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MULTISCALE COMPUTATIONAL METHODS. Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi. Computational bottlenecks:. Elementary particles Physics standard model. Chemistry, materials science. Schrödinger equation. Molecular dynamics forces.
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MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi
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
Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing
Two-particle Lennard-Jones potential Particle distance r0 00 + external forces…
small step Moving one particle at a timefast local ordering r0 slow global move
Numerical solution of a partial differential equation (PDE) e.g., approximating Laplace eq. on a fine grid
fine grid h u= average ofu's approximating Laplace eq.
u given on the boundary h e.g., u= average ofu's Solution algorithm: approximating Laplace eq. Point-by-point RELAXATION
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
Scale-born obstacles: • Many variablesn gridpoints / particles / pixels / … • Interacting with each otherO(n2) • Slowness Slowly converging iterations / Slow Monte Carlo / Small time steps / … • due to • Localness of processing 2. Attraction basins
E(r) r Optimizationmin E(r) multi-scale attraction basins
Macromolecule ~ 10-15 second steps
E ri tijkl rij rl rj Potential Energy Lennard-Jones Electrostatic Bond length strain Bond angle strain torsion hydrogen bond rk
t Macromolecule Dihedral potential G2 G1 T t 0 -p p + Lennard-Jones + Electrostatic ~104Monte Carlo passes for one T Gi transition
E(r) r Optimizationmin E(r) multi-scale attraction basins
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
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
, error Approximation Residual equation: Relaxation of linear systems Ax=b Eigenvectors: Relaxation: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
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
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
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)
When relaxation slows down: the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth The error can be approximated on a coarser grid
h LhUh=Fh LU=F 2h L2hU2h=F2h 4h L4hU4h=F4h
Local relaxation approximation smooth h 2h LhUh=Fh L2hU2h=F2h
~ ~ ~ ~ = + h h 2 2 h h u u v v new old TWO GRID CYCLE Fine grid equation: 1. Relaxation Approximate solution: Smooth error: Residual equation: residual: 2. Coarse grid equation: Approximate solution: 3. Coarse grid correction: 4. Relaxation