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MULTISCALE COMPUTATIONAL METHODS

MULTISCALE COMPUTATIONAL METHODS. Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi. Poisson equation:. given. Approximating Poisson equation:. given. Solving PDE : Influence of pointwise relaxation on the error. Error of initial guess.

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MULTISCALE COMPUTATIONAL METHODS

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

  2. Poisson equation: given 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 smoothingslow 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. h LhUh=Fh LU=F 2h L2hU2h=F2h 4h L4hU4h=F4h

  6. smooth Local Relaxation h LhUh = Fh approximation 2h R2h = ( Fh -Lh ) L2hU2h = F2h L2hV2h = R2h 4h L4hV4h = R4h

  7. h 2h . . . h0/4 h0/2 h0 * * * * interpolation (order m) of corrections Full MultiGrid (FMG) algorithm 1 4 4 4 2 4 4 4 3 multigrid cycle V interpolation (order l+p) to a new grid residual transfer enough sweeps or direct solver relaxation sweeps * algebraic error < truncation error

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

  9. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  10. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

  11. h 2h . . . h0/4 h0/2 h0 * * * * interpolation (order l+p) to a new grid interpolation (order m) of corrections residual transfer enough sweeps or direct solver relaxation sweeps * algebraic error < truncation error Full MultiGrid (FMG) algorithm

  12. smooth Local Relaxation h LhUh = Fh approximation Full Approximation scheme (FAS): U2h = + V2h 2h L2hV2h = R2h L2hU2h = F2h Fine-to-coarse defect correction 4h L4hU4h = F4h

  13. Multigrid solversCost: 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 Withinonesolver

  14. h 2h . . . h0/4 h0/2 h0 * * * * interpolation (order l+p) to a new grid interpolation (order m) of corrections residual transfer enough sweeps or direct solver relaxation sweeps * algebraic error < truncation error Full MultiGrid (FMG) algorithm

  15. smooth Local Relaxation h LhUh = Fh approximation Full Approximation scheme (FAS): U2h = + V2h 2h L2hV2h = R2h L2hU2h = F2h Truncation error estimator Fine-to-coarse defect correction 4h L4hU4h = F4h

  16. Multigrid solversCost: 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) (1977,1982) FAS (1975) Withinonesolver

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

  18. h 2h . . . h0/4 h0/2 h0 * * * * interpolation (order m) of corrections Full MultiGrid (FMG) algorithm 1 4 4 4 2 4 4 4 3 multigrid cycle V interpolation (order l+p) to a new grid residual transfer enough sweeps or direct solver relaxation sweeps * algebraic error < truncation error

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

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

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

  22. Multigrid solversCost: 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. Compressible Navier-Stokes(on the viscous scale) h-principal L Central Cauchy-Riemann Central (Navier-) Stokes

  25. Multigrid solversCost: 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

  26. Multigrid solversCost: 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. ALGEBRAIC MULTIGRID (AMG) 1982 Ax = b

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

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

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

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

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