Geometric (Classical) MultiGrid

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

Linear (2nd order) interpolation in 1D

x1 x2x

F(x)

)()()( 212

11

12

2 xFxxxx

xFxxxx

xF

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 UUUUyyyy

Uyyyy

U ......;12

02

12

10

(Ul,Vl) (Ur,Vr)

lr Uxxxx

Uxxxx

yxU12

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 ,1Ni 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 x0)U()U( 10

hihUL 0UUU

211 2

hiii 11 Ni

00 NUU

fine grid

h

u = average of u's

approximating Laplace eq.2 2

2 2 0u ux y

u given on the boundary

h

e.g., u = average of u's

approximating Laplace eq.2 2

2 2 0u ux 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 relaxationSolve 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

LU=Fh

2h

4h

LhUh=Fh

L2hU2h=F2h

L4hU4h=F4h

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?

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

Why additional relaxations are needed?

The coarse grid correction

Interpolate and add

Why additional relaxations are needed?

The coarse grid correction

Interpolate and add

Why additional relaxations are needed?

The coarse grid correction

Interpolate and add

Why additional relaxations are needed?

The coarse grid correction

Interpolate and add

Why additional relaxations are needed?

The coarse grid correction

Interpolate and add

Why additional relaxations are needed?

Interpolate and add => high oscillatory component emerges

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

NNOfuku

(1977,1982)