MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA

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

Major scaling bottlenecks: computing Elementary particles (QCD) Schrödinger equation molecules condensed matter Molecular dynamics protein folding, fluids, materials Turbulence, weather, combustion,… Inverse problems da, control, medical imaging Vision, recognition

Scale-born obstacles: Many variables n gridpoints / particles / pixels / … Interacting with each other O(n 2 ) Slowness Slow Monte Carlo / Small time steps / … Slowly converging iterations / due to 1.Localness of processing

0 r0r0 Particle distance Two-particle Lennard-Jones potential + external forces…

small step Moving one particle at a time fast local ordering slow global move r0r0

e.g., approximating Laplace eq. Numerical solution of a partial differential equation (PDE) on a fine grid

fine grid h u = average of u's approximating Laplace eq.

u given on the boundary h e.g., u = average of u's approximating Laplace 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

Scale-born obstacles: Many variables n gridpoints / particles / pixels / … Interacting with each other O(n 2 ) Slowness Slow Monte Carlo / Small time steps / … Slowly converging iterations / due to 1.Localness of processing 2. Attraction basins

Fluids Gas/Liquid 1.Positional clustering Lennard-Jones r0r0 2.Electrostatic clustering Dipoles Water: 1& 2

r E(r) Optimization min E(r) multi-scale attraction basins

~ second steps Macromolecule

Potential Energy Lennard-Jones Electrostatic Bond length strain Bond angle strain torsion hydrogen bond rkrk E  ijkl riri rjrj rlrl r ij

Macromolecule      + Lennard-Jones ~10 4 Monte Carlo passes for one T G i transition G1G1 G2G2 T Dihedral potential + Electrostatic

r E(r) Optimization min E(r) multi-scale attraction basins

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

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: Fast convergence of high modes Eigenvectors:

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 smoothing slow solution

When relaxation slows down: DISCRETIZED PDE'S the error is smooth Along characteristics the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth

When relaxation slows down: DISCRETIZED PDE'S GENERAL SYSTEMS OF LOCAL EQUATIONS the error is smooth Along characteristics The error can be approximated by a far fewer degrees of freedom (coarser grid) the error is a sum of low eigen-vectors ELLIPTIC PDE'S the error is smooth

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

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

h2h Local relaxation approximation smooth L h U h =F h L 2h U 2h =F 2h

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