1. Basic Iterative Methods and Multigrid Ideas Hector D. Ceniceros Department of Mathematics UCSB Math 104 B Material Based on “A Multigrid Tutorial” by W. L. Briggs and the “Multigrid Tutorial” by Livne.

2. The problem: Solve the large linear system Iterative methods take the general matrix form: Convergence iff Iterative Methods Matrix form is good for analysis:

3. Examples of Iterative Methods In actual application of these methods we don’t use these matrices Instead we use equivalent forms for the iterations.

4. Example 1

5. From Jacobi to GS Gauss-Seidel: Use the approximations as they are available:

6. SOR We can write SOR in the equivalent form Updated residual

7. Example 1 (SOR)

8. Example 2: A BVP Introduce a grid and use finite differences

9. 2 æ ö + c - 2 h 1 f æ ö v 1 ç ÷ æ ö 1 ç ÷ ç ÷ 2 ç ÷ - + c - f 1 2 h 1 v ç ÷ 2 2 ç ÷ ç ÷ ç ÷ ç ÷ v ç ÷ 2 1 f - + c - 1 2 h 1 3 3 ç ÷ ç ÷ ç ÷ 2 ç ÷ ç ÷ h ç ÷ v ç ÷ ç ÷ ç ÷ f - 2 N 2 - 2 - + c - N ç ÷ 1 2 h 1 ç ÷ ç ÷ v è ø - 1 N ç ÷ f è ø - 2 1 N - + c 1 2 h è ø A Tridiagonal System =

10. Basic Loop for Jacobi Jacobi. v0 stores the old (k) values. v stores the new (k+1) values j index is shifted by one. … while ( dif > TOL & iter <= MAX_ITER) for j=2:N v(j) =(v0(j-1) + v0(j+1) + f(j))/d; end iter=iter+1; dif=norm(v-v0,inf)/norm(v,inf); v0=v; end d=2+c*h^2

11. Basic Loop for GS … while ( dif > TOL & iter <= MAX_ITER) for j=2:N v(j) =(v(j-1) + v0(j+1) + f(j))/d; end iter=iter+1; dif=norm(v-v0,inf)/norm(v,inf); v0=v; end

12. Basic Loop for SOR …. while ( dif > TOL & iter <= MAX_ITER) for j=2:N v(j) =(1-w)*v0(j) + (w/d)*(f(j) + v(j-1) + v0(j+1)); end iter=iter+1; dif=norm(v-v0,inf)/norm(v,inf); v0=v; end

13. K=3 K=1 K=6 Performance of the Basic Iterative Methods We consider the BVP with f=0 and c=0. In this case the Exact solution of Ax=b is the x=0. We use Fourier modes as initial guesses

14. Error vs Iterations Jacobi Gauss-Seidel SOR Error decreases fast for large k but slowly for small k!

15. Error reduction stalls Jacobi: Initial guess: p p p j 6 j 3 2 j æ æ ö æ ö æ ö ö 1 = + + v s i n s i n s i n ç ç ÷ ç ÷ ç ÷ ÷ 0 3 N N N è è ø è ø è ø ø | | | | e ¥

16. p p p 2 j 1 6 j 3 2 j æ ö æ ö æ ö 1 1 = + + v s i n s i n s i n ç ÷ ç ÷ ç ÷ j N 2 N 2 N è ø è ø è ø Jacobi, GS Relaxation Smooth the Error • Initial error: • Error after 35 iteration sweeps: Many relaxation schemes have the smoothing property: oscillatory modes of the error are eliminated effectively, but smooth modes are damped very slowly.

17. Observation Toward Multigrid • Many relaxation schemes have the smoothing property, where oscillatory modes of the error are eliminated effectively, but smooth modes are damped very slowly. • This might seem like a limitation, but by using coarse grids we can use the smoothing property to good advantage. What is smooth on fine grid looks rougher on a coarser grid

18. Smooth Error is (relatively) more Oscillatory on Coarse Grid • A smooth function: • Can be represented by linear interpolation from a coarser grid:

19. Another Observation for Multigrid Let v be an approximation to the solution of Au=f, where the residual r=f-Av. The the error e=u-v satisfies Ae=r. Thus if we have an approximation to the error e we can “correct” the approximation v by v’ = v +e This is called error or residual (coarse-grid) correction

20. Relaxing Ae=r • After relaxing on Au=f on the fine grid, the error will be smooth. On the coarse grid, however, this error appears more oscillatory, and relaxation will be more effective. • Therefore we go to a coarse grid and relax on the residual equation Ae=r, with an initial guess of e=0.

21. Restrict Coarse-grid Correction Correct h h h Relax on = A u f h h h ¬ + u u e h h h h = - r f A u Compute Interpolate h 2 h h » e I e 2 h 2 h 2 h 2 h = A e r Solve

22. Multigrid • Coarse grid correction at multiple levels (nested iteration). • When multigrid converges (A positive definite, for example) is hard to beat. It yields the solution to Ax=b in O(N) operations, which is optimal.