MULTISCALE COMPUTATIONAL

METHODS

Achi BrandtThe Weizmann Institute of ScienceUCLA

www.wisdom.weizmann.ac.il/~achi

• 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

Scale-born obstacles:

• Many variables n gridpoints / particles / pixels / …

• Interacting with each other O(n2)

• Slowness

Slow Monte Carlo / Small time steps / …Slowly converging iterations /

due to

1. Localness of processing

0

0r0 Particle distance

Two-particleLennard-Jonespotential

+ external forces…

small step

Moving one particle at a time

fast local ordering

slow global move

r0

e.g.,

approximating Laplace eq.2 2

2 20

u u

x y

Numerical solution of a partial differential

equation (PDE)

on a fine grid

fine grid

h

u = average of u's

approximating Laplace eq.2 2

2 20

u u

x y

u given on the boundary

h

e.g., u = average of u's

approximating Laplace eq.2 2

2 20

u u

x y

Point-by-point RELAXATIONSolution algorithm:

Solving PDE: Influence of pointwiserelaxation 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 variables n gridpoints / particles / pixels / …

• Interacting with each other O(n2)

• Slowness

Slow Monte Carlo / Small time steps / …Slowly converging iterations /

due to

1. Localness of processing

2. Attraction basins

r

E(r)

Optimization min E(r)

multi-scale attraction basins

~ 10-15 second steps

Macromolecule

Potential Energy

S rr ,126

NBji ij

ij

ij

ij BALennard-Jones

S r

NB , j i ij

qqji Electrostatic

Bond length strain

Bond angle strain

)(1SV

DA,,,

ιjκlnijkl ncos ljki

torsion

DHA

HBAH,D, HA

HA

HA

HA 4

1210

S r

D

r

Ccos

hydrogen bond

rk

)r,...,r,r( n21E

2

,

)rr(S

S N

ijijj i

ij

2

,,

)(SKBA

ijkijk kji

ijk coscos

ijkl

ri

rjrl

rij ijk

Macromolecule

+ Lennard-Jones

~104 Monte Carlo passes

for one T Gi transition

G1 G2T

Dihedral potential

+ Electrostatic

r

E(r)

Optimization min E(r)

multi-scale attraction basins

Scale-born obstacles:

• Many variables

• Interacting with each other O(n2)

Slow Monte Carlo / Small time steps / …

1. Localness of processing

2. Attraction basins

Removed by multiscale algorithms

• Multiple solutions

• SlownessSlowly converging iterations /

n gridpoints / particles / pixels / …

Inverse problems / Optimization

Statistical sampling Many eigenfunctions

Solving PDE: Influence of pointwiserelaxation on the error

Error of initial guess Error after 5 relaxation sweeps

Error after 10 relaxations Error after 15 relaxations

Fast error smoothingslow solution

Relaxation of linear systems

Ax=b

Approximation x~, error xxe ~Residual equation: rxbe :~

iii vv A max21

i

iie ve iii

ie vr

Relaxation: Fast convergence of high modes

ii

i ee

max

1

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 pointwiserelaxation 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:

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

LhUh=Fh

L2hU2h=F2h

L4hU4h=F4h

h

2h

Localrelaxation

approximation

hu~

hV hh u~U smooth

hh u~LF hhVhLhR

h2Vh2L h2R

LhUh=Fh

L2hU2h=F2h

h2Vh2L h2R

TWO GRID CYCLE

Approximate solution:hu~

hhh u~UV hhh RVL

hhhh u~LFR

Fine grid equation: hhh FUL

2. Coarse grid equation: hhh RVL 22

hh2

hold

hnew uu h2v~~~ h

h2

Residual equation:

Smooth error:

1. Relaxation

residual:

h2v~Approximate solution:

3. Coarse grid correction:

4. Relaxation