Geometric (Classical) MultiGrid

Hierarchy of graphs Apply grids in all scales: 2x2, 4x4, …, n 1/2 xn 1/2 Coarsening Interpolate and relax Solve the large systems of equations by multigrid! G1G1 G2G2 G3G3 GlGl G1G1 G2G2 G3G3 GlGl

Linear (2 nd order) interpolation in 1D x1x1 x2x2 x F(x)

i S(i) (U lb,V lb ) (U rt,V rt )(U lt,V lt ) (U rb,V rb ) (x 2,y 2 )(x 1,y 2 ) (x 2,y 1 )(x 1,y 1 ) (x 0,y 0 ) Bilinear interpolation C(S(i))={rb,rt,lb,lt}

i S(i) (U lb,V lb ) (U rt,V rt )(U lt,V lt ) (U rb,V rb ) (x 2,y 2 )(x 1,y 2 ) (x 2,y 1 )(x 1,y 1 ) (x 0,y 0 ) (U l,V l )(U r,V r )

From (x,y) to (U,V) by bilinear intepolation

Linear scalar elliptic PDE (Brandt ~1971)  1 dimension Poisson equation  Discretize the continuum x0x0 x1x1 x2x2 xixi x N-1 xNxN x=0x=1 Grid: Let local averaging

Linear scalar elliptic PDE  1 dimension Laplace equation  Second order finite difference approximation => Solve a linear system of equations Not directly, but iteratively => Use Gauss Seidel pointwise relaxation

fine grid h u = average of u's approximating Laplace eq.

u given on the boundary h e.g., u = average of u's approximating Laplace eq. Point-by-point RELAXATION Solution algorithm:

Exc#9: Error calculations 1.Use Taylor expansion to calculate the error when U’’(x) is approximated by 2.Find a,b,c,d and e such that This is a higher order approximation for U’’(x) than the one in exercise 1.

Exc#10: Gauss Seidel relaxation Solve the 1D Laplace equation U’’(x)=0, 0<x<1 by Gauss Seidel relaxation. Start with the approximations 1. U i = random(0,1), 2. U i = sin(  x), where U 0 = U N = 0 for N=32. Plot the L2 norm of the error and of the residual versus the number of iterations k=1,…,100, where the L2 norm of a vector v is and the residual of LU=F is R=F-LU Do you see a difference in the asymptotic behavior between the 2 norms? Which case converges faster 1. or 2., explain

Influence of (pointwise) Gauss-Seidel relaxation on the error Poisson equation, uniform grid Error of initial guess Error after 5 relaxation Error after 10 relaxations Error after 15 relaxations

The basic observations of ML  Just a few relaxation sweeps are needed to converge the highly oscillatory components of the error => the error is smooth  Can be well expressed by less variables  Use a coarser level (by choosing every other line) for the residual equation  Smooth component on a finer level becomes more oscillatory on a coarser level => solve recursively  The solution is interpolated and added

h2h Local relaxation approximation smooth L h U h =F h L 2h U 2h =F 2h

LU=F h 2h 4h L h U h =F h L 2h U 2h =F 2h L 4h U 4h =F 4h

TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h2 v ~~~ h old h new uu  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation

Why additional relaxations are needed?

A smooth approximation is obtained after relaxation on the finer level

Why additional relaxations are needed? A smooth approximation is obtained after relaxation on the finer level The coarse grid correction

Why additional relaxations are needed? The coarse grid correction Interpolate and add

Why additional relaxations are needed? The coarse grid correction Interpolate and add

Why additional relaxations are needed? The coarse grid correction Interpolate and add

Why additional relaxations are needed? The coarse grid correction Interpolate and add

Why additional relaxations are needed? The coarse grid correction Interpolate and add

Why additional relaxations are needed? Interpolate and add => high oscillatory component emerges

TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h old h new uu h2 v ~~~  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation by recursion MULTI-GRID CYCLE Correction Scheme

interpolation (order m) of corrections relaxation sweeps residual transfer enough sweeps or direct solver *... * h0h0 h 0 /2 h 0 /4 2h h V-cycle: V     

Multigrid solvers Cost: operations per unknown Linear scalar elliptic equation (Achi Brandt ~1971)

Multigrid solvers Cost: operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) AMG Several coupled PDEs* (1980) Non-elliptic: high-Reynolds flow Highly indefinite: waves Many eigenfunctions (N) Near zero modes Gauge topology: Dirac eq. Inverse problems Optimal design Integral equations Full matrix Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) FAS (1975) Within one solver (1977,1982)