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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. u given on the boundary. h. e.g., u = function of u 's and f. Solution algorithm:.

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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. u given on the boundary h e.g., u= function ofu's andf Solution algorithm: approximating Poisson eq. Point-by-point RELAXATION

  4. 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

  5. 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

  6. h LhUh=Fh LU=F 2h L2hU2h=F2h 4h L4hU4h=F4h

  7. ~ ~ ~ ~ = + h h 2 2 h h u u v v new old MULTI-GRID CYCLE TWO GRID CYCLE Fine grid equation: 1 1. Relaxation Approximate solution: Smooth error: Residual equation: 2 residual: 2. Coarse grid equation: 3 4 Approximate solution: by recursion 5 3. Coarse grid correction: 6 4. Relaxation

  8. 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

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

  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 Full matrix Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986) (1977,1982) FAS (1975) Withinonesolver

  11. Scale-born obstacles: • Many variables n gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … 1. Localness of processing 2. Attraction basins • Multiple solutions Inverse problems / Optimization Many eigenfunctions Statistical sampling Removed by multiscale algorithms

  12. Computational bottlenecks: • Elementary particles Physics standard model • Chemistry, materials science Schrödinger equation Molecular dynamics forces • (Turbulent) flows Partial differential equations • Vision: recognition • Seismology • Tomography (medical imaging) • Graphs: data mining,… • VLSI design

  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) 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

  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. Approximate solution: Error: Residual equation: 2. Coarse grid eq. ~ ~ ~ 3. = + h h 2 h u u v new old Two Grid Cycle for solving 1. Fine grid relaxation Full Approximatioin Scheme (FAS): defect correction Goto 1

  16. 3 1 Correction h LhUh = Fh LU = F 4 2 interpolation of changes 2h L2hU2h = F2h Fine-to-coarse defect correction Truncation error estimator 4h L4hU4h = F4h

  17. Coarse-Grid Aliasing

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