Post on 22-Jan-2016
description
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