Docstoc

Slide 1 - IPAM

Document Sample
Slide 1 - IPAM Powered By Docstoc
					         MULTISCALE
        COMPUTATION:

     From Fast Solvers
  To Systematic Upscaling




A. Brandt
The Weizmann Institute of Science
UCLA

www.wisdom.weizmann.ac.il/~achi
Major scaling bottlenecks:
computing
Elementary particles (QCD)

Schrödinger equation
  molecules
  condensed matter

Molecular dynamics
 protein folding, fluids, materials

Turbulence, weather, combustion,…

Inverse problems
  da, control, medical imaging

Vision, recognition
  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
 1. Localness of processing
 Moving one particle at a
         time


        fast local ordering




small
step



        slow global move
    Solving PDE: Influence of pointwise
                relaxation on the error




    Error of initial guess     Error after 5 relaxation
                                                sweeps




Error after 10 relaxations   Error after 15 relaxations



      Fast error smoothing
                slow solution
  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
 1. Localness of processing
 2. Attraction basins
 Optimization                          min E(r)




E(r)




                                                  r
       multi-scale attraction basins
  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
 1. Localness of processing
 2. Attraction basins



  Removed by
    multiscale processing
    Solving PDE: Influence of pointwise
                relaxation on the error




    Error of initial guess     Error after 5 relaxation
                                                sweeps




Error after 10 relaxations   Error after 15 relaxations



      Fast error smoothing
                slow solution
         h
LU = F        LhUh = Fh




         2h
              L2hV2h = R2h
              L2hU2h = F2h




         4h
              L4hV4h = R4h


         
  Multigrid solvers
  Cost: 25-100 operations per unknown
• Linear scalar elliptic equation   (~1971)
  Multigrid solvers
  Cost: 25-100 operations per unknown
• Linear scalar elliptic equation    (~1971)
• Nonlinear                         FAS (1975)
                             h
        LU = F                              LhUh = Fh




                             2h
                                                        2h 
                                                               
                                                                   2h
2h     h,approximate    2h                  2h   2h     2h
U =U                   +V                    U  F
                                           L V =R
                             Fine-to-coarse defect correction




                             4h
                                                        4h 
                                                               
                                                                   4h
                                            4h   4h
                                           L U =F


                             
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
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
    and differet physics; e.g. atomistic

    “Quasicontiuum” method [B., 1992]
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE) AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
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
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
1D Wave Equation: u”+k2u=f

Non-local components:

e iwx, w ≈ ±k
Slow to converge in local processing


                         2p/w
                       wavelength


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:
                              2h
     A1 + 2 i k A1′ = f1 =       rh(x) e-ikx
                              h
2D Wave Equation: Du+k2u=f
                                              w2
Non-local:                   (a3,b3)               (a2,b2)
                         (a4,b4)                       (a1,b1)
ei(w1 x + w2 y)                                k
                                                             w1
w12 + w22 ≈ k2            (a5,b5)
                                                        (a8,b8)

                     O(H)                          (a7,b7)
                                    (a6,b6)


On coarser grid (meshsize H):

                  cH
    v(x)   A r (x, y) ei (a r x  b r y)
                  r 1

Fully efficient multigrid solver
Tends to Geometrical Optics
Radiation Boundary Conditions:
              directly on coarsest level
        Generally: LU=F

Non-local part of U has the form
        m

       Σ
       r=1
             Ar(x) φr(x)


L φr ≈ 0
Ar(x) smooth

{φr } found by local processing
Ar represented on a coarser grid
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
       N eigenfunctions
Electronic structures (Kohn-Sham eq):

   V( x)ψi ( x)  i ψi ( x)
                 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
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations                  Full matrix
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
Integro-differential Equation
    Lu ( x)   G ( x, y ) u ( y ) dy
     differential
     nn
 Au f ,            dense     A


Multigrid solver

Distributive relaxation:
       1st order                   
       2nd order                    2   


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

V(x) 
                Ω
                     G(x,  u( ) d           x  '




    G(x,                Transform     Complexity

    ix                                O(n logn)
e                          Fourier

    - x 
e                          Laplace      O(n logn)
    - | x -  |2 /  2                   O(n)
e                          Gauss
   1
                           Potential     O(n)
| x - |

G(x, *Exp(ik Waves                  O(n logn)
           G(x,y) ~ 1 / | x – y |




              Glocal
Gsmooth




               s                        |x-y|




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

  s ~ next coarser scale



  O(n)                 not static!
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics            Monte-Carlo

Massive parallel processing
*Rigorous quantitative analysis               (1986)
Discretization       Lattice   L  L 
      for accuracy   ε : L ~ εq



Monte Carlo cost           ~   L Ld   z   2


         “volume factor”
         “critical slowing down”




Multiscale                     ~ ε 2

        Multigrid moves

        Many sampling cycles
        at coarse levels
    Multigrid solvers
    Cost: 25-100 operations per unknown
•   Linear scalar elliptic equation (~1971)*
•   Nonlinear                             FAS (1975)
•   Grid adaptation                      (1977,1982)
•   General boundaries, BCs*
•   Discontinuous coefficients
•   Disordered: coefficients, grid (FE)        AMG
•   Several coupled PDEs*                     (1980)
•   Non-elliptic: high-Reynolds flow
•   Highly indefinite: waves              u  k 2u  f
•   Many eigenfunctions (N)              O( N log N )
•   Near zero modes
•   Gauge topology: Dirac eq.
•   Inverse problems
•   Optimal design                   Within one solver
•   Integral equations
•   Statistical mechanics

Massive parallel processing
*Rigorous quantitative analysis               (1986)
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
    and differet physics; e.g. atomistic

    “Quasicontiuum” method [B., 1992]
UPSCALING:
Derivation of coarse equations
 in small windows




   Repetitive systems
  e.g., same equations everywhere
  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
 1. Localness of processing
 2. Attraction basins



  Removed by
    multiscale processing
   Systematic Upscaling

1. Choosing coarse variables


2. Constructing coarse-level
   operational rules
      equations
      Hamiltonian
Macromolecule




~ 10-15 second steps
  Systematic Upscaling
1. Choosing coarse variables

  Criterion:
          Fast convergence of
          “compatible relaxation”
  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
Macromolecule
Macromolecule




 Two orders of magnitude faster simulation
Fluids




 £   Total mass
 £   Total momentum
 £   Total dipole moment
 £   average location
1




    1




2
Windows


 Fine level :
 density ~ 1

Coarser level
  Larger density fluctuations
                      Fine level :
                      density ~  2

 Fine level
 density : 3


Still coarser level
Fluids




   Total mass:

          Summing   m(x)
Lower Temperature T

Summing also
    i w x
e            m(x)               v
                                    u

       w  u  2p , w  v  0


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 by
 Multiscale 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 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, …
         Upscaling Projects
• QCD (elementary particles):
  Renormalization multigrid                                      Ron
  BAMG solver of Dirac eqs. M x  f                   Livne, Livshits

  Fast update of M 1, det M                              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
THANK YOU

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:1
posted:10/5/2011
language:English
pages:58