MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute
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 Error after 10 relaxations Error after 5 relaxation sweeps Error after 15 relaxations Fast error smoothing slow solution
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
h Lh. Uh=Fh LU=F 2 h L 2 h. U 2 h=F 2 h 4 h L 4 h. U 4 h=F 4 h
Local Relaxation h approximation L h. U h = F h smooth 2 h R 2 h = ( Fh -Lh 2 h L 2 h. V U 2 h = R F 2 h ) 4 h L 4 h. V 4 h = R 4 h
Full Multi. Grid (FMG) algorithm h 2 h . . h 0/4 . h 0/2 h 0 * * 14 442 4 4 43 multigrid cycle V 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
Multigrid solvers Cost: 25 -100 operations per unknown • Linear scalar elliptic equation (~1971)
Multigrid solvers Cost: 25 -100 operations per unknown • • • • Linear scalar elliptic equation (~1971)* Nonlinear FAS (1975) Grid adaptation (1977, 1982) 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 Within one solver Optimal design Integral equations Full matrix Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986)
Multigrid solvers Cost: 25 -100 operations per unknown • • • • Linear scalar elliptic equation (~1971)* Nonlinear FAS (1975) Grid adaptation (1977, 1982) 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 Within one solver Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986)
Full Multi. Grid (FMG) algorithm h 2 h . . h 0/4 . h 0/2 h 0 * * 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
Local Relaxation h approximation L h. U h = F h smooth Full Approximation scheme (FAS): U 2 h = 2 h 2 h L 2 h. U V 2 h = F R 2 h + V 2 h Fine-to-coarse defect correction 4 h L 4 h. U 4 h = F 4 h
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 Within one solver Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis
Full Multi. Grid (FMG) algorithm h 2 h . . h 0/4 . h 0/2 h 0 * * 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
Local Relaxation h approximation L h. U h = F h smooth Full Approximation scheme (FAS): U 2 h = 2 h 2 h L 2 h. U V 2 h = F R 2 h + V 2 h Truncation error estimator Fine-to-coarse defect correction 4 h L 4 h. U 4 h = F 4 h
Multigrid solvers Cost: 25 -100 operations per unknown • • • • Linear scalar elliptic equation (~1971)* Nonlinear FAS (1975) Grid adaptation (1977, 1982) 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 Within one solver Optimal design Integral equations Statistical mechanics Massive parallel processing *Rigorous quantitative analysis (1986)
Local patches of finer grids • Same fast solver FMG
Full Multi. Grid (FMG) algorithm h 2 h . . h 0/4 . h 0/2 h 0 * * 14 442 4 4 43 multigrid cycle V 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
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
Finer level with local coordinate transformation With anisotropic further refinement x, y r, s Boundary or tracked layer
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 and differet physics; e. g. atomistic “Quasicontiuum” method [B. , 1992]
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)
Stokes
h-principal L Compressible Navier-Stokes (on the viscous scale) Central Cauchy-Riemann Central (Navier-) Stokes
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
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
ALGEBRAIC MULTIGRID (AMG) Ax = b 1982
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)
ALGEBRAIC MULTIGRID (AMG) 1982 Ax = b Coarse variables - a subset Criterion: Fast convergence of “compatible relaxation” Relax Ax = 0 Keeping coarse variables = 0
ALGEBRAIC MULTIGRID (AMG) 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 1982
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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