MULTISCALE%20COMPUTATION:%20From%20Fast%20Solvers%20To%20Systematic%20Upscaling

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MULTISCALE COMPUTATIONFrom Fast SolversTo
Systematic Upscaling

• A. Brandt
• The Weizmann Institute of Science
• UCLA
• www.wisdom.weizmann.ac.il/achi

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Major scaling bottleneckscomputing

• 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

Slowly converging iterations /
Slow Monte Carlo / Small time steps /

• due to
• Localness of processing

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Moving one particle at a timefast local
ordering
slow global move
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Fast error smoothingslow solution
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Scale-born obstacles

• Many variables n
  gridpoints / particles / pixels /

• Interacting with each other O(n2)

• Slowness

Slowly converging iterations /
Slow Monte Carlo / Small time steps /

• due to
• Localness of processing

2. Attraction basins
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Fluids Gas/Liquid

• Positional clustering
• Lennard-Jones

• Electrostatic clustering
• Dipoles

Water 1 2
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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

Slowly converging iterations /
Slow Monte Carlo / Small time steps /

• due to
• Localness of processing

2. Attraction basins
Removed by multiscale processing
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Fast error smoothingslow solution
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h
LU F
LhUh Fh
2h
L2hU2h F2h
L2hV2h R2h
4h
L4hV4h R4h
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Full MultiGrid (FMG) algorithm
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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

FAS (1975)
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h
LhUh Fh
LU F
2h
L2hV2h R2h
U2h Uh,approximate V2h
L2hU2h F2h
Fine-to-coarse defect correction
4h
L4hU4h F4h
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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)

(1977,1982)
FAS (1975)
Within one 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
• Statistical mechanics
• Massive parallel processing
• Rigorous quantitative analysis (1986)

(1977,1982)
FAS (1975)
Within one solver
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Local patches of finer grids

• Same fast solver

• Each level correct the equations of the next
  coarser level

• Each patch may use different coordinate system
  and anisotropic grid

Quasicontiuum method B., 1992
and differet physics e.g. atomistic

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

(1977,1982)
FAS (1975)
Within one 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
• Statistical mechanics
• Massive parallel processing
• Rigorous quantitative analysis (1986)

(1977,1982)
FAS (1975)
Within one solver
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ALGEBRAIC MULTIGRID (AMG)
1982
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Classical ALGEBRAIC MULTIGRID (AMG)
1982
Ax b, "M matrix"
aii
aij
aij aji ? 0 (i 1,,n)

• Relaxation ? approximation

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Classical ALGEBRAIC MULTIGRID (AMG)
1982
Ax b, "M matrix"
aii
aij
aij aji ? 0 (i 1,,n)
Coarse variables - a subset
( xc)3(a23 x2 a34 x4)/(a23 a34)
AMG Cycle
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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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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)

(1977,1982)
FAS (1975)
Within one solver
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Non-local components eiwx, w k Slow to
converge in local processing
The error after relaxation v(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
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2D Wave Equation Duk2uf Non-local

ei(w1 x w2 y)
w12 w22 k2
On coarser grid (meshsize H)

• Fully efficient multigrid solver
• Tends to Geometrical Optics
• Radiation Boundary Conditions
• directly on coarsest level

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Generally LUF
Non-local part of U has the form
m
S
Ar(x) fr(x)
r 1
L fr 0 Ar(x) smooth fr 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)

(1977,1982)
FAS (1975)
Within one solver
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N eigenfunctions
Electronic structures (Kohn-Sham eq)
i 1, , N electrons
O (N) gridpoints per yi
O (N2 ) storage
Orthogonalization
O (N3 ) operations
Multiscale eigenbase
1D Livne
O (N log N) storage operations
V Vnuclear V(y)
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)

(1977,1982)
FAS (1975)
Within one solver
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Integro-differential Equation

• differential
• , dense

Multigrid solver Distributive relaxation 1st
order 2nd order
Solution cost one fast transform(one fast
evaluation of the discretized integral transform)
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Integral Transforms
G(x,x)
Transform
Complexity

O(n logn)
Fourier
Laplace
O(n logn)
O(n)
Gauss
Potential
O(n)
G(x,x)
Exp(ikx)
O(n logn)
Waves
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1 / x y
G(x,y) Gsmooth(x,y) Glocal(x,y)
s next coarser scale
O(n) not static!
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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)

(1977,1982)
FAS (1975)
Within one solver
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Discretization Lattice
for accuracy
Monte Carlo cost
volume factor critical slowing down
Multigrid moves
Many sampling cycles at coarse levels
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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)

(1977,1982)
FAS (1975)
Within one solver
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Local patches of finer grids

• Same fast solver

• Each level correct the equations of the next
  coarser level

• Each patch may use different coordinate system
  and anisotropic grid

Quasicontiuum method B., 1992
and differet physics e.g. atomistic

• Each patch may use different coordinate system
  and anisotropic grid and different
• physics e.g. Atomistic

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Repetitive systemse.g., same equations everywhere

• UPSCALING
• Derivation of coarse equationsin small windows

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Scale-born obstacles

• Many variables n
  gridpoints / particles / pixels /

• Interacting with each other O(n2)

• Slowness

Slowly converging iterations /
Slow Monte Carlo / Small time steps /

• due to
• Localness of processing
• Attraction basins

Removed by multiscale processing
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A solution value is NOT generally determined
just by few local equations
A coarse equation IS generally determined just
by few local equations
? O (N) operations
The coarse equation can be derived ONCE for all
similar neighborhoods
? operations ltlt N
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Systematic Upscaling

1. Choosing coarse variables

• Constructing coarse-level operational rules
• equations
• Hamiltonian

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Macromolecule
10-15 second steps
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Systematic Upscaling

• Choosing coarse variablesCriterion Fast
  convergence of compatible relaxation

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Systematic Upscaling

• Choosing coarse variablesCriterion Fast
  equilibration of compatible Monte Carlo
• OR Fast convergence of
• compatible relaxation
• Local dependence on coarse variables
• Constructing coarse-level operational rules
• Done locally
• In representative windows
• fast

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Systematic Upscaling

• Choosing coarse variablesCriterion Fast
  equilibration of compatible Monte Carlo
• Local dependence on coarse variables
• Constructing coarse-level operational rules
• Done locally
• In representative windows
• fast

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Macromolecule
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Potential Energy
Lennard-Jones
Electrostatic
Bond length strain
Bond angle strain
torsion
hydrogen bond
rk
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Macromolecule
Two orders of magnitude faster simulation
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Macromolecule
Dihedral potential
G2
G1
T
t
-p
0
p
Lennard-Jones
Electrostatic
104 Monte Carlo passes for one T
Gi transition
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Fluids
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• 1

1

• 2

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Hierarchy of Coarser levels

• Total mass at scale s at point x
• summed recursively
• densities at all scales
• Summing recursivelydensity variations at any
  scaleaveraged to all higher scales

s
s -1
s
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Windows

• Coarser level
• Larger density fluctuations

Still coarser level
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Fluids
Summing
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Lower Temperature T

• Summing also

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 derive
• from 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

• Efficient transitions between meta-stable
  configurations.

Applicable to fluids, solids, macromolecules,
electronic structures, elementary particles,
turbulence,
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Upscaling Projects

• QCD (elementary particles) Renormalization
  multigrid RonBAMG
  solver of Dirac eqs. Livne,
  Livshits
• Fast update of , det
  Rozantsev
• (3n 1) dimensional Schrödinger eq.
  FilinovReal-time Feynmann path integrals
  Zlochin multiscale electronic-density
  functional
• DFT electronic structures Livne,
  Livshits molecular dynamics
• Molecular dynamicsFluids
  Ilyin, Suwain, MakedonskaPolymers
  , proteins Bai,
  KlugMicromechanical structures
  Ghoniem defects, dislocations, grains
• Navier Stokes Turbulence
  McWilliams Dinar, Diskin

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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

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Cure Multiscale Computation

• Define a Coarser system
• Derive equations (or probabilistic rules)
  governing the coarse system
• Move similarly to a still-coarser system
  etc.

• Small computational volumes at each scale
• No need to sum far interactions
• No slowness
• Leading to macroscopic equations
  (or tabled rules) of the
  material

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Exact Quantum Mechanics
n masses m1, , mn
located at r1, , rn rj(xj, yj,
zj)
Forces
potential V(r1, , rn)
Classical r(t)
Probability amplitude y(r1, , rn, t)
Approximations Born-Oppenheimer Hartree-Fock
Local density perturbations
Direct Numerical Real-time path integrals
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F cycle