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
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
+ Lennard-Jones
~104 Monte Carlo passes
for one T Gi transition
G1 G2T
Dihedral potential
+ Electrostatic
Windows
Coarser level
Larger density fluctuations
Still coarser level
1~density
:level Fine
2~density
:level Fine
3:density
level Fine
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