geometric (classical) multigrid. hierarchy of graphs apply grids in all scales: 2x2, 4x4, …, n 1/2...
TRANSCRIPT
Hierarchy of
graphs
Apply grids in all scales: 2x2, 4x4, … , n1/2xn1/2
Coarsening Interpolate and relax
Solve the large systems of equations by multigrid!
G1
G2
G3
Gl
G1
G2
G3
Gl
i
S(i)
(Ulb,Vlb)
(Urt,Vrt)(Ult,Vlt)
(Urb,Vrb)
(x2,y2)(x1,y2)
(x2,y1)(x1,y1)
(x0,y0)
Bilinear interpolation
C(S(i))={rb,rt,lb,lt}
i
S(i)
(Ulb,Vlb)
(Urt,Vrt)(Ult,Vlt)
(Urb,Vrb)
(x2,y2)(x1,y2)
(x2,y1)(x1,y1)
(x0,y0)
lbltlrbrtr UUUUyy
yyU
yy
yyU ......;
12
02
12
10
(Ul,Vl) (Ur,Vr)
lr Uxx
xxU
xx
xxyxU
12
02
12
1000 ),(
From (x,y) to (U,V) by bilinear intepolation
])~~(
)~~[(),(
])()[(),(
))((
2
))((
2
))(())((,
22
,
jscpjpjpj
iscpipipi
jscpjpjpj
iscpipipi
jiij
jijiji
ij
VyVy
UxUxaVUE
yyxxayxE
Linear scalar elliptic PDE (Brandt ~1971)
1 dimension Poisson equation
Discretize the continuum
LU )(xx F)(U 10 x
0)U()U( 10
x0 x1 x2 xi xN-1 xN
x=0 x=1h
Grid: ihxN
h i ,1
Ni 0
h
Let ihi FF local
averaging),( ixU )( ixFi
hi UU
Linear scalar elliptic PDE 1 dimension Laplace equation
Second order finite difference approximation
=> Solve a linear system of equationsNot directly, but iteratively=> Use Gauss Seidel pointwise relaxation
LU 0 )(U x 10 x
0)U()U( 10
hihUL 0
UUU
2
11 2
hiii 11 Ni
00 NUU
u given on the boundary
h
e.g., u = average of u's
approximating Laplace eq.2 2
2 20
u u
x y
Point-by-point RELAXATIONSolution algorithm:
Exc#9: Error calculations
1. Use Taylor expansion to calculate the error when U’’(x) is approximated by
2. Find a,b,c,d and e such that
This is a higher order approximation for U’’(x) than the one in exercise 1.
2
)()(2)(
h
hxUxUhxU
)()()2()()()()2( 4hOxUhxeUhxdUxcUhxbUhxaU
Exc#10: Gauss Seidel relaxation
Solve the 1D Laplace equation U’’(x)=0, 0<x<1 by Gauss Seidel relaxation.
Start with the approximations 1. Ui = random(0,1) ,
2. Ui = sin(x) , where U0 = UN = 0 for N=32.Plot the L2 norm of the error and of the residualversus the number of iterations k=1,…,100, wherethe L2 norm of a vector v isand the residual of LU=F is R=F-LUDo you see a difference in the asymptotic behavior
between the 2 norms?Which case converges faster 1. or 2. , explain
21
1
22 ]
1[||||
n
iivn
v
Influence of (pointwise) Gauss-Seidelrelaxation on the error
Poisson equation, uniform grid
Error of initial guess Error after 5 relaxation
Error after 10 relaxations Error after 15 relaxations
The basic observations of ML Just a few relaxation sweeps are needed to
converge the highly oscillatory components of the error
=> the error is smooth Can be well expressed by less variables Use a coarser level (by choosing every other
line) for the residual equation Smooth component on a finer level becomes
more oscillatory on a coarser level=> solve recursively The solution is interpolated and added
h
2h
Local relaxation
approximation
hu~
hV hh u~U smoothhh u~LF hhVhL
hR
h2Vh2L h2R
LhUh=Fh
L2hU2h=F2h
h2Vh2L h2R
TWO GRID CYCLE
Approximate solution:hu~
hhh u~UV hhh RVL
hhhh u~LFR
Fine grid equation: hhh FUL
2. Coarse grid equation: hhh RVL 22
hh2
h2v~~~ hold
hnew uu h
h2
Residual equation:Smooth error:
1. Relaxation
residual:
h2v~Approximate solution:
3. Coarse grid correction:
4. Relaxation
Why additional relaxations are needed?
A smooth approximation is obtained after relaxation on the finer level
Why additional relaxations are needed?
A smooth approximation is obtained after relaxation on the finer level
The coarse grid correction
TWO GRID CYCLE
Approximate solution:hu~
hhh u~UV hhh RVL
hhhh u~LFR
Fine grid equation: hhh FUL
2. Coarse grid equation: hhh RVL 22
hh2
hold
hnew uu h2v~~~ h
h2
Residual equation:Smooth error:
1. Relaxation
residual:
h2v~Approximate solution:
3. Coarse grid correction:
4. Relaxation
1
2
34
5
6
by recursion
MULTI-GRID CYCLE
Correction Scheme
interpolation (order m)of corrections relaxation sweeps
residual transfer
ν ν enough sweepsor direct solver*
.. .
*
1ν
1ν
1ν
2ν
2ν
2ν
Vcyclemultigrid
h0
h0/2
h0/4
2h
h
V-cycle: V
Multigrid solversCost: 25-100 operations per unknown
• Linear scalar elliptic equation (Achi Brandt ~1971)
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)