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7/26/2019 Approximate solution of defect equation
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Approximate solution of defectequation
Ifum
is any approximation ofu and
rm =f Aum
is its residual, then the residual equation Aem = rm is
equivalent to the original equation: By solving for the
correction em, we obtain the solution u=um + em.
If we use, however, anapproximationAof A, such thatAem =rmcan be solved more easily, we obtain an iterative process
of the form
rm =f Aum, Aem =rm, um+1 =um +em (m= 0, 1, 2...)
This process is obviously equivalent to the general iterationwhere
Q= I (A)1A . Vice versa, ifQ is given, this yields an approximationAofA
according to A= A(I Q)1 .
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Splitting, Preconditioning
An equivalent way of constructing Q is to start with a splitting
A=A R, Aum+1
=Ru
m
+ f .Here
Q= (A)1R=I (A)1A . A third, also equivalent, approach is based on the idea
of preconditioning. Here the original equation Au = f is
replaced by an equivalent equation
CAu= Cf
where C is an invertible matrix. C is called a (left)
preconditioner of A. The identification with the above
terminology is by
(A)1 =C.In other words, the inverse (
A)1 of any (invertible)
approximationAis a left preconditioner and vice versa. Richardsons iteration for the preconditioned system (with
= 1)
um+1 =um + C(f Aum) = (I CA)um + Cfis equivalent to the general iteration withQ= I CA.
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Two ways of improving theconvergence
of basic iterative methods
For any approximation ui
h of the solution uh, we denote theerror byeih:=uh uih, and the defect (or residual) by
rih:=bh AhuihThe defect equation
Aheih=r
ih
is equivalent to the original equation, since
uh=uih+ e
ih.
This leads to the procedure
uih rih=bh Ahuih Aheih=rih uh=uih+ eih .This procedure is not a meaningful numerical process.
However, if Ah is approximated by a simpler operatorAh such that A1h exists, eih in Aheih = rih gives a newapproximation
ui+1h :=uih+eih.
The procedural formulation then looks like
uih rih=bh Ahuih Aheih=rih ui+1h =uih+eih .
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Iteration matrix
The iteration operator of this method is given by
Qh=Ih (Ah)1Ah: G(h) G(h),whereIhdenotes the identity on G(h). We have
u
i+1
h =Qhu
i
h+sh with sh= (A)1
h bh (i= 0, 1, 2, . . .) .For the errors, it follows that
ei+1h =Qheih=Ih (Ah)1Aheih (i= 0, 1, 2, . . .)
This represents a general class of iterative schemes. For
example, with
Ah=Dh,the Jacobi scheme is regained; withAhthe lower triangularpart ofAhwe obtain Gauss-Seidel.
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1. Coarse grid correction
The idea is to use an appropriate approximation AH of Ahon a coarser grid H, for instance the grid with mesh size
H= 2h. This means that the defect equation is replaced by
AHeiH=riH.AH :G(H) G(H), dimG(H) < dimG(h) and AH1exists.
AsriH andeiHare grid functions on the coarser grid H, weneed two (linear) transfer operators
IHh : G(h) G(H), IhH : G(H) G(h)
IHh is used to restrictrihtoH:
ri
H :=IH
h ri
h ,
andIhH is used to interpolate (or prolongate) the correctioneiHback toh: eih:=IhHeiH .
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for Poissons equation
Smoothing iteration: lexicographic Gauss-Seidel method
Coarse grid discretization:
AHuH= 1
H2
0 1 0
1 4 10 1 0
H
.
The simplest example for a restriction operator is the
injection operator
rH
(P) =IHh
rh
(P) :=rh
(P) for P
H
h
,
A fine and a coarse grid with the injection operator are
presented:
h
h
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Multigrid components
Prolongation operator:
y
2h
2h
x
Ih2he2h(x, y) =
e2h(x, y) for1
2
[e2h(x, y+ h) +e2h(x, y h)] for
12 [e2h(x+ h, y) +e2h(x h, y)] for14
[e2h(x+ h, y+ h) +e2h(x+ h, y h)+e2h(x h, y+ h) +e2h(x h, y h)] for
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Altogether, one coarse grid correction step (calculating
ui+1h fromuih) proceeds as follows:
Coarse grid correctionuih ui+1h Compute the defect rih=fi Ahuih Restrict the defect (finetocoarse transfer) riH=I
Hh r
ih
Solve exactly onH AHeiH=riH Interpolate the correction eih=IhHeiH Compute a new approximation ui+1h =u
ih+eih
The associated iteration operator is given by
Ih BhAh with Bh=IhHAH1IHh .
Taken on its own, the coarse grid correction process is of nouse: It is not convergent! We have
Ih IhHAH1IHh Ah 1 .
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high frequency components,
not visible on2h
low frequency components,
visible also on2h
High frequency components cannot be corrected on acoarse grid !
Coarse grid correction makes sense, if low frequencies are
dominating the error.
We can decompose the sum into partial sums:
p1
k,l=1 k,l
k,l
=highk,lk,l +lowk,lk,l
where lowk,lk,l =p/21k,l=1
k,lk,l
and
highk,l
k,l =p1
k,lp/2max(k,l)k,l
k,l .
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Point-wise Gauss-Seidel:
ui,j =1
4[h2bi,j+ u
n+1i1,j+u
ni+1,j+ u
n+1i,j1+ u
ni,j+1]
The effect on the erroren =un uhis a local averaging effect:eni,j =
1
4[en+1i1,j+ e
ni+1,j+ e
n+1i,j1+ e
ni,j+1]
We have found already
||en+1|| (Q)||en||, (n ).
Analysis of the smoothing effect requires consideration ofeigenvalues/-vectors ofQ, which are closely related toA. Look
at the Fourier expansion of the error:
eh(x, y) =p1k,l=1
k,lsin kx sin ly=p1k,l=1
k,lk,l
The fact that this error becomes smooth means that the high
frequency components, i.e.,k,lsin kx sin ly withk orl large
become small after a few iterations, whereas the low frequency
components
k,lsin kx sin ly withkandl small
hardly change.
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The two grid iteration
Correction scheme
It is necessary to combine the two processes of smoothingand of coarse grid correction.
Consider a linear problem Ahuh=bh on grid Gh(1) 1 smoothing steps
on the fine grid: uh =S1(u0h, bh)(2) computation of residuals
on the fine grid: rh:=bh Ahuh(3) restriction of residuals
from fine to coarse: rH :=IHh rh
(4) solution of the
coarse grid problem: AHeH=rH(5) prolongation of corrections
from coarse to fine: eh:=IhHeH
(6) adding the corrections to thecurrent fine grid approximation: uh =uh+ eh
(7) on the fine grid: u1h =S2(uh, bh) Steps (1) and (7) arepre and postsmoothing,
steps (2)...(6) form thecoarse grid correction cycle.
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3. Multigrid
since 1973
Iterative methods likeJacobiandGauss-Seidelconvergeslowly on fine grids, however, theysmooththe erroruh u
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Smootherrors can beapproximated well on
coarser grids (with
much less grid points)
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Multigrid is a O(N)- method !
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Multigrid components
Choice of coarse grid The choice of grid depends on the smoothness of the error.
Grid coarsening is particularly simple for structured grids.
For irregular finite volume/ finite element grids coarse grids
are chosen based on the connections in the matrix. In
this case, it is better to say that coarse matrices are
constructed.
It is possible to determine, based on matrix properties
(M-matrix, for example), where the error will be smooth
and accordingly how to coarsen algebraically (algebraic
multigrid, AMG).
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2. Krylov subspace accleration
Basic Iterative solution Methods
We computed the iterates by the following recursion:ui+1 =ui + B1(b Aui) =ui + B1ri
Writing out the first steps of such a process we obtain:
u0,
u1 = u0 + (B1r0),
u
2
= u
1
+ (B
1
r
1
) =u
0
+B
1
r
0
+ B
1
(b Au0
AB1
r
0
),= u0 + 2B1r0 B1AB1r0,...
This implies that
ui u0 + span
B1r0, B1A(B1r0), . . . , (B1A)i1(B1r0)
.
The subspace Ki(A; r0) := span r0, Ar0, . . . , Ai1r0 is calledtheKrylov-spaceof dimension i corresponding to matrix A
and initial residualr0.
ui calculated by a basic iterative method is an element of
u0 + Ki(B1A; M1r0).
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Recombination of Iterants
The acceleration of a basic iterative method by iterantrecombination starts from successive approximations
u1h, u2h, . . . , u
mh, from previous iterations.
In order to find an improved approximation uh,, we consider
a linear combination of them + 1 latest approximationsumih , i= 0, ,m,
uh,=umh + mi=1
i(umih umh) ,
(assumingm m) withi= 1. For linear equations, the corresponding residual, rh, = fh
Lhuh,, is given by
rh,=rmh +
m
i=1 i(rmih
rmh) ,
wherermih =fh Lhumih . To obtain an improved approximation uh,, parameters iare
determined such that residualrh, is minimized.
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Minimizerh,, i.e.
||rmh +
m
i=1i(r
mih
rmh)
|| ,
with respect to theL2-norm | | | |2. This is a classical defect minimization problem. In principle,
the optimal coefficientsican be determined by a (Gram-
Schmidt) orthonormalization process.
Here, however, it is also possible to solve the system of linear
(normal) equations
H
12...
m
=
12...
m
,
where the matrixH= (hik)is defined by
hik = < rmih , r
mkh > < rmh, rmih >
< rmh, rmkh >+ < rmh, rmh > i= 1, . . . , m, k= 1, . . . ,m ,with the standard Euclidean inner product< ., . >and
i=< rmh, r
mh > < rmh, rmih > .
The work for solving the minimization problem is small.
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The Chebyshev method
Supposeu
1
, . . . , u
k
have been obtained via a basic iterativemethod, and we wish to determine coefficients j(k),
j = 0, . . . , ksuch that
yk =k
j=0j(k)u
j
is animprovementofuk.
If u0 = . . .= uk = u, then it is reasonable to insist that yk = u.Hence we require
kj=0
j(k) = 1,
Consider how to choose the j(k)so that the error yk u is
minimized. Since error e(k+1) = Qke0 where ek = uk u. Thisimplies that
yk u= kj=0
j(k)(uj u) = k
j=0j(k)Q
je0.
Using the 2-norm we look for j(k) such thatyk u2 isminimal.
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The Chebyshev method
To simplify this minimization we use the following inequality:
yk u2 pk(Q)2u0 u2wherepk(z) =
kj=0
j(k)zj andpk(1) = 1.
Minimize
pk(Q)
2for all polynomials satisfyingpk(1) = 1.
Assumption that Q is symmetric with eigenvalues i that
satisfy n . . . 1
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The Chebyshev method
The solution of this problem is obtained by Chebyshev
polynomials. These polynomials cj(z) can be generated bythe following recursion
c0(z) = 1,
c1(z) =z,
cj(z) = 2zcj1(z) cj2(z).These polynomials satisfy
|cj(z)
| 1 on [
1, 1] but grow
rapidly in magnitude outside this interval. As a consequencethe polynomial
pk(z) =ck1 + 2 z
ck
1 + 2 1
satisfies pk(1) = 1, since1 + 2 1 = 1 + 2 1, and tends tobe small on[, ]. The last property can be explained by the
fact that1 1 + 2 z
1 for z [, ] so the
numerator is less than 1 in absolute value, whereas the
denominator is large in absolute value since 1 + 2 1 >1.
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-method
This leads to
yk u2 pk(Q)2u0 u2 u u02|ck
1 + 2 1
|.
Calculation ofyk is costly, since all u0, . . . , uk should be kept
in memory. Furthermore, one needs to add k + 1 vectors,
which costs fork 5more than one matrix vector product. Using the recursion of the Chebyshev polynomials it is
possible to derive athree term recurrenceamong the yk.
Vectorsyk can be calculated as:
y0 =u0
solvez0 fromBz0 =b Ay0 theny1 is given byy1 =y0 + 22z
0
solvezk fromBzk =b Ayk theny(k+1) is given by
y(k+1) =4 2 2
ck
1 + 2 1
ck+1
1 + 2 1
yk y(k1) + 2
2 zk+y(k1)
The Chebyshev semi-iterative method associated withBy(k+1) = (B
A)yk + b.
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The semi-iterative Chebyshevmethod
Theory
Note that only 4 vectors are needed in memory and the
extra work consists of the addition of 4 vectors.
Acceleration is effective with good lower and upper bounds
ofand. These parameters may be difficult to obtain.
Assumption in deriving the Chebyshev acceleration: the
iteration matrixB1(B A)is symmetric. Thus, analysis doesnot apply to the SOR iteration matrix B1
(B
A). To repair
this Symmetric SOR (SSOR) is proposed. In SSOR one SOR
step is followed by a backward SOR step. In this backward
step the unknowns are updated in reversed order.
Suppose that the matrix B1A is symmetric and positive
definite and that the eigenvaluesi are ordered as follows
0< 1 2 . . . n. It is then possible to prove the followingtheorem:
If the Chebyshev method is applied and B1Ais symmetric
positive definite then
yk u2 2
K2(B1A) 1K2(B1A) + 1
k
u0 u2.
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The Chebyshev method
Proof SinceB1A= B1(B (B A)) =IB1(B A) =IQwe see that the eigenvalues satisfy the following relation:
i= 1 i or i= 1 i.This leads to the inequality:
yk u2 u u02
|ck
1 + 2 (1(11))(11)(1n)
|.
So it remains to estimate the denominator. Note that
ck
1 + 2(1 (1 1))(1 1) (1 n)
= ckn+ 1
n 1= ck
1 + 1n
1 1n
.The Chebyshev polynomial can also be given by
ck(z) =1
2
z+
z2 1
k+
z
z2 1k
.
This expression can be used to show that
ck
1+1n1
1n
> 12
1+1n
11n
+
1+1n1
1n
2 1k
=
= 12
1+1n+21n1
1n
k = 121+1n1
1n
k .
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The Chebyshev method
The condition numberK2(B1A)is equal to n1 . This leads to
yk u2 2
K2(M1A) 1K2(M1A) + 1
k
u0 u2.
Chebyshev type methods which are applicable to a wider
range of matrices are given in the literature.
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