MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling

MULTISCALE COMPUTATION:

From Fast SolversTo Systematic Upscaling

A. BrandtThe Weizmann Institute of ScienceUCLA

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

Major scaling bottlenecks:computing

Elementary particles (QCD)

Schrödinger equationmoleculescondensed matter

Molecular dynamicsprotein folding, fluids, materials

Turbulence, weather, combustion,…

Inverse problemsda, control, medical imaging

Vision, recognition

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 distance

Two-atomLennard-Jonespotential

r0

small step

small step

Moving one particle at a time

fast local ordering

slow global move

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




• Slowness


due to


2. Attraction basins

Fluids Gas/Liquid

1. Positional clustering

Lennard-Jones

r0

2. Electrostatic clustering

Dipoles

Water: 1& 2

r

E(r)

Optimization min E(r)

multi-scale attraction basins

Macromolecule

+ Lennard-Jones

~104 Monte Carlo passes

for one T Gi transition

G1 G2T

Dihedral potential

+ Electrostatic

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




• Slowness


due to



Removed by multiscale processing

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

LU = F

h

2h

4h

LhUh = Fh

L2hU2h = F2h

L4hV4h = R4h

L2hV2h = R2h

interpolation (order l+p)to a new grid

interpolation (order m)of corrections

relaxation sweeps

algebraic error< truncation error

residual transfer

ν νenough sweepsor direct solver*

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

Full MultiGrid (FMG) algorithm

..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

Vcyclemultigrid

h0

h0/2

h0/4

2h

h

Multigrid solversCost: 25-100 operations per unknown

• Linear scalar elliptic equation (~1971)


• Linear scalar elliptic equation (~1971)• Nonlinear FAS (1975)

LU = F

h

2h

4h

LhUh = Fh

L4hU4h = F4h

h2

h4

Fine-to-coarse defect correction

L2hV2h = R2hU2h = Uh,approximate +V2h L2hU2h = F2h


• 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

)log(

2

NNO

fuku

(1977,1982)


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs*

(1980)



(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

• Same fast solver

Local patches of finer grids

• Each level correct the equations of the next coarser level

• Each patch may use different coordinate system and anisotropic grid

“Quasicontiuum” method [B., 1992]

• Each patch may use different coordinate system and anisotropic grid and different

physics; e.g. Atomistic

and differet physics; e.g. atomistic





(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

ALGEBRAIC MULTIGRID (AMG) 1982

ALGEBRAIC MULTIGRID (AMG) 1982

Coarse variables - a subset

1. “General” linear systems

2. Variety of graph problems

Graph problems

Partition: min cut

Clustering bioinformatics

Image segmentation

VLSI placement Routing

Linear arrangement: bandwidth, cutwidth

Graph drawing low dimension embedding

Coarsening: weighted aggregation

Recursion: inherited couplings (like AMG)

Modified by properties of coarse aggregates

General principle: Multilevel objectives





(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

2h

h

2wavelength

Non-local components:

eix, ≈ ±kSlow to converge in local processing

The error after relaxationv(x) = A1(x) eikx + A2(x) e-ikx

A1(x), A2(x) smooth

Ar(x) are represented on coarser grids:

A1 + 2 i k A1′ = f1 = rh(x) e-ikx

1D Wave Equation: u”+k2u=f

k

8,8)

1,1)

2,2)3,3)

4,4)

5,5)

6,6)7,7)O(H)

2D Wave Equation: Du+k2u=f

Non-local:

ei( x + 2 y)

+

≈ k2

On coarser grid (meshsize H):

Fully efficient multigrid solverTends to Geometrical OpticsRadiation Boundary Conditions: directly on coarsest level

cH

1r

v(x) y)(x,A ry)x( rre i

Σr = 1

m

Ar(x) φr(x)

Generally: LU=F

Non-local part of U has the form

L φr ≈ 0

Ar(x) smooth

{φr } found by local processing

Ar represented on a coarser grid





(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

N eigenfunctions

Electronic structures (Kohn-Sham eq):

)(ψ)(ψ)(V xxx iii i i = 1, …, = 1, …, NN = # electrons= # electrons

O (N) gridpoints per i

O (N2 ) storage

Orthogonalization O (N3 ) operations

O (N log N) storage & operations

Multiscale eigenbase 1D: Livne

V = Vnuclear + V()One shot solver



• 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


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

Integro-differential Equation

differential

, dense

2

dyyuyxGxLu )(),()(

fuAnn

A

Multigrid solver

Distributive relaxation:1st order2nd order

Solution cost ≈ one fast transform(one fast evaluation of the discretized integral transform)

Integral Transforms

Ω

d )u( G(x, V(x) 'x

|-x|

1

/|-x|-e

x-e

ixe

22

G(x, Transform

Fourier

Laplace

Gauss

Potential

Complexity

n logn)

n logn)

n)

n)

G(x,Exp(ik Waves n logn)

Glocal

G(x,y)

Gsmooth

s |x-y|

G(x,y) = Gsmooth(x,y) + Glocal(x,y)

s ~ next coarser scale

~ 1 / | x – y |

O(n) not static!



(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 Monte-Carlo


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

Discretization Lattice LL

for accuracy :ε qε ~L

Monte Carlo cost ~dL

“volume factor”

“critical slowing down”

Multiscale ~ 2ε

Multigrid moves

2zL

Many sampling cyclesat coarse levels



(1980)



(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

• Same fast solver

Local patches of finer grids

• Each level correct the equations of the next coarser level

• Each patch may use different coordinate system and anisotropic grid

“Quasicontiuum” method [B., 1992]

• Each patch may use different coordinate system and anisotropic grid and different

physics; e.g. Atomistic

and differet physics; e.g. atomistic

Repetitive systemse.g., same equations everywhere

UPSCALING:

Derivation of coarse equationsin small windows




• Slowness


due to



Removed by multiscale processing

Systematic Upscaling

1. Choosing coarse variables

2. Constructing coarse-level operational rules

equations

Hamiltonian

Macromolecule

~ 10-15 second steps



Criterion: Fast convergence of “compatible

relaxation”



Criterion: Fast equilibration of “compatible Monte Carlo”

OR: Fast convergence of

“compatible relaxation”

Local dependence on coarse variables

2. Constructing coarse-level operational rules

Done locally

In representative “windows” fast

Macromolecule

Macromolecule

Two orders of magnitude faster simulation

Fluids

£ Total mass£ Total momentum£ Total dipole moment£ average location

1

1

2

Windows

Coarser level

Larger density fluctuations

Still coarser level

1~density

:level Fine

2~density

:level Fine

3:density

level Fine

Fluids

Total mass:

)(xmSumming

Lower Temperature T

Summing also

0 ,2 vwuw

)(xme xwi v

u

Still lower T:More precise crystal direction and

periods determined at coarser spatial levels

Heisenberg uncertainty principle:

Better orientational resolution at larger spatial scales

Optimization byMultiscale annealing

Identifying increasingly larger-scale

degrees of freedom

at progressively lower temperatures

Handling multiscale attraction basins

E(r)

r


Rigorous computational methodology to derivefrom physical laws at microscopic (e.g., atomistic) level

governing equations at increasingly larger scales.

Scales are increased gradually (e.g., doubled at each level)

with interscale feedbacks, yielding:

• Inexpensive computation : needed only in some small “windows” at each scale.

• No need to sum long-range interactions

Applicable to fluids, solids, macromolecules, electronic structures, elementary particles, turbulence, …

• Efficient transitions between meta-stable configurations.

Upscaling Projects• QCD (elementary particles):

Renormalization multigrid Ron

BAMG solver of Dirac eqs. Livne, Livshits Fast update of , det Rozantsev

• (3n +1) dimensional Schrödinger eq. Filinov

Real-time Feynmann path integrals Zlochin

multiscale electronic-density functional

• DFT electronic structures Livne, Livshits

molecular dynamics

• Molecular dynamics:

Fluids Ilyin, Suwain, Makedonska

Polymers, proteins Bai, Klug

Micromechanical structures Ghoniem defects, dislocations, grains

• Navier Stokes Turbulence McWilliams

Dinar, Diskin

1MfxM

M

THANK YOU

MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling

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