffvvvvAIIAIIAIIAIIA1321132112321NNNyyNyyyyyy−−−−1hy2hhhhhhhhhhhhhhAyxxxyxxxyxxxyxiö÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷øσ++−⋅⋅⋅⋅⋅⋅⋅⋅⋅−σ++−−σ++−−σ++æçççççççççççççççè=111111111111112222222222222213 of 119Iterative Methods for LinearSystems•Consider Au = f where A is NxN and let v be anapproximation to u.• Two important measures:– The Error: with norms– The Residual: withvue ,−=||=||||ee∞xami=|||| ee212åiNi =vAfr−=||||r∞||||r214 of 119Residual correction• Since and , we can write as which means that , which is theResidual Equation:• Residual Correction:vue ,−=vAfr−=fuA=fevA=)+(vAfeA−=reA=evu+=15 of 119Relaxation Schemes• Consider the 1D model problem• Jacobi Method (simultaneous displacement): Solvethe ith equation for holding other variablesfixed:==−≤≤=−+− uuNifhuuuNiiii +− 02110112viNifhvvvidloidloiweni)(+)(−)(−≤≤)++(= 112121116 of 119In matrix form, the relaxation is• Let where D is diagonal and L and Uare the strictly lower and upper parts of A.• Then becomes• Let , then the iteration is:fuA=fuULuD+)+(=fDuULDu +)+(=−−11ULDRJ)+(=−1ULDA)−−(==)−−(fuULDfDvRv−)()( dloJwen+=117 of 119The iteration matrix and theerror• From the derivation,• the iteration is• subtracting,•or•hencefDuULDu +)+(=−−11fDuRu +=J−1fDvRv−)()( dloJwen+=1e)()( dloJwen= eRvRuRvu −=−)()( dloJJwenfDvRfDuRvu )+(−+=−−)(−)( dloJJwen1118 of 119Weighted Jacobi Relaxation• Consider the iteration:•Letting A = D-L-U, the matrix form is:


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