multiscale computational methods
DESCRIPTION
MULTISCALE COMPUTATIONAL METHODS. Achi Brandt The Weizmann Institute of Science UCLA www.wisdom.weizmann.ac.il/~achi. Poisson equation:. given. Approximating Poisson equation:. given. Solving PDE : Influence of pointwise relaxation on the error. Error of initial guess. - PowerPoint PPT PresentationTRANSCRIPT
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MULTISCALE COMPUTATIONAL
METHODS
Achi BrandtThe Weizmann Institute of ScienceUCLA
www.wisdom.weizmann.ac.il/~achi
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0u 1u2u
3u4u
given
02
043214
Fh
uuuuu
Poisson equation:2 2
2 2
u uF
x y
Approximating Poisson equation:
given
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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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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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LU=F
h
2h
4h
LhUh=Fh
L2hU2h=F2h
L4hU4h=F4h
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hLhUh = Fh
LocalRelaxation
hu~approximation
hV hh u~U smooth
2h
4h
L2hU2h = F2h
L4hV4h = R4h
L2hV2h = R2h R2h = ( Fh -Lh )hh2 hu~
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interpolation (order l+p)to a new grid
interpolation (order m)of corrections
relaxation sweeps
algebraic error< truncation error
residual transfer
ν νenough sweepsor direct solver*
2ν
Full MultiGrid (FMG) algorithm
..
.
*
1ν
1ν
1ν
2ν
2ν
2ν
h0
h0/2
h0/4
2h
h
**
1ν
1ν
1ν2ν
*
2ν
2ν
2ν
multigrid cycle V
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Multigrid solversCost: 25-100 operations per unknown
• Linear scalar elliptic equation (~1971)
![Page 9: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/9.jpg)
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)
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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 11: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/11.jpg)
interpolation (order l+p)to a new grid
interpolation (order m)of corrections
relaxation sweeps
algebraic error< truncation error
residual transfer
ν νenough sweepsor direct solver*
**
1ν
1ν
1ν2ν
*
2ν
2ν
2ν
Full MultiGrid (FMG) algorithm
..
.
*
1ν
1ν
1ν
2ν
2ν
2ν
Vcyclemultigrid
h0
h0/2
h0/4
2h
h
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h
2h
4h
LhUh = Fh
L4hU4h = F4h
h2
h4
Fine-to-coarse defect correction
L2hV2h = R2h
Full Approximation scheme (FAS):
U2h = + V2h L2hU2h = F2h
LocalRelaxation
hu~approximation
hV hh u~U smooth
hu~hh2
![Page 13: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/13.jpg)
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
Massive parallel processing*Rigorous quantitative analysis
Within one solver
![Page 14: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/14.jpg)
interpolation (order l+p)to a new grid
interpolation (order m)of corrections
relaxation sweeps
algebraic error< truncation error
residual transfer
ν νenough sweepsor direct solver*
**
1ν
1ν
1ν2ν
*
2ν
2ν
2ν
Full MultiGrid (FMG) algorithm
..
.
*
1ν
1ν
1ν
2ν
2ν
2ν
Vcyclemultigrid
h0
h0/2
h0/4
2h
h
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h
2h
4h
LhUh = Fh
L4hU4h = F4h
h2
h4
Fine-to-coarse defect correction
L2hV2h = R2h
Full Approximation scheme (FAS):
U2h = + V2h L2hU2h = F2h
LocalRelaxation
hu~approximation
hV hh u~U smooth
Truncation error estimator
hu~hh2
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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 17: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/17.jpg)
• Same fast solver FMG
Local patches of finer grids
• Each patch may use different coordinate system and anisotropic grid and different
physics; e.g. Atomistic
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interpolation (order l+p)to a new grid
interpolation (order m)of corrections
relaxation sweeps
algebraic error< truncation error
residual transfer
ν νenough sweepsor direct solver*
2ν
Full MultiGrid (FMG) algorithm
..
.
*
1ν
1ν
1ν
2ν
2ν
2ν
h0
h0/2
h0/4
2h
h
**
1ν
1ν
1ν2ν
*
2ν
2ν
2ν
multigrid cycle V
![Page 19: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/19.jpg)
• Same fast solver FMG,
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
• Same fast solver FMG, FAS
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x , y
)(
)(
sy
sx
r , s
Finer level with local coordinate transformation
Boundary or tracked layer
With anisotropic further refinement
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• Same fast solver FMG,
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
• Same fast solver FMG, FAS
![Page 22: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/22.jpg)
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* (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)
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Stokes
0
0
0
yx
y
x
vu
Pv
Pu
0
0
0
0
0
0
p
v
u
yx
y
x
2 yyxxL
L
det
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h-principal LLu f Ldet
0
v
u
Riemann
Cauchy
xy
yx
Compressible Navier-Stokes(on the viscous scale)
uk3
2
Central Cauchy-Riemannh2
Central (Navier-) Stokeshh
Q2
Stokes
2D
0
0
0
yx
y
x
20
p
v
u
StokesNavier
ibleIncompress
2D
Q
0
0
0
yx
y
x
Q
Q
1Q u R
p
v
u
Euler
2D
1
0
00
0
0
0
0
0
0
0
0
0 y
x
y
y
x
x
P
u
P
u
u
u
22
23
au
xu
PPa p2
2
p
v
u
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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) • 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
Massive parallel processing*Rigorous quantitative analysis
![Page 26: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/26.jpg)
![Page 27: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/27.jpg)
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
Massive parallel processing*Rigorous quantitative analysis
![Page 28: MULTISCALE COMPUTATIONAL METHODS](https://reader035.vdocuments.mx/reader035/viewer/2022062323/5681595b550346895dc69966/html5/thumbnails/28.jpg)
ALGEBRAIC MULTIGRID (AMG) 1982
Ax = b
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When relaxation slows down:
DISCRETIZED PDE'S
GENERAL SYSTEMS OF LOCAL EQUATIONS
the error is smooth
Along characteristics
The error can be approximated
by a far fewer degrees
of freedom (coarser grid)
the error is a sum of low eigen-vectors
ELLIPTIC PDE'S
the error is smooth
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ALGEBRAIC MULTIGRID (AMG) 1982
Coarse variables - a subset
Criterion: Fast convergence of “compatible relaxation”
Ax = b
Relax Ax = 0Keeping coarse variables = 0
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ALGEBRAIC MULTIGRID (AMG) 1982
Coarse variables - a subset
1. “General” linear systems2. Variety of graph problems
General procedures for deriving:* Interpolations
Ax = b
* Restriction* Coarse-level equations
Generalizations:
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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
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