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 process

Not optimization !

Multigrid solversCost: 25-100 operations per unknown

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

Massive parallel processing*Rigorous quantitative analysis

(1986)

u given on the boundary

h

u = function of u's and f Probability distribution of u = function of u's and f

Point-by-point RELAXATIONPoint-by-point MONTE CARLO

Discretization Lattice LL

for accuracy :ε qε ~L

“volume factor”

Multiscale cost ~2ε

Multigrid cycles

Many sampling cyclesat coarse levels

“critical slowing down”

Monte Carlo cost ~ dL 2zL

Statistical samples

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

Repetitive systemse.g., same equations everywhere

UPSCALING:

Derivation of coarse equationsin small windows

Vs.

COARSENING:

For acceleration

Or surrogate problems

Etc.

A solution value is NOTgenerally determined just by few local equations

N unknowns O (N) solution operations

UPSCALING:

The coarse equation can be derived ONCE for all similar neighborhoods

# operations << N

A coarse equation ISgenerally determined just by few local equations

Systematic Upscaling

1. Choosing coarse variables

2. Constructing coarse-level operational rules

equations

Hamiltonian

Macromolecule

~ 10-15 second steps

ALGEBRAIC MULTIGRID (AMG) 1982

Coarse variables - a subset

Criterion: Fast convergence of “compatible relaxation”

Ax = b

Relax Ax = 0Keeping coarse variables = 0

Systematic Upscaling

1. Choosing coarse variables

Criterion: Fast convergence of “compatible relaxation”

2. Constructing coarse-level operational rules

(equations / Hamiltonian)

Done locally

Local dependence on coarse variables

OR: Fast equilibration of

In representative “windows”

“compatible Monte Carlo”

Macromolecule

~ 10-15 second steps

Macromolecule

Two orders of magnitude faster simulation

Macromolecule

+ Lennard-Jones

~104 Monte Carlo passes

for one T Gi transition

G1 G2T

Dihedral potential

+ Electrostatic

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

Systematic Upscaling

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 ??? defects, dislocations, grains

• Navier Stokes Turbulence McWilliams

Dinar, Diskin

1M

fxM

M

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