Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
An Introduction to Multigrid Techniques
BOBBY PHILIP
Computer Science and Mathematics Division
Oak Ridge National Laboratory, U.S.A.
[email protected]
CIMPA Research School
Indian Institute of Science,
July 12, 2013
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
1
This research was conducted in part under the auspices of the Office of
Advanced Scientific Computing Research, Office of Science, U.S. Department
of Energy under Contract No. DE-AC05- 00OR22725 with UT-Battelle, LLC.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Elliptic Model Examples
−∇ · ∇φ = f
−∇ · D(x)∇φ = f
−∇ · D(φ)∇φ = f
defined in the interior of a domain Ω with boundary ∂Ω.
Boundary conditions:
I Dirichlet
I Neumann
I Robin
I Others..
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Discretization and Grids
Discretization:
I Finite difference
I Finite volume
I Finite element
I Spectral element
I Mimetic schemes
I ....
Types of grids:
I Structured uniform grids
I Block structured grids (locally uniform)
I Unstructured grids
Numerical approximation results in sparse discrete linear and
nonlinear systems of coupled equations.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Terminology
I
Let v h denote an approximate to the exact solution, u h
I
The error: e h = u h − v h
I
The residual: r h = f h − Ah v h
I
The error satisfies the residual equation: Ah e h = r h
r h = f h − Ah u h = Ah u h − Ah v h = Ah (u h − v h ) = Ah e h
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Stationary iterative processes
Splitting A as A = M − N yields the iterative process
u n+1 = u n + M −1 r n
Examples include (damped) Jacobi and Gauss-Seidel
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Smoothers
Error after 1 smoothing steps
Error after 10 smoothing steps
0.9
0.7
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0.1
0
35
35
30
30
25
25
30
20
30
20
25
15
20
25
15
15
10
20
15
10
10
5
10
5
5
0
5
0
0
0
Error after 100 smoothing steps
Error after 50 smoothing steps
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
35
35
30
30
25
30
20
25
15
20
25
30
20
25
15
20
15
10
10
5
5
0
0
BOBBY PHILIP
15
10
10
5
5
0
0
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Coarse Grid Correction
Smooth error can be represented well on a coarser mesh:
Error after 10 smoothing steps
Error on a coarse grid
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
18
35
16
30
14
25
30
20
25
15
20
15
10
10
5
16
14
10
12
8
10
6
8
6
4
4
2
5
0
12
2
0
0
0
Smooth error appears more oscillatory on a coarser mesh.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Grid Aliasing
Oscillatory error gets aliased into smooth error on a coarser mesh:
sin(30*pi*x)*sin(30*pi*y) on 32x32 grid
sin(30*pi*x)*sin(30*pi*y) on 16x16 grid
25
20
25
15
20
15
10
10
10
10
5
5
0
5
5
0
0
BOBBY PHILIP
0
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Two Grid Algorithm
This suggests the following two grid algorithm:
I
(Pre) Smooth on Ah uh = fh
I
Compute the residual: rh = fh − Ah uh . (Note: Ah eh = rh )
I
Solve the coarse residual equation: AH eH = Rrh
I
Correct the fine grid approximation: uh ←− uh + PeH
I
(Post) Smooth on Ah uh = fh
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Multigrid µ-Cycle
Algorithm: uh ←− MG µ(uh , fh , ν1 , ν2 )
if (Ωh coarsest grid) then
uh ←− (Ah )−1 fh
else
Pre-smooth: ν1 times on Ah uh = fh with initial guess uh
Restrict residual: fH ←− R(fh − Ah uh )
Initial guess:
uH ←− 0
Correct:
uH ←− MG µ(uH , fH , ν1 , ν2 ) µ times
Update:
uh ←− uh + PuH
Post-smooth: ν2 times on Ah uh = fh with initial guess uh
endif
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Smooth
Restrict
Update
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: W-Cycle
Ωh Smooth
ΩH
Smooth
Restrict
Smooth
Smooth
Restrict
Update
Smooth
Update
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Review: Multigrid components
I
Grid Coarsenings
I
Smoothers
I
Restriction operators
I
Coarse grid operators
I
Prolongation operators
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Full Approximation Scheme (FAS)
I
The Full Approximation Scheme is a multigrid method for
nonlinear problems
I
Discrete nonlinear problem: Ah (uh ) = fh on the finest level,
where Ah is a nonlinear operator
I
Error for a given approximation is uh − vh
I
Nonlinear residual is rh = fh − Ah (vh ) = Ah (uh ) − Ah (vh )
I
Given these ingredients how do we construct a multilevel
algorithm?
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS:Two Grid Algorithm
I
Smooth (nonlinearly) on Ah (vh ) = fh with initial guess vh
I
Compute residual: rh = fh − Ah (vh )
I
Restrict the residual: fH = Rrh
I
Solve the coarse residual equation
I
Correct the fine grid approximation
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS:Two Grid Algorithm
I
What is the coarse residual equation?
I
rH
= fH − AH (vH )
= AH (uH ) − AH (vH )
= AH (vH + eH ) − AH (vH )
I
Coarse residual equation:
AH (vH + eH ) − AH (vH ) = rH
I
I
Approximate vH = R̂vh and rH = Rrh
Now we have:
AH (R̂vh + eH ) − AH (R̂vh ) = Rrh
I
or equivalently:
ABOBBY
) = Introduction
AH (R̂vhto) +
Rrh
H (wH
PHILIP
Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
I
(Pre) Smooth (nonlinearly) on Ah (vh ) = fh with initial guess
vh
I
Compute residual: rh = fh − Ah (vh )
I
Restrict the residual: fH = Rrh
I
Restrict the solution: vH = R̂vh
I
Solve the coarse residual equation
AH (wH ) = AH (R̂vh ) + Rrh
with initial guess vH = R̂vh
I
Compute correction: eH = wH − R̂v h
I
Update fine grid approximation: vh = vh + PeH
I
(Post) Smooth on Ah (vh ) = fh with initial guess vh
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS Components
I
Nonlinear Smoothers
I
I
I
Nonlinear Jacobi-Newton
Nonlinear Gauss-Seidel-Newton
Coarse Grid Operator
I
I
Rediscretization
Nonlinear Galerkin Discretization
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS: τ -correction
I
Right hand side for coarse grid correction: AH (R̂vh ) + Rrh
I
Can be written as: Rfh + AH (R̂vh ) − R(Ah vh )
I
τ -correction: τhH = AH (R̂vh ) − R(Ah vh )
I
(h, H) truncation error estimate for adaptive mesh refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS: FMG
Algorithm: uh ←− FAS − FMG (fh ; µ, ν1 , ν2 )
if (Ωh coarsest grid) then
uh ←− (Ah )−1 fh
else
Restrict RHS:
Coarse approximation:
Initial guess:
Solve:
endif
fH ←− Rfh
uH ←− FAS − FMG (fH , ν)
uh ←− ΠhH uH
uh ←− FASµ(uh , fh , ν1 , ν2 )
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS-FMG-Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS-FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS-FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS-FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
FAS-FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
Many flavors:
I
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
Many flavors:
I
I
h-refinement: variants include: structured AMR, block
structured AMR, cell based AMR (unstructured)
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
Many flavors:
I
I
I
h-refinement: variants include: structured AMR, block
structured AMR, cell based AMR (unstructured)
p-refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
Many flavors:
I
I
I
I
h-refinement: variants include: structured AMR, block
structured AMR, cell based AMR (unstructured)
p-refinement
r -refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Mesh Refinement
I
Adaptive Mesh Refinement (AMR) is a numerical technique
to introduce local grid resolution only where required.
I
Benefits are significant savings in memory and computational
cost.
I
Obviously not as easy to implement as a single grid
calculation.
Many flavors:
I
I
I
I
I
h-refinement: variants include: structured AMR, block
structured AMR, cell based AMR (unstructured)
p-refinement
r -refinement
combinations of the above, e.g., hp-refinement.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Uniform Grid Refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Uniform Grid Refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Degrees
freedom
Error
MU
WU
of
400
1600
6400
25600
10.88
4/3
7/2
0.589
16/3
14
0.062
64/3
56
0.016
256/3
224
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Grid Refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Adaptive Grid Refinement
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Number
of
refinement
levels
Degrees
of
freedom
Error
MU
WU
1
2
3
4
400
800
1200
1600
10.88
4/3
7/2
0.319
16/3
10
0.02
28/3
13
0.001
40/3
16
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Structured Adaptive Mesh Refinement
Structured adaptive mesh refinement (SAMR) represents a locally
refined mesh as a union of logically rectangular meshes.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Hierarchical Structure of SAMR Grids
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
AMR Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
AMR Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
AMR Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
AMR Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
AMR Cycle
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Inverting Elliptic Components
Choices:
I
Algebraic - ILU etc
I
Level by Level Solver
I
Multigrid
I
Multilevel Methods
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Inverting Elliptic Components
Choices:
I
Algebraic - ILU etc
I
Level by Level Solver
I
Multigrid
I
Multilevel Methods
Fast Adaptive Composite (FAC) Grid method:
I
Extension of multigrid to work on AMR grids.
I
Uses smoothing only on local patches.
I
Level independent convergence rate
I
V-cycle version optimal
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Compute residual: r 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Smooth: A3 e 3 = r 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Update: u 3 ← u 3 + e 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Update: u 3 ← u 3 + e 3
Compute residual:r 2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Update: u 3 ← u 3 + e 3
Smooth: A2 e 2 = r 2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Solve/smooth: A1 e 1 = r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + I12 e 1
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Compute residual: r 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Smooth: A2 e 2 = r 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 3 ← u 3 + I23 e 2
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Compute residual: r 3
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Smooth: A3 e 3 = r 3
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
Update: u 3 ← u 3 + e 3
Update: u 3 ← u 3 + e 3
Update: u 2 ← u 2 + e 2
Update: u 2 ← u 2 + e 2
Update: u 1 ← u 1 + e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Compute residual: r 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Ωh2
Compute residual: r 3
Compute residual:r 2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Ωh2
Ωh1
Compute residual: r 3
Compute residual:r 2
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Ωh2
Ωh1
Smooth: A3 e 3 = r 3
Smooth: A2 e 2 = r 2
Solve: A1 e 1 = r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Ωh3
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Form Correction: e 3 ← e 3 + Pe 2
Ωh3
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Breaking Synchrony
Ωhc
Update: u 3 ← u 3 + e 3
Ωh3
Update: u 2 ← u 2 + e 2
Ωh2
Update: u 1 ← u 1 + e 1
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Compute residual:r 2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Compute residual:r 2
Ωh2,r
Ωh1
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
Compute residual: r 3
Copy residual: rr3 ← r2
Compute residual:r 2
Copy residual:rr2 ← r1
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
Solve: A3 e 3 = r 3
Solve: A2r er2 = rr2
Solve: A2 e 2 = r 2
Solve: A1r er1 = rr1
Solve: A1 e 1 = r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Correct: e 3 ← e 3 − Per3
Ωh3,r
Ωh2
Correct: e 2 ← e 2 − Per2
Ωh2,r
Ωh1
Correct: e 1 ← e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Ωh3
Ωh3,r
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Form Correction: e 3 ← e 3 + Pe 2
Ωh3
Ωh3,r
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFAC
Ωhc
Update: u 3 ← u 3 + e 3
Ωh3
Ωh3,r
Update: u 2 ← u 2 + e 2
Ωh2
Ωh2,r
Update: u 1 ← u 1 + e 1
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Compute residual:r 2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Compute residual: r 3
Ωh3,r
Ωh2
Compute residual:r 2
Ωh2,r
Ωh1
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Ωh3,r
Ωh2
Ωh2,r
Ωh1
Compute residual: r 3
Copy residual: rr3 ← r2
Compute residual:r 2
Copy residual:rr2 ← r1
Compute residual: r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Ωh3,r
Smooth: A2r er2 = rr2
Ωh2
Ωh2,r
Smooth: A1r er1 = rr1
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Smooth A3 e 3 = r 3 , initial guess Pe2,r
Ωh3,r
Ωh2
Smooth A2 e 2 = r 2 , initial guess Pe2,r
Ωh2,r
Ωh1
Solve A1 e 1 = r 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Correct: e 3 ← e 3 − Per3
Ωh3,r
Ωh2
Correct: e 2 ← e 2 − Per2
Ωh2,r
Ωh1
Correct: e 1 ← e 1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Ωh3
Ωh3,r
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Form Correction: e 3 ← e 3 + Pe 2
Ωh3
Ωh3,r
Form Correction: e 2 ← e 2 + Pe 1
Ωh2
Ωh2,r
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Ωhc
Update: u 3 ← u 3 + e 3
Ωh3
Ωh3,r
Update: u 2 ← u 2 + e 2
Ωh2
Ωh2,r
Update: u 1 ← u 1 + e 1
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms
I
Fast Adaptive Composite (FAC) Grid method:
I
I
Extension of multigrid to work on AMR grids.
Uses smoothing only on local patches.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms
I
Fast Adaptive Composite (FAC) Grid method:
I
I
I
Extension of multigrid to work on AMR grids.
Uses smoothing only on local patches.
AFAC: Asynchronous FAC.
I
I
I
I
Uses restricted grids (coarsenings of each refinement level).
Uses direct solvers or multigrid for each level/restricted level.
Convergence rate the square root of that for FAC.
Good parallelism.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms
I
Fast Adaptive Composite (FAC) Grid method:
I
I
I
AFAC: Asynchronous FAC.
I
I
I
I
I
Extension of multigrid to work on AMR grids.
Uses smoothing only on local patches.
Uses restricted grids (coarsenings of each refinement level).
Uses direct solvers or multigrid for each level/restricted level.
Convergence rate the square root of that for FAC.
Good parallelism.
AFACx:
I
I
I
I
Uses restricted grids.
Uses smoothers only
Cheaper than AFAC
Convergence rate comparable to AFAC.
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
Software
I
BoxMG, parallel open source geometric black box multigrid
solver (LANL)
I
SMG, PFMG, BoomerAMG, parallel open source multigrid
solvers (LLNL)
I
PETSc parallel multigrid solver (ANL)
I
LAMG, parallel algebraic multigrid solver (LANL)
I
SAMRSolvers, Multilevel FAC, AFAC, AFACx solvers,
ORNL/Philip
BOBBY PHILIP
Introduction to Multigrid
Review
Full Approximation Scheme (FAS)
Structured Adaptive Mesh Refinement
Conclusion
References
I
Multigrid Tutorial, Briggs, Henson, McCormick, SIAM
I
Multigrid, Trottenberg, Oosterlee, Schuller
I
An Introduction to Multigrid Methods, Wesseling
I
Multigrid Methods and Applications, Hackbusch
I
Multigrid methods, Bramble
I
Multigrid Adaptive Methods for PDEs, McCormick
BOBBY PHILIP
Introduction to Multigrid