Introduction to Multigrid Methods Chapter 9: Multigrid Methodology and Applications Gustaf Söderlind Numerical Analysis, Lund University Textbooks: A Multigrid Tutorial, by William L Briggs. SIAM 1988 A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles. Cambridge 1996 Matrix-based multigrid: Theory and Applications, by Yair Shapira. Springer 2008 Multi-Grid Methods and Applications, by Wolfgang Hackbusch, 1985 c Gustaf Söderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2011-2012 Introduction to Multigrid Methods p.1/16
1. Geometric multigrid General (simplified) structure of Multigrid for A h u h = f h 1. Pre-smoothing u new h u h M(A h u h f h ) 2. Restrict r 2h R 2h h (A hu new h f h ) 3. Solve A 2h e 2h = r 2h 4. Prolong and correct v h u new h P h 2h e 2h 5. Post-smoothing u h v h M(A h v h f h ) If e h Im(P h 2h ), how does A h act on Im(P h 2h )? Introduction to Multigrid Methods p.2/16
Variational properties. The operator A 2h Error residual equation A h e h = r h (fine) Restricted to coarse grid Assume error approximation Error residual equation Galerkin condition Assume smoothing restriction Prolongation and restriction R 2h h A he h = R 2h h r h e h = P h 2h e 2h R 2h h A hp h 2h e 2h = R 2h h r h A 2h := R 2h h A hp h 2h R 2h h = I2h h F π P h 2h = 2(R2h h )T (coarse) Definition These are the variational properties and A 2h := 2R 2h h A h(r 2h h )T (The constant is 2 only in 1D) Introduction to Multigrid Methods p.3/16
Restriction operation in 2D 0 0 0 Plain injection 0 1 0 0 0 0 The stencil is applied on even/even grid points Introduction to Multigrid Methods p.4/16
Smoothing restriction in 2D 1/16 1/8 1/16 2 nd order lowpass filter 1/8 1/4 1/8 1/16 1/8 1/16 For this restriction, we have R = P T /4 and P = 4R T The stencil is applied on even/even grid points Introduction to Multigrid Methods p.5/16
Notes on restriction and prolongation In order to have the variational property, the smoothing restriction is necessary A smoothing restriction is generally beneficial to the properties of the multigrid method providing extra smoothing to that of the basic iterative scheme The variational property is advantageous in particular when FEM is used Introduction to Multigrid Methods p.6/16
Geometric multigrid in R d The process and its technical details are referred to as the geometric multigrid method The essential feature is that grid properties determine how to go from one grid to the next With a smoothing restriction (2 nd order LP filter) and the Galerkin condition one has P = 2 d R T A c = 2 d RA f R T for problems in d dimensions, i.e., with domain in R d Introduction to Multigrid Methods p.7/16
Geometric multigrid applications Applications in differential equations, integral equations, image processing These problems work with discretizations, meshing the computational geometry (domain) Embedded mesh hierarchy provide the multigrid sequence, with Galerkin + LP filter as the preferred choice, together with V- or W-cycle iteration Convergence properties are linked to the properties of the differential operator and its boundary conditions Introduction to Multigrid Methods p.8/16
Variational multigrid Special variant of geometric multigrid used with the finite element method Utilizes variational formulation (weak form) of the problem and the linearity of elements to achieve a simple structure P = R T A c = RA f P Often combined with domain decomposition Introduction to Multigrid Methods p.9/16
3. Multigrid in integral equations Multigrid can be used to solve Fredholm integral equations of the 1st kind (not well-conditioned) Deconvolution: given f, find u f(x) = In operator form 1 0 k(x y)u(y) dy = ( k u ) (x) Ku = f Discrete case: K Toeplitz Image processing, denoising, both gray scale and RGB Introduction to Multigrid Methods p.10/16
Deconvolution In the case of infinite intervals f(x) = k(x y)u(y) dy Solution in terms of Fourier transforms ˆf and ˆk u(x) = ˆf(ω) ˆk(ω) e2πiωx dω implies strong connections to Fourier transforms With multigrid hierarchical solution techniques available in all cases Introduction to Multigrid Methods p.11/16
Image processing. Denoising grayscales Sequence of nonlinear diffusion equations u [i] α u [i] 1 + u[i 1] x 2 + u [i 1] y 2 k i = z Continuous grayscale image with noise, to be discretized over grid consisting of all pixels. z contains grayscale data, sequence u [i] (x k,y l ) successively denoised Discretize in standard way, choose parameters α,k i to tune the denoising; α should be small to avoid introducing extra blur, but big enough to allow efficient denoising Introduction to Multigrid Methods p.12/16
Denoising RGB images Nonlinear diffusion in three channels R α (F(T(u)) R) = z R G α (F(T(u)) G) = z G B α (F(T(u)) B) = z B These are solved iteratively from u [0] (R [0], G [0], B [0] ) = z R [i] α (F(T(u [i 1] )) R [i] ) = z R G [i] α (F(T(u [i 1] )) G [i] ) = z G B [i] α (F(T(u [i 1] )) B [i] ) = z B Introduction to Multigrid Methods p.13/16
The Helmholtz equation The Helmholtz equation u βu = f is often called indefinite when β > 0 as it will approach a singular problem With Dirichlet data on the unit square, the eigenvalues are λ k,l [ β] = π 2 (k 2 + l 2 ) β so ellipticity is lost when β > 2π 2 MG methods often work less well for non-elliptic problems Introduction to Multigrid Methods p.14/16
Anisotropic elliptic equations Directional imbalance εu xx u yy = f With Dirichlet data on the unit square, the eigenvalues are λ k,l = π 2 (εk 2 + l 2 ) Easily compensated in the discrete equation by choosing ε x y, but only when the direction is aligned with the coordinate axes. If anistotropy is oblique, reformulate as D u = F Introduction to Multigrid Methods p.15/16
Large eigenvalue problems Associated with elliptic boundary value problems are eigenvalue problems Ax = λx, where A is the same matrix as in the differential equation Technically a hierarchy of grids could be used, but as convergence is less associated with grid properties than with eigenvalue separation, and iterative methods are always needed, the eigenvalue problems are solved using a special iterative method directly Lanczos method Introduction to Multigrid Methods p.16/16