MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling
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Transcript of MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling
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MULTISCALE COMPUTATION:
From Fast SolversTo Systematic Upscaling
A. BrandtThe Weizmann Institute of ScienceUCLA
www.wisdom.weizmann.ac.il/~achi
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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
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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
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small step
Moving one particle at a time
fast local ordering
slow global move
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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
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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
![Page 7: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/7.jpg)
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Macromolecule
+ Lennard-Jones
~104 Monte Carlo passes
for one T Gi transition
G1 G2T
Dihedral potential
+ Electrostatic
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r
E(r)
Optimization min E(r)
multi-scale attraction basins
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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
Removed by multiscale processing
![Page 11: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/11.jpg)
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
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LU = F
h
2h
4h
LhUh = Fh
L2hU2h = F2h
L4hV4h = R4h
L2hV2h = R2h
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Multigrid solversCost: 25-100 operations per unknown
• Linear scalar elliptic equation (~1971)
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Multigrid solversCost: 25-100 operations per unknown
• 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)
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Multigrid solversCost: 25-100 operations per unknown
• 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)
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LU = F
h
2h
4h
LhUh = Fh
L4hU4h = F4h
h2
h4
Fine-to-coarse defect correction
L2hV2h = R2hU2h = Uh,approximate +V2h L2hU2h = F2h
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Multigrid solversCost: 25-100 operations per unknown
• 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)
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• 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
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Multigrid solversCost: 25-100 operations per unknown
• 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)
![Page 20: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/20.jpg)
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Multigrid solversCost: 25-100 operations per unknown
• 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)
![Page 22: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/22.jpg)
ALGEBRAIC MULTIGRID (AMG) 1982
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ALGEBRAIC MULTIGRID (AMG) 1982
Coarse variables - a subset
1. “General” linear systems
2. Variety of graph problems
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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
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SWA
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Data: Filippi
TaggedTagged Our resultsOur results
Detected Lesions
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Multigrid solversCost: 25-100 operations per unknown
• 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)
![Page 28: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/28.jpg)
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
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Multigrid solversCost: 25-100 operations per unknown
• 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)
![Page 30: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/30.jpg)
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
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Multigrid solversCost: 25-100 operations per unknown
• 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 Full matrix• Statistical mechanics
Massive parallel processing*Rigorous quantitative analysis
(1986)
FAS (1975)
Within one solver
)log(
2
NNO
fuku
(1977,1982)
![Page 32: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/32.jpg)
Integro-differential Equation
differential
, dense
2
dyyuyxGxLu )(),()(
fuAnn
A
Multigrid solver
Distributive relaxation:1st order2nd order
Solution cost ≈ one fast transform (one matrix multiply)
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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)
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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!
Gsmooth(x,y) tranferred directly to coarser
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Multigrid solversCost: 25-100 operations per unknown
• 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 Monte-Carlo
Massive parallel processing*Rigorous quantitative analysis
(1986)
FAS (1975)
Within one solver
)log(
2
NNO
fuku
(1977,1982)
![Page 36: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/36.jpg)
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
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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
Removed by multiscale processing
![Page 38: MULTISCALE COMPUTATION: From Fast Solvers To Systematic Upscaling](https://reader030.fdocuments.in/reader030/viewer/2022032606/56812cd4550346895d91923e/html5/thumbnails/38.jpg)
Repetitive systemse.g., same equations everywhere
UPSCALING:
Derivation of coarse equationsin small windows
Small scale ratio at a time
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Systematic Upscaling
1. Choosing coarse variables
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
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Macromolecule
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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
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Macromolecule
Two orders of magnitude faster simulation
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Fluids
£ Total mass£ Total momentum£ Total dipole moment£ average location
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Windows
Coarser level
Larger density fluctuations
Still coarser level
1~density
:level Fine
2~density
:level Fine
3:density
level Fine
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Fluids
Total mass:
)(xmSumming
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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
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Optimization byMultiscale annealing
Identifying increasingly larger-scale
degrees of freedom
at progressively lower temperatures
Handling multiscale attraction basins
E(r)
r
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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.
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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.
Real-time Feynmann path integrals Zlochin
multiscale electronic-density functional
• DFT electronic structures Livne, Livshits, Carter
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
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THANK YOU
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Aggregating Regions Adaptively
e.g., by similarity of
• densities astrophysics
• heights epitaxial growth
• color image segmentationcolor variances at all scaleselongation continuation deblurringshapes recognition
…