6. Multigrid & Krylov Methods. June 1, 2010
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1 June 1, 2010 Scientific Computing II, Tobias Weinzierl page 1 of 27
2 Outline of This Session A recapitulation of iterative schemes Lots of advertisement Multigrid Ingredients Multigrid Analysis Scientific Computing II, Tobias Weinzierl page 2 of 27
3 6.1. Recapitulation What is a stationary scheme? Which stationary schemes do you know? What is the iteration matrix? When does a Gauß-Seidel converge: what is the criterion for the iteration matrix? What happens, if the stiffness matrix is not sufficiently diagonal dominant? Why does a Gauß-Seidel scheme converge faster than a Jacobi-type scheme? What means in-situ in the context of a Gauß-Seidel and Jacobi solver? What is the complexity of the LR decomposition? Why should you never invert a matrix? How do these relaxation schemes work? What is the difference of underrelaxation and overrelaxation? What means overrelaxation for a Jacobi method? What are the bad guys? What is a residual? What is its semantics? What is the computationally expensive part of the stationary schemes? Scientific Computing II, Tobias Weinzierl page 3 of 27
4 What is the idea of red-black colouring? Why is red-black Gauß-Seidel really a Gauß-Seidel scheme? Does Jacobi depend on the enumeration order of the unknowns? Does Gauß-Seidel depend on the enumeration order of the unknowns (example)? How do iteration matrix and eigenfrequencies interplay? How do you determine the spectral radius in terms of Fourier modes? What is the convergence order of Jacobi and Gauß-Seidel? Is Gauß-Seidel always faster than Jacobi? How does relaxation modify the solver s behaviour in terms of Fourier modes? Scientific Computing II, Tobias Weinzierl page 4 of 27
5 CSE/COME Trip June 26 For the CSE and COME students (as well as the lecturers, professors, and some very special guests) Registration: till June 1, 2010, at Christa Halfar s office Deadline: today! Cost: 5 euros Scientific Computing II, Tobias Weinzierl page 5 of 27
6 Exam Takes place on July 21, 2010 Is announced at TUMOnline You have to register via TUMOnline (CSE students, too) Scientific Computing II, Tobias Weinzierl page 6 of 27
7 KAUST MAC/KAUST workshop on July 14, 2010 Carreer opportunities (Ph.D.) See webpage and register ( Scientific Computing II, Tobias Weinzierl page 7 of 27
8 6.2. Multigrid We know... that the convergence factor of standard stationary iteration schemes is in 1 O(h 2 ). that certain error modes are reduced faster than others (spectral picture of overall behaviour). that smooth error modes become rougher on coarser grids. It might thus make sense to perform only a fixed number of iterations on the finest grid. to remove the error then on a coarser grid. to continue recursively. Scientific Computing II, Tobias Weinzierl page 8 of 27
9 Correction Schemes Au (0) = f Au (1) = f If we want to solve the error equation afterwards, we have to solve Ae (2) = Scientific Computing II, Tobias Weinzierl page 9 of 27
10 Correction Schemes Au (0) = f Au (1) = f If we want to solve the error equation afterwards, we have to solve Ae (2) = A (u ( ) u (1)) Scientific Computing II, Tobias Weinzierl page 9 of 27
11 Correction Schemes Au (0) = f Au (1) = f If we want to solve the error equation afterwards, we have to solve Ae (2) = A (u ( ) u (1)) = f A u (1) Scientific Computing II, Tobias Weinzierl page 9 of 27
12 Correction Schemes Au (0) = f Au (1) = f If we want to solve the error equation afterwards, we have to solve Alternative iteration formulation: Perform µ 1 standard iterations. Store away approximation u (µ 1). Introduce variabel e for all unknowns. Compute res for each unknown. Ae (2) = A (u ( ) u (1)) = f A u (1) = res (1) Solve error equation (correction scheme). Sum up u (µ 1) and e. Scientific Computing II, Tobias Weinzierl page 9 of 27
13 6.3. Restriction What Does a Coarse Grid Look Like? For the coarse grid, just neglect every second grid point (standard coarsening). Project a solution from the fine grid to the coarse grid (restriction): Copy solution from grid points which belong to both grids (injection). Apply averaging stencil. Scientific Computing II, Tobias Weinzierl page 10 of 27
14 A Coarse Grid Correction Schemes A e = f A u (1) = res Idea: Error e is smooth, i.e. project residual to coarser grid. Solve there a coarse grid correction equation: A 2h e 2h = R (f A u (1)) Then, project correction error e down to fine grid (prolongation) before we sum up u and e. e h = P e 2h u u + e h Scientific Computing II, Tobias Weinzierl page 11 of 27
15 Example: Restriction Residual: Scientific Computing II, Tobias Weinzierl page 12 of 27
16 6.4. Prolongation Prolongation: Project coarse solution (correction) to fine grid. Again, different variants. However, use shape function layout: Then: P = R T for full-weightening. Scientific Computing II, Tobias Weinzierl page 13 of 27
17 Prolongation and Restriction Matrices Scientific Computing II, Tobias Weinzierl page 14 of 27
18 6.5. Galerkin Multigrid What Does the Coarse Grid Operator Look Like? Idea: If the correction on the coarse grid represents the null error on the fine grid, the correction equation should not induce a change of the overall solution. If we project such a solution e ( ) down to the fine grid, apply the stencil, and restrict it back to the 2h coarse grid, nothing happens. The coarse grid operator should mirror this behaviour. A 2h e ( ) 2h = R A h P e ( ) 2h Scientific Computing II, Tobias Weinzierl page 15 of 27
19 Ingredients of Standard Galerkin Multigrid Given is A. Fix a prolongation P fitting to our problem (this is the art of Galerkin-MG). Set R = P T. Set A 2h = R A P. The Standard V (µ 1, µ 2 ) cycle: Apply µ 1 smoothing steps. Compute res on fine grid. Restrict res on coarse grid. This is the coarse grid right-hand side. Recursion yields an approximation of error e on coarse grid. Prolongate error e on fine grid. Sum up old approximation and error. Apply µ 2 smoothing steps. Scientific Computing II, Tobias Weinzierl page 16 of 27
20 Exercise: Coarse Grid Operator for Poisson Equation Scientific Computing II, Tobias Weinzierl page 17 of 27
21 Aliasing High frequencies appear to be smooth modes on coarse grids. In multiscale spectral analysis, we have to study coarse and smooth modes separately. Scientific Computing II, Tobias Weinzierl page 18 of 27
22 Remarks Finding an appropriate P is the art, and, often, standard coarsening fails. We do not solve the equation on the coarser grids, but the correction equation. Choosing an appropriate smoother (not the solver) also is art. Scientific Computing II, Tobias Weinzierl page 19 of 27
23 6.6. Two-grid Analysis (1D) Notation Convergence analysis of multigrid methods is difficult and has occupied researchers for several decades. (Briggs, A Multigrid Tutorial) We analyse a 1D-Poisson Equation given by the standard stencil. We analyse full-weightening. We analyse standard coarsening. We study the error/solution modes in terms of eigenfrequencies of A: with wk h (i) = (sin(ikπh)) k n low frequency (representable on the coarse grid), 2 k > n high frequency (not representable on the coarse grid). 2 We assume that the coarse grid problem is solved exactly. We do no postsmoothing, i.e. µ := µ 1, µ 2 = 0. Scientific Computing II, Tobias Weinzierl page 20 of 27
24 Operators Involved Smoothing operations: e h C µ e h. Determine coarse right-hand side: Coarse grid equation: b 2h = R res = R (b A u h ) = R (b A u ( ) + A e h ) = R A e h A 2h e 2h = R res e 2h = A 1 2h R A e h Prolongate correction and sum it up (without smoothing): With smoothing: u h u h + Pe 2h e h u ( ) u h = u ( ) u h,fine Pe 2h = e h P A 1 2h R A e h e h (id P A 1 2h R A)Cµ e h Scientific Computing II, Tobias Weinzierl page 21 of 27
25 Restriction Analysis Full Weightening Smooth modes, i.e. k n 2 : wk h (i) = sin(ikπh) Rw h k = cos2 ( kπh 2 Oscillatory modes, i.e. k > n 2, k := n k: This is Rw h k = sin2 ( kπh 2 ) w 2h k ) w 2h k Scientific Computing II, Tobias Weinzierl page 22 of 27
26 Restriction Analysis Full Weightening Smooth modes, i.e. k n 2 : wk h (i) = sin(ikπh) Rw h k = cos2 ( kπh 2 Oscillatory modes, i.e. k > n 2, k := n k: Rw h k = sin2 ( kπh 2 ) w 2h k ) w 2h k This is aliasing, i.e. oscillatory modes induce smooth modes on 2h (complementary modes). Example follows Briggs: A Multigrid Tutorial Scientific Computing II, Tobias Weinzierl page 22 of 27
27 Restriction Analysis Injection q m h = (sin(mπhi)) i=1,2,...,n 1 Scientific Computing II, Tobias Weinzierl page 23 of 27
28 Restriction Analysis Injection For coarse grid points: q m h = (sin(mπhi)) i=1,2,...,n 1 Rq m h = (sin(mπhi)) i=2,4,...,n 1 = (sin(mπ2hi)) i=1,..., N = { q m 2h if m N q N+1 m 2h if m > N For the other grid points, i.e. k > N+1 1: 2 sin(mπ2hi) = sin((m (N + 1) + (N + 1))π2hi) = sin((m (N + 1))π2hi) cos((n + 1)π2hi) + }{{} =cos(2πi)=1 cos((m (N + 1))π2hi) sin((n + 1)π2hi) }{{} =sin(2πi)=0 = sin((n + 1 m)π2hi). Scientific Computing II, Tobias Weinzierl page 23 of 27
29 The inverse of the PDE Operator (A h e) i = 1 h 2 (e i+1 2e i + e i 1 ) A h qh m =... = 2(cos(mπh) 1) A 1 h qm h = h 2 h 2 2(cos(mπh) 1) q m h Scientific Computing II, Tobias Weinzierl page 24 of 27
30 Prolongation Analysis Interpolation operator: Pw 2h k = cos 2 ( kπh 2 wk 2h (i) = sin(ikπ2h) ) ( ) kπh wk h sin2 wk 2h 2 Interpolation induces both smooth and oscillatory modes. Example follows Briggs: A Multigrid Tutorial Insights: Maximum eigenvalue is bounded and independent of h. Interpolation induces both smooth and oscillatory modes, i.e. we need a postsmoothing. Two-grid analysis already is extremely complicated (use computer algebra or numerical methods). Maximum eigenvalue is bounded and independent of h. However, inverse of A 2h usually is not available. Scientific Computing II, Tobias Weinzierl page 25 of 27
31 Outlook I Nested V-Cycles Still, we start with an arbitrary initial guess u (0). What could be a better guess? Scientific Computing II, Tobias Weinzierl page 26 of 27
32 Outlook I Nested V-Cycles Still, we start with an arbitrary initial guess u (0). What could be a better guess? Insight: It does not make sense to solve problem better than the discretisation error allows us to to (see algebraic error). However, we need higher order interpolation to end up with an F-cycle. Scientific Computing II, Tobias Weinzierl page 26 of 27
33 Outlook II Full Approximation Storage On the coarser levels, we have solely a correction available (not an approximation). However, an approximation would be of use (for non-linear operators A, e.g.). What is a trivial coarse representation of the solution? Scientific Computing II, Tobias Weinzierl page 27 of 27
34 Outlook II Full Approximation Storage On the coarser levels, we have solely a correction available (not an approximation). However, an approximation would be of use (for non-linear operators A, e.g.). What is a trivial coarse representation of the solution? However, we now have to modify the right-hand side of the correction equation. Scientific Computing II, Tobias Weinzierl page 27 of 27
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