1. MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi

2. given Poisson equation: Approximating Poisson equation: given

3. Solving PDE : Influence of pointwise relaxation on the error Error of initial guess Error after 5 relaxation sweeps Error after 10 relaxations Error after 15 relaxations Fast error smoothing slow solution

4. When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S (e.g., Poisson equation) the error is smooth The error can be approximated on a coarser grid

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

6. h L h U h = F h Local Relaxation approximation smooth 2h 4h L 2h U 2h = F 2h L 4h V 4h = R 4h L 2h V 2h = R 2h R 2h = ( F h -L h )

7. interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h ***  multigrid cycle V

8. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)

9. Multigrid solvers Cost: 25-100 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)

10. Multigrid solvers Cost: 25-100 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 Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) FAS (1975) Within one solver (1977,1982)

11. interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * *** Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h

12. h 4h L h U h = F h L 4h U 4h = F 4h Fine-to-coarse defect correction L 2h V 2h = R 2h Full Approximation scheme (FAS): U 2h = + V 2h L 2h U 2h = F 2h Local Relaxation approximation smooth

13. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) Several coupled PDEs* Non-elliptic: high-Reynolds flow Highly indefinite: waves Many eigenfunctions (N) Near zero modes Gauge topology: Dirac eq. Inverse problems Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis Within one solver

14. interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * *** Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h

15. h 4h L h U h = F h L 4h U 4h = F 4h Fine-to-coarse defect correction L 2h V 2h = R 2h Full Approximation scheme (FAS): U 2h = + V 2h L 2h U 2h = F 2h Local Relaxation approximation smooth Truncation error estimator

16. Multigrid solvers Cost: 25-100 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 Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) FAS (1975) Within one solver (1977,1982)

17. Same fast solver FMG Local patches of finer grids Each patch may use different coordinate system and anisotropic grid and different physics; e.g. Atomistic

18. interpolation (order l+p) to a new grid interpolation (order m) of corrections relaxation sweeps algebraic error < truncation error residual transfer enough sweeps or direct solver * Full MultiGrid (FMG) algorithm... * h0h0 h 0 /2 h 0 /4 2h h ***  multigrid cycle V

19. Same fast solver FMG, Local patches of finer grids Each level correct the equations of the next coarser level Each patch may use different coordinate system and anisotropic grid Same fast solver FMG, FAS

20. x, y r, s Finer level with local coordinate transformation Boundary or tracked layer With anisotropic further refinement

21. Same fast solver FMG, Local patches of finer grids Each level correct the equations of the next coarser level Each patch may use different coordinate system and anisotropic grid “Quasicontiuum” method [B., 1992] Each patch may use different coordinate system and anisotropic grid and different physics; e.g. Atomistic and differet physics; e.g. atomistic Same fast solver FMG, FAS

22. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) 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 Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986)

23. Stokes

24. h-principal L Compressible Navier-Stokes (on the viscous scale) Central Cauchy-Riemann Central (Navier-) Stokes

25. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) Several coupled PDEs* Non-elliptic: high-Reynolds flow Highly indefinite: waves Many eigenfunctions (N) Near zero modes Gauge topology: Dirac eq. Inverse problems Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis

27. Multigrid solvers Cost: 25-100 operations per unknown Linear scalar elliptic equation (~1971)* Nonlinear Grid adaptation General boundaries, BCs* Discontinuous coefficients Disordered: coefficients, grid (FE) Several coupled PDEs* Non-elliptic: high-Reynolds flow Highly indefinite: waves Many eigenfunctions (N) Near zero modes Gauge topology: Dirac eq. Inverse problems Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis

28. ALGEBRAIC MULTIGRID (AMG) 1982 Ax = b

29. When relaxation slows down: DISCRETIZED PDE'S GENERAL SYSTEMS OF LOCAL EQUATIONS the error is smooth Along characteristics The error can be approximated by a far fewer degrees of freedom (coarser grid) the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth

30. ALGEBRAIC MULTIGRID (AMG) 1982 Coarse variables - a subset Criterion: Fast convergence of “compatible relaxation” Ax = b Relax Ax = 0 Keeping coarse variables = 0

31. ALGEBRAIC MULTIGRID (AMG) 1982 Coarse variables - a subset 1. “General” linear systems 2. Variety of graph problems General procedures for deriving: * Interpolations Ax = b * Restriction * Coarse-level equations Generalizations:

32. Graph problems Partition: min cut Clustering bioinformatics Image segmentation VLSI placement Routing Linear arrangement: bandwidth, cutwidth Graph drawing low dimension embedding