MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA

given Poisson equation: Approximating Poisson equation: given

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

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

Relaxation of linear systems Ax=bAx=b Approximation, error Residual equation: Relaxation sweep: Eigenvectors: Fast convergence of high modes

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

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

TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h old h new uu h2 v ~~~  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation by recursion MULTI-GRID CYCLE

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

Multigrid solvers Cost: operations per unknown Linear scalar elliptic equation (~1971)

Multigrid solvers Cost: 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)

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

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

Multigrid solvers Cost: 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)

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

Two Grid Cycle for solving Approximate solution: Error: Residual equation: 1. Fine grid relaxation Goto 1 2. Coarse grid eq. 3. h old h new uu h2 v ~~~  Full Approximatioin Scheme (FAS): defect correction

LU = F h 2h 4h L h U h = F h L 4h U 4h = F 4h Fine-to-coarse defect correction L 2h U 2h = F 2h Correction Truncation error estimator interpolation of changes

Coarse-Grid Aliasing