Transcript Slide 1
Mutigrid Methods
for Solving Differential Equations
Ferien Akademie ’05 – Veselin Dikov
Multigrid Methods
Agenda
● Model problem
● Relaxation. Smoothing property
● Elements of Multigrid
● Multigrid schemes
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Model Problem
● 1D boundary problem of steady state
temperature a long a uniform rod
u" ( x) u ( x) f ( x),
u (0) u (1) 0
0 x 1,
0
● Discretization in n points, step h = 1/n
v j 1 2v j v j 1
2
h
v0 v n 0
v j f j ,
Ferien Akademie ’05
1 j n 1,
Veselin Dikov
Multigrid Methods
Model Problem
T
● Av = f, where v v1 ,...,vn1 and f f1 ,..., fn1
T
2 h 2
1
1
A 2
h
● Stencil notation
1
2 h 2
.
1
.
.
.
.
1
1 2 h 2
1
A 2 (1 2 h 2 1)
h
● A is Symmetric positive definite
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Agenda
● Model problem
● Relaxation. Smoothing property
● Elements of Multigrid
● Multigrid schemes
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Iterative Methods
● Iterative vs Direct methods
More about iterative methods
● Jacobi and Gauss-Seidel methods
● Smoothing property
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Smoothing Property
● Error along the domain
After 35 sweeps with
weighted Jacobi
Error was smoothed
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Smoothing Property
● Smoothing property explained in four steps
1. Fourier modes
vk , j
jk
sin
,
n
0 j n,
1 k n 1
● k – wave number
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
k=1
k=2
k=7
k = 12
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Smoothing Property
● Smoothing property explained in four steps
1. Fourier modes
● smooth modes -
n
1 k
2
n
k n 1
● oscillatory modes 2
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
● f = 0, σ = 0
Au = 0
● exact solution: u = 0
● error: e = u – v = -v
we can trace the error!
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
3. Weighted Jacobi relaxation
● wJacobi step
v(k 1) R v(k ) c
2 1
1 2 1
R I A I
.
.
.
2
2
.
.
1
1 2
● error
e( m) R me(0)
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
3. Weighted Jacobi relaxation
4. Three experiments
● we relax with wJacobi with ω = 2/3 on
initial guesses respectively:
( 0)
( 0)
( 0)
v
v
,
v
v
and
v
v6
1
3
e
# iterations
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
3. Weighted Jacobi relaxation
4. Three experiments
● repeat the experiment with:
e
ω = 2/3 and initial guess
1
1
(0)
v v1 v3 v6
2
2
# iterations
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Smoothing Property
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
3. Weighted Jacobi relaxation
4. Three experiments
● Explanation
● Rω has the same eigenvectors as A and
they are the same as the wave vectors
● Recall that for the error e(m) = Rme(0)
● Eigenvalues of Rω ?
Ferien Akademie ’05
Veselin Dikov
Smoothing Property
Multigrid Methods
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
Eigenvalue
3. Weighted Jacobi relaxation
4. Three experiments
● Explanation
wavenumber k
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Smoothing Property
● Smoothing property explained in four steps
1. Fourier modes
2. Modified model problem
3. Weighted Jacobi relaxation
4. Three experiments
● Explanation
● Smoothing property
● Fast damping of oscillatory error modes
● Common for all iterative methods
● How to overcome the bad performance
effect over smooth error modes?
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Agenda
● Model problem
● Relaxation. Smoothing property
● Elements of Multigrid
● Multigrid schemes
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Elements of Multigrid
● Element I: A smooth wave looks more
oscillatory on a coarser grid
● Aliasing: k looks like (n-k)
Ferien Akademie ’05
Veselin Dikov
Elements of Multigrid
Multigrid Methods
● Element II: Nested Iterations
finest grid
Relax
Relax
coarsest grid
Relax
transfer the coarse grid
result to the finer grid for
the initial guess
● Problems?
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Elements of Multigrid
● Element III: Correction scheme
● Residual equation: Ae = r
● The scheme:
» Relax on Au = f on to obtain an
h
approximation v .
» Compute r f Avh .
2h
» Relax on Ae = r on to obtain an
2h
approximation to the error, e .
h
h
2h
v
v
e
» Correct the approximation
.
h
Ferien Akademie ’05
Veselin Dikov
Elements of Multigrid
Multigrid Methods
● Element IV: Interpolation and restriction
● Interpolation I 2hh :
v2h j v 2j h ,
1 2h
v
v j v 2j h1 ,
2
● Restriction I h2 h :
h
2 j 1
n
0 j 1.
2
2h
h
Injection: v j v2 j .
1 h
h
h
v
v
2
v
v
Full weighting:
2 j 1
2j
2 j 1 ,
4
● Variational property:
2h
j
I
h
2h
,
c I
2h T
h
Ferien Akademie ’05
n
1 j 1.
2
cR
Veselin Dikov
Multigrid Methods
Agenda
● Model problem
● Relaxation. Smoothing property
● Elements of Multigrid
● Multigrid schemes
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Two-Grid
● Two-Grid = Corr.Scheme+Interpolation+Restriction
v h MG v h , f h
h
» Relax 1 times on Ah u h f h on with initial
h
guess v
» Compute r h f h Ah v h and restrict r 2h I h2h r h.
2h 2h
2h
» Solve A e r on 2 h.
h
h
h
» Interpolate e h I 2hh e 2h and correct v v e .
h
» Relax 2 times on Ah u h f h on with initial
h
v
guess
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Two-Grid -> V-Cycle
● Two-Grid Scheme
● V-Cycle = Recursive Two-Grid Scheme
V-Cycle
Ferien Akademie ’05
W-Cycle
Veselin Dikov
Multigrid Methods
Full Multigrid(FMG)
● FMG = V-Cycle + nested iterations
FMG
Ferien Akademie ’05
Veselin Dikov
Multigrid Methods
Costs
● V-Cycle costs
d
1
1
1
2
n
Storage 2n d 1 d 2 d ... L
2
2 1 2d
2
1
1
1
2
Computational cost 2
1
...
WU
d
2d
L
d
2
2 1 2
2
● FMG computational costs
1
1
2
2 1
1
...
WU
2
d
d
2d
L
2
2 1 2 d
1 2 2
● Speedup because working on smaller domains
Ferien Akademie ’05
Veselin Dikov