Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
An Introduction to Multigrid Techniques
BOBBY PHILIP
Computer Science and Mathematics Division
Oak Ridge National Laboratory, U.S.A.
CIMPA Research School
Indian Institute of Science,
July 10, 2013
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Overview
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Multigrid
Smoothing
Coarse Grid Correction
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Some 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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Numerical methods for linear systems
I
Linear systems
I
Direct methods
I
I
I
I
Iterative methods
I
I
I
I
Gaussian elimination O(N 3 ), (O(N 7/3 ) when banded, in 3D)
Cyclic reduction O(Nlog (N))
Fourier methods O(Nlog (N)))
Simple stationary iterations (Jacobi, Gauss-Seidel etc)
Krylov subspace methods O(N 4/3 log ())
Full Multigrid (FMG) O(N)
Nonlinear systems
I
I
I
Newton-Krylov methods
Newton-Multigrid
Full Approximation Scheme (FAS)
BOBBY PHILIP
Introduction to Multigrid
Estimated Required Solution Time (s)
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Cholesky
Jacobi
GS
Band Cholesky
CG−MIC(0)
Optimal
20
10
15
10
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
age of the universe
10
10
60 years
5
10
one minute
0
10
−5
10
0
10
3
10
6
10
Problem Size
(Courtesy, Scott Maclachlan, Tufts)
BOBBY PHILIP
Introduction to Multigrid
9
10
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
To introduce linear multigrid we are going to work with:
I
Poissons equation with homogenous Dirichlet boundary
conditions
I
Uniform single grid
I
Vertex centered finite difference discretization
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Discretize on a uniform mesh:
For simplicity assume nx = ny = n, hx = hy .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Discretize on a uniform mesh:
(0, ny + 1)
(nx + 1, ny + 1)
(0, 0)
(nx + 1, 0)
For simplicity assume nx = ny = n, hx = hy .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Discretize on a uniform mesh:
(0, ny + 1)
(nx + 1, ny + 1)
(0, 0) hx
(nx + 1, 0)
For simplicity assume nx = ny = n, hx = hy .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
Discretize on a uniform mesh:
(0, ny + 1)
(nx + 1, ny + 1)
hy
(0, 0) hx
(nx + 1, 0)
For simplicity assume nx = ny = n, hx = hy .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
At any interior point (i, j) discretize using the standard 5 point
finite difference stencil:
(i, j + 1)
(i − 1, j)
(i, j)
(i + 1, j)
(i, j − 1)
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
h
h − uh
−ui−1,j
+ 2ui,j
i+1,j
h2
+
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
h
h − uh
−ui,j−1
+ 2ui,j
i,j+1
h2
= fi,jh ,
with 1 ≤ i, j ≤ n.
or equivalently
h − uh
h
h
h
4ui,j
i−1,j − ui+1,j − ui,j−1 − ui,j+1
h2
= fi,jh ,
with 1 ≤ i, j ≤ n.
This yields the linear system
Ah u h = f h .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion

1
A = 2
h
h





B
−I
·
BOBBY PHILIP
Elliptic Model Problems
Discretizations and Grids
Linear and Nonlinear Solvers
−I
B
·
·

−I
·
·
−I




−I 
B
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
I
Hard to invert A directly
I
Easier to solve a ”nearby” system Mv h = f h where M ≈ A
I
e h = u h − v h =⇒ Ah e h = r h
I
Approximate e h using M instead of A in residual equation
I
Leads to the iteration vn+1 = vn + M −1 rn
I
Error propagation:
en+1 = en − M −1 rn
= en − M −1 Aen
= (I − M −1 A)en
= (I − M −1 A)n+1 e0
I
Iteration converges iff ρ(I − M −1 A) < 1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Stationary iterative processes
Can also be considered as splitting A as:
A = M − N.
This yields the iterative process:
Mu n+1 = Nu n + f ,
or
u n+1 = M −1 Nu n + M −1 f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Stationary iterative processes
Rewriting the previous iteration we have:
u n+1 = M −1 (M − A)u n + M −1 f
= u n − M −1 Au n + M −1 f
= u n + M −1 (f − Au n )
= u n + M −1 r n
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Stationary iterative processes
Let A = D − L − U
Common examples of stationary iterations:
I
Jacobi:
u n+1 = u n + D −1 r n
e n+1 = (I − D −1 A)e n
I
Damped Jacobi:
u n+1 = u n + ωD −1 r n
I
Gauss-Seidel:
u n+1 = u n + (D − L)−1 r n
u n+1 = (I − (D − L)−1 A)e n
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
2D Poisson example
I
Iteration matrix for damped Jacobi:
2
(I − ωD −1 Ah ) = (I − ωh4 Ah )
I
Error damping for ω-Jacobi : e m = (I −
I
Eigenfunctions: φk,l (x, y ) = sin(kπx)sin(lπy ),
(x, y ) ∈ Ωh , k, l = 1, 2, · · · , n − 1
I
2 lπh
Eigenvalues: λk,l = 1 − ω(sin2 ( kπh
2 ) + sin ( 2 ))
I
Spectral radius ≈ 1 − ωπh
2
Pn−1
P
0
m
e = k,l=1 αk,l φk,l =⇒ e m = n−1
k,l=1 αk,l λk,l φk,l
I
ωh2 h m 0
4 A ) e
2
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Damped Jacobi with ω = 4/5 on k = 1, l = 1 error mode
Error before Jacobi smoothing steps
Error after 5 smoothing steps
1
1
0.9
0.9
0.8
0.8
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
35
35
30
30
25
25
30
20
25
15
30
20
20
10
25
15
15
20
15
10
10
5
10
5
5
0
5
0
0
0
Convergence rate: krk k/krk−1 k = 0.996
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Damped Jacobi with ω = 4/5 on k = 16, l = 16 error mode
Error before Jacobi smoothing steps
Error after 5 smoothing steps
−4
x 10
1
4
0.8
3
0.6
2
0.4
1
0.2
0
0
−0.2
−1
−0.4
−2
−0.6
−3
−0.8
−1
−4
35
35
30
30
25
25
30
20
30
20
25
15
15
10
25
15
20
20
15
10
10
5
10
5
5
0
5
0
0
0
Convergence rate: krk k/krk−1 k = 0.2
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Damped Jacobi with ω = 4/5 on
sin(πx)sin(πy ) + sin(16πx)sin(16πy ) + sin(25πx)sin(25πy ) mixed
error mode
Error before Jacobi smoothing steps
Error after 5 smoothing steps
3
1
2.5
0.9
2
0.8
1.5
0.7
1
0.6
0.5
0.5
0
0.4
−0.5
0.3
−1
0.2
−1.5
0.1
−2
0
35
35
30
30
25
25
30
20
25
15
20
15
10
30
20
25
15
20
15
10
10
5
5
0
0
10
5
5
0
BOBBY PHILIP
0
kr k
1.30277e+02
2.00318e+01
8.33307e+00
3.49242e+00
1.48631e+00
6.79236e-01
3.93535e-01
3.17248e-01
3.00978e-01
2.97096e-01
2.95470e-01
2.94248e-01
2.93099e-01
2.91968e-01
Introduction to Multigrid
Conv. Factor
1.000
0.407
0.416
0.419
0.426
0.457
0.579
0.806
0.949
0.987
0.995
0.996
0.996
0.996
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
I
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
For elliptic equations
ρ(I − M −1 A) ≈ (1 − O(h2 )) −→ 1
as the mesh is refined.
I
Hence, stationary iterative methods such as damped Jacobi,
G-S, etc are not good solvers.
I
However, they are excellent smoothers of error.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Try to obtain a coarse approximation to the error
After smoothing, the information available about the error is in the
residual.
Ah eh = rh
Since eh is smooth it can be approximated on a coarser grid.
The coarse approximation to eh is obtained by solving
AH eH = rH
where H = 2h typically and AH and rH are defined on a coarse grid.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Two Grid Algorithm
I
The process of
I
I
I
I
Restriction to the coarser grid
Solving the coarse residual equation
Prolongation of the correction to the fine grid
Updating the current fine grid approximation
is called Coarse Grid Correction (CGC).
I
Solving the coarse residual equation can be accomplished by
smoothing and CGC from a still coarser grid.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Restriction: R
Restriction operator R transfers data from Ωh to ΩH
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Restriction: R
Restriction operator R transfers data from Ωh to ΩH
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Restriction: R
Restriction operator R transfers data from Ωh to ΩH
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Restriction: R
Restriction operator R transfers data from Ωh to ΩH
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Interpolation: P
I
Interpolation operator P transfers data from ΩH to Ωh
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Interpolation: P
I
Interpolation operator P transfers data from ΩH to Ωh
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Interpolation: P
I
Interpolation operator P transfers data from ΩH to Ωh
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Interpolation: P
I
Interpolation operator P transfers data from ΩH to Ωh
I
Operator dependent interpolation for more difficult probems
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Coarse Grid Operators
I
Rediscretization on the coarse grid
I
Variational Galerkin coarse grid operators: AH = RAh P
I
Operators based partially on Galerkin that try and maintain
sparsity
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FEM: Canonical Projection and Restriction Operators
I
I
I
Let φjh , j = 1, 2, · · · , Nh , nodal basis for ‘fine’ space Vh ,
φjH , j = 1, 2, · · · , NH , basis for ‘coarse’ subspace VH
P h
i
φjH = N
i=1 pij φh , j = 1, 2, · · · , NH .
For vH ∈ VH :
vH
=
NH
X
vHj φjH
j=1
=
I
j=1
vHj
Nh
X
pij φih
i=1
Nh X
NH
Nh
X
X
(
pij vHj )φih =
[PvH ]i φih
i=1 j=1
I
=
NH
X
i=1
In vector form vh = PvH .
R = Pt
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FEM: Canonical Coarse Grid Operators
I
Matrix form: Solve AH vH = R(fh − Ah uh )
I
Variational form: For uh ∈ Vh , find v ∈ VH such that
a(uh + v , w ) = f (w ) ∀w ∈ VH
I
a(v , w ) = f (w ) − a(uh , w )
P H s s
Choose w = φjH , j = 1, 2, · · · , NH , v = N
s=1 vH φH
I
Some error prone linear algebra leads to:
(RAh P)vH
= R(fh − Ah uh )
AH vH
= R(fh − Ah uh )
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Problems with varying anisotropy
I
Example from Klaus Stüben.
I
The underlying PDE is:
−(aux )x − (buy )y + cuxy = f (x, y )
defined on a unit square with full Dirichlet boundary
conditions.
I
The problem is defined such that a = b = 1 everywhere
except in the upper left quarter of the unit square where
b = 103 and the lower right quarter where a = 103
I
To split the domain into four regions with varying anisotropies,
c = 0 except in the upper right quarter where c = 2.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Error with different smoothers
0.7
0.4
0.6
0.5
0.3
0.4
0.2
0.3
0.2
0.1
0.1
0
0.8
0.2
0.6
0.4
0.4
0.6
0.8
0.2
BOBBY PHILIP
0.8
0.2
0.6
0.4
0.4
0.6
0.8
Introduction to Multigrid
0.2
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Block Smoother
Block smoother created 285 blocks on the 961 × 961 system.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
BOBBY PHILIP
0.6
0.8
1
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Convergence Results
Pointwise
kr k
Conv. Fac.
1.704e+007
1.00000
2.580e+006
0.15138
7.049e+005
0.27318
3.736e+005
0.53000
2.666e+005
0.71357
BOBBY PHILIP
Block
kr k
Conv. Fac.
1.704e+007
1.00000
2.908e+003
0.00017
9.142e+002
0.31444
6.060e+002
0.66285
4.764e+002
0.78607
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Smoothers and Coarse Grids
I
Smoothers
I
I
I
I
I
I
Point Smoothers
Line and Plane Smoothers
Block Smoothers
ILU smoothers
Algebraic smoothers
Coarse Grids
I
I
I
I
Uniform
Semicoarsening
Multiple coarsenings
Algebraic coarsenings
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Multigrid V-Cycle
Algorithm: uh ←− MGV (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 ←− MGV (uH , fH , ν1 , ν2 )
Update:
uh ←− uh + PuH
Post-smooth: ν2 times on Ah uh = fh with initial guess uh
endif
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Ω4h
Restrict
Smooth
Restrict
Smooth
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Ω4h
Restrict
Smooth
Restrict
Smooth
Restrict
Ω8h
Smooth/Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Ω4h
Restrict
Smooth
Restrict
Smooth
Smooth
Restrict
Update
Ω8h
Smooth/Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Ω4h
Restrict
Smooth
Smooth
Restrict
Smooth
Update
Smooth
Restrict
Update
Ω8h
Smooth/Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
V-Cycle
Ωh Smooth
ΩH
Ω4h
Smooth
Restrict
Smooth
Update
Smooth
Restrict
Smooth
Update
Smooth
Restrict
Update
Ω8h
Smooth/Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Restrict
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Restrict
Smooth
Smooth
Smooth
Restrict
Update
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
W-Cycle
Ωh Smooth
ΩH
Smooth
Restrict
Smooth
Smooth
Restrict
Update
Smooth
Update
Ω4h
Solve
BOBBY PHILIP
Solve
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Simplistic Nested Iteration
Algorithm: u h ←− SNI (f h ; ν)
if (Ωh coarsest grid) then
u h ←− (Ah )−1 f h
else
Restrict RHS:
f H ←− IhH f h
Coarse approximation: u H ←− SNI (f H , ν)
Smooth: ν times on Ah u h = f h with initial guess ΠhH u H
endif
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
h
f H = ΠH
hf
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
Ω4h
h
f H = ΠH
hf
H
f 4h = Π4h
H f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
Ω4h
h
f H = ΠH
hf
H
f 4h = Π4h
H f
Ω8h
4h
u 8h = (A8h )−1 Π8h
4h f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
Ω4h
h
f H = ΠH
hf
H
f 4h = Π4h
H f
Smooth
Initial guess
Ω8h
4h
u 8h = (A8h )−1 Π8h
4h f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
Ω4h
h
f H = ΠH
hf
Smooth
H
f 4h = Π4h
H f
Initial guess
Smooth
Initial guess
Ω8h
4h
u 8h = (A8h )−1 Π8h
4h f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Nested Iteration
Ωh
ΩH
Ω4h
Smooth
Initial guess
Smooth
h
f H = ΠH
hf
H
f 4h = Π4h
H f
Initial guess
Smooth
Initial guess
Ω8h
4h
u 8h = (A8h )−1 Π8h
4h f
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Simplistic Nested Iteration
I
Nested iteration attempts to provide a good initial guess for
uh .
I
Note that we are solving for the solution u h , and not the error
I
The interpolation operator ΠhH is higher order than in the MG
cycles
I
Simplistic Nested Iteration can have problems with aliasing
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Full Multigrid
Algorithm: uh ←− 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 ←− FMG (fH , ν)
uh ←− ΠhH uH
uh ←− MG µ(uh , fh , ν1 , ν2 )
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FMG-Cycle
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
FMG-Cycle
Ωh
ΩH
Ω4h
Ω8h
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Full Multigrid
I
MG µ cycles are required in FMG to tackle oscillatory and
smooth error introduced by interpolation
I
FMG can provide a solution within truncation error
I
FMG is optimal in work done
I
FMG is more forgiving than a V- or W-cycle
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Costs
I
Storage costs
I
I
I
I
I
In d dimensions each coarse grid has 2−d grid points as the
next finer grid
vh and fh need to be stored on each level
Total cost : 2N d (1 + 2−d + 2−2d + 2−3d · · · 2−kd )
This works out to 2, 4/3, 8/7 in 1, 2, 3 dimensions respectively
Computation cost
I
I
I
I
1WU is cost of one smoothing cycle on finest level
Ignore cost of interpolation and restriction (typically estimated
at 20% of a WU)
For 1 V(1,1) cycle cost: 2(1 + 2−d + 2−2d + 2−3d · · · 2−kd )
WU’s
Works out to 4, 8/3, 16/7 WU’s in 1, 2, 3 dimensions
respectively
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Error Propagation: Two Grid Algorithm
Recall the 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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Error Propagation: Two Grid Algorithm
I
(Pre) Smooth: eh = (Ih − (Mh )−1 Ah )ν1 eh0 ≡ Shν1 eh0
I
Compute residual: rh = Ah Shν1 eh0
I
Solve coarse residual equation: eH = (AH )−1 RAh Shν1 eh0
I
Correct fine grid approximation:
eh = (Ih − P(AH )−1 RAh )Shν1 eh0
ν1 0
−1
(Post) Smooth: eh1 = Shν2 A−1
h − P(AH ) R Ah Sh eh
I
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Convergence Analyses: Smoothing Property
Hackbusch introduced a measure of smoothness given by:
µ(ν) =
||Ah Shν ||
||Ah ||
Definition A multigrid method is said to possess the smoothing
property if there exists a function η(ν) such that for sufficiently
large ν
||Ah Shν || ≤ η(ν)h−α
where α > 0 and η(ν) → 0 as ν → ∞. Alternatively, this is also
defined using the expression
||Ah Shν || ≤ η(ν)||Ah ||.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Convergence Analyses: Approximation Property
Hackbusch also introduced a measure for the coarse grid correction:
Definition A multigrid method is said to possess the
approximation property if there exists a constant CA > 0 such that
−1
β
||A−1
h − P(AH ) R|| ≤ CA h
where β > 0. This could be roughly interpreted as a measure of
how well the coarse grid correction (P(AH )−1 Rrh ) approximates
the error remaining after smoothing (A−1
h rh ).
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
Convergence Analyses
I
Local Mode/Fourier Analysis (Brandt)
I
Multilevel Subspace Splittings (Oswald, Griebel, Xu, Bramble,
Pasciak )
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
I
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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 = Rvh and rH = Rrh
Now we have:
AH (Rvh + eH ) − AH (Rvh ) = Rrh
I
or equivalently:
AH (wH ) = AH (Rvh ) + Rrh
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Smoothing
Coarse Grid Correction
Full Approximation Scheme (FAS)
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
AH (wH ) = AH (Rvh ) + Rrh
I
Compute correction: eH = wH − Rv h
I
Update fine grid approximation: vh = vh + PeH
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Hierarchical Structure of SAMR Grids
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Inverting Elliptic Components
Choices:
I
Algebraic - ILU etc
I
Level by Level Preconditioner
I
Multigrid
I
Multilevel Methods
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: Inverting Elliptic Components
Choices:
I
Algebraic - ILU etc
I
Level by Level Preconditioner
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Compute residual: r 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Smooth: A3 e 3 = r 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: FAC
Ωhc
Ωh3
Update: u 3 ← u 3 + e 3
Ωh2
Ωh1
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
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
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
"!#
#
! $!%
!%
&'( BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
Compute residual: rc = fc − Ac uc = Aec .
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
ek ẽk , k = 2, 3, . . . , J.
Interpolate: ek0 ← P
k
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
I
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
ek ẽk , k = 2, 3, . . . , J.
Interpolate: ek0 ← P
I
Smooth/Solve: Ak ek = rk , k = 1, 2, . . . , J.
I
k
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
I
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
ek ẽk , k = 2, 3, . . . , J.
Interpolate: ek0 ← P
I
Smooth/Solve: Ak ek = rk , k = 1, 2, . . . , J.
I
Adjust fine grid corrections: ek ← ek − ek0 , k = 2, 3, . . . , J.
I
k
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
I
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
ek ẽk , k = 2, 3, . . . , J.
Interpolate: ek0 ← P
I
Smooth/Solve: Ak ek = rk , k = 1, 2, . . . , J.
I
Adjust fine grid corrections: ek ← ek − ek0 , k = 2, 3, . . . , J.
P
Form composite grid correction: ec = Jk=1 Pkc ek .
I
I
k
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Multilevel Algorithms: AFACx
Algorithm: AFACx for Ax = f
I
I
Compute residual: rc = fc − Ac uc = Aec .
Restrict rc :
I
I
rk = Rkc rc , k = 1, 2, . . . , J.
e kc rc , k = 2, 3, . . . , J.
r˜k = R
I
Smooth: Aek ẽk = r˜k , k = 2, 3, . . . , J.
ek ẽk , k = 2, 3, . . . , J.
Interpolate: ek0 ← P
I
Smooth/Solve: Ak ek = rk , k = 1, 2, . . . , J.
I
I
Adjust fine grid corrections: ek ← ek − ek0 , k = 2, 3, . . . , J.
P
Form composite grid correction: ec = Jk=1 Pkc ek .
I
Update current approximation: uc ← uc + ec .
I
k
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
Structured Adaptive Mesh Refinement
Conclusion
Software
I
BoxMG, parallel open source geometric black box multigrid
solver (LANL)
I
SMG, PFMG, parallel open source geometric black box
multigrid solver (LLNL)
I
PETSc parallel multigrid solver (ANL)
I
BoomerAMG, parallel algebraic multigrid solver (LLNL)
I
LAMG, parallel algebraic multigrid solver (LANL)
I
http://www.mgnet.org, maintained by Craig Douglas
BOBBY PHILIP
Introduction to Multigrid
Overview
Multigrid
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