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MULTISCALE COMPUTATIONAL METHODS

Explore the efficiency of Multi-Grid Cycle and Full MultiGrid for solving elliptic PDEs, with emphasis on error estimation and grid adaptation. Learn how these methods tackle computational bottlenecks and overcome algorithmic obstacles. Discover their applications in diverse fields from physics to VLSI design.

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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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