MULTISCALE COMPUTATIONAL METHODS

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MULTISCALE COMPUTATIONAL METHODS

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

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Poisson equation
given
Approximating Poisson equation
given
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u given on the boundary
h
e.g., u function of u's and f
Solution algorithm
approximating Poisson eq.
Point-by-point RELAXATION
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Fast error smoothingslow solution
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Relaxation of linear systems
Axb
Eigenvectors
Relaxation sweep
Fast convergence of high modes
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When relaxation slows down
the error is a sum of low eigen-vectors
ELLIPTIC PDE'S (e.g., Poisson equation)
the error is smooth
The error can be approximated on a coarser grid
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h
LhUhFh
LUF
2h
L2hU2hF2h
4h
L4hU4hF4h
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TWO GRID CYCLE
MULTI-GRID CYCLE
Fine grid equation
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1. Relaxation
Approximate solution
Smooth error
Residual equation
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residual
2. Coarse grid equation
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4
Approximate solution
by recursion
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3. Coarse grid correction
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4. Relaxation
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Full MultiGrid (FMG) algorithm
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4
4
3
4
4
4
2
1
multigrid cycle V
interpolation (order lp) to a new grid
residual transfer
enough sweeps or direct solver
relaxation sweeps

algebraic error lt truncation error
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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 Full
  matrix
• Statistical mechanics
• Massive parallel processing
• Rigorous quantitative analysis (1986)

(1977,1982)
FAS (1975)
Within one solver
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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 /
1. Localness of processing
2. Attraction basins

• Multiple solutions

Inverse problems / Optimization
Many eigenfunctions
Statistical sampling
Removed by multiscale algorithms
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Computational bottlenecks

• Elementary particles
• Physics standard model

• Chemistry, materials science

Schrödinger equation
Molecular dynamics forces

• (Turbulent) flows
• Partial differential equations

• Vision recognition

• Seismology
• Tomography (medical imaging)
• Graphs data mining,
• VLSI design

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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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Full MultiGrid (FMG) algorithm
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Two Grid Cycle for solving
1. Fine grid relaxation
Full Approximatioin Scheme (FAS)
defect correction
Goto 1
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h
LhUh Fh
LU F
4
2
interpolation of changes
2h
L2hU2h F2h
Fine-to-coarse defect correction
Truncation error estimator
4h
L4hU4h F4h
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Coarse-Grid Aliasing