Computing,
Suppl.
5,
I -
32
(1984)
by Springer-Verlag 1984
The Defect Correction Approach
K.
Bohmer,
Marburg,
P.
Hemker,
Amsterdam,
and
H.
J.
Stetler,
Wien
Abstract
This is
an
introductory
survey
of
the defect correction
approach
which may serve as a unifying
frame
of
reference for the
subsequent
papers
on
special subjects.
I.
Introduction
There
are
many
ways to introduce defect corrections. In this expository article
we
motivate the defect correction
approach
from its basic idea:
For
a given
mathematical
problem
and
a given
approximate
solution,
- define the
defect
as a quantity which indicates how well the problem has been
solved,
- use this information in a
simplified
rersion
of
the problem to
obtain
an
appropriate
correction
quantity,
apply this correction to the
approximate
solution to
obtain
a new (better)
approximate
solution.
Naturally,
the
procedure
may now
be
repeated.
Of
course, this
fundamental
approach
has been used in
mathematics
since long.
We
give some examples in
Chapter
2.
In
Chapter
3 we formalize the general defect
correction principle
and
describe several processes which implement it.
Since defect corrections
are
especially powerful in
combination
with discretizations
of
analytic problems, in
Chapter
4 we review discretization
methods
and
asymptotic
expansions for their local
and
global discretization errors. In
Chapter
5,
we establish
the general framework for the combination of defect corrections with discretization
methods,
and
we survey a variety
of
algorithms
of
this kind.
The
powerful multigrid
approach
is
interpreted
as a particularly interesting application
of
the defect
correction principle in
Chapter
6.
2
K.
Bohmer,
P.
Hemker,
and
H.
J.
Stetter:
2.
Historical Examples
of
Defect Correction
Prototypes
of
defect correction are
the
classical procedures for the calculation
of
a
zero of a nonlinear function in one variable: An
approximation
z
of
the
solution
z*
of
the problem
F(z)=O
(2.1)
is
substituted into
F:
the value of
F
(Z)
defines the defect.
The
simplified version
of
(2.1)
which yields the correction
of
z
is some local linearization
F(z):=F
(ZJ+
m(z-Z)
=0,
(2.2)
where
m:::::
F'
(z*);
see Fig.
1.
Newton's method
is
a
more
refined case where
m
is
updated during the iteration.
Fig. I
Another well-known prototype
is
"'iterative refinement"
("Nachiteration")
in
the
numerical solution of linear algebraic equations
Az=b.
(2.3)
After an approximate solution
:,
with
an
unknown round-off
contamination,
has
been obtained from a direct solution procedure, its defect
d:=Az-h
is
computed
(with special care). Then the matrix decomposition
of
the previous solution process
is
used once more to compute a correction
LI
z
from
A
Llz=d.
(2.4)
Here the use of the
numerically
transfimned
A
represents the
"simplified version"
of
(2.3);
if(2.4) could
be
solved exactly it would naturally yield the exact correction.
If
z
is
a discrete approximate solution
of
an
analytic problem, the defect
formation
becomes a non-trivial
part
of
the procedure. In the late forties,
L.
Fox
([17], [18])
considered the discretization of a second-order
boundary
value
problem
-z"(t)+p(t)
z(t)=q(t)
on
(a,b),
z(a)
and
z(b)
given,
(2.5)
by
central second-order differences
on
an
equidistant grid
G
in
[a,
b].
He
suggested
(though not
in
these terms) that a defect
of
an
approximate
solution
z
= {
z
(tv),
t
,.E
G}
might be defined
via
substitution of
z
into a discretization of
(2.5)
which included
The Defect Correction Approach
3
4-th
order
differences. This defect d =
{d"}
could be used as inhomogeneity in the
problem for the correction function
L1
z
-Jz"(t)+p(t)
L1z(t)=d(t)
on
(a,b),
L1z(a)=L1z(b)=0;
(2.6)
(2.6)
could then
be
solved again by the basic
(="simplified")
second-order
discretization method. Fox considered the recursive application
of
this approach,
with
the
inclusion
of
differences
of
higher
and
higher order into the computation of
the defect
d.
He
and
others applied this method to a variety of problems, see e.g.
[19],
[20].
Fox's
approach
was later
put
into a more general, abstract frame-work
by
Pereyra
([41]-[44])
and
effectively implemented; see Section5.2.1 and
Pereyra's
paper
in this volume.
A further generalization of the defect correction principle and
an
increase
of
the
interest in the subject were initiated by the presentation of a paper
"On
the
estimation of errors propagated in the numerical solution of ordinary differential
equations" by P.
E.
Zadunaisky
at
the 1973 Dundee Conference
on
Numerical
Analysis. Zadunaisky's heuristic technique turned
out
to permit
an
interpretation in
terms
of
defect correction which represented a novel realization
of
the old idea; see
Section 5.2.2. This brings us to the contemporary view
of
the subject.
3. General Defect Correction Principles
3.1 Basic Defect Correction Processes
We wish to "solve" the equation
Fz=y,
(3.1)
where F : D c
E--+
fJ
c
Eis
a bijective continuous, generally nonlinear operator;
E,
E
are Banach spaces.
The
domain D
and
the range
f>
are closed subsets depending on
F;
fJ
contains an
appropriate
neighbourhood
of
y. Hence, for every y E
f>
there
exists,
in
D, exactly one solution of
Fz
=
y;
the solution
of
the given problem (3. l)will
be called z*.
We
assume
that
(3.1)
cannot
be solved directly, but
that
the defect
d(i):=Fz-y
(3.2)
may be evaluated for
"approximate
solutions" z ED. Furthermore, we assume that
we
can readily solve the approximate problem
Fz=y
(3.3)
for
y E
/5,
i.e. that
we
can evaluate the solution
operator
G of (3.3). G :
D--+
D
is
an
approximate inverse
of
F such
that
(in some appropriate sense)
G F
z-::::;
z for i E D
(3.4)
and
FGy-::::;y
for yeD.
(3.5)
4
K.
Bohmer,
P.
Hemker, and
H.J.
Stetter:
Let us now assume
that
we
know some
approximation
z
ED for
z*
and
that
we
have
computed its defect (3.2
).
In the general (nonlinear) case, there
are
two
ways
to
use
this information for
the
computation of a (hopefully better)
approximation
z
by
means
of
solving problems of type (3.3); see Fig. 2:
(A)
(8)
Fig. 2
y+
d(:Z}
d(:Z)
y
(A)
We compute the change
.dz
in
the solution of(3.3) when the right
hand
side y is
changed by
d(z).
We then use
.dz
as a correction for
z,
i.e.
we
transfer
the
observed
change to
our
target problem (3.1):
::
=z-.d:=z-[G
(y+d(ZJ)-G'
.vJ
z:=z-GFz+Gy.
(3.6)
(Bl
We generate an equation
(3.3)
with solution
z
and
change its
right-hand
side
T
=
F:
by
d
(:).We
then take the solution of this modified equation
as
z,
i.e.
we
again
transfer the effect observed
for
(3.3)
to
our
target problem (3.1):
T:=T-d(z)=l-
F
G'T+
y,
::=G
l=
G
[(F-F)z+
y].
(3.7)
Note
that
it
is
the existence of
G
and
not of
F
=
a-
1
which is essential,
as
is
immediately clear from (3.6) and (3.7).
In
some respect, versions
(A)
and
(B)
appear
dual
to each other.
In both approaches, the arising problems with modified
right-hand
sides
are
often
called
nei?1hbori11y
problems.
ln
some applications, the
operator
F
-
F
in
(3.7)
is
much simpler
than
either
For
F
so
that
there
is
an
advantage
in
using
approach
(8).
The success of the basic defect correction steps
(3
.6)
or
(3.7)
depends
on the
contractii·ity
of
the operations
(I
-
G
F) :
D->
D
or(/
-
F
G)
:
fj
_,
fj
resp., since (3.6)
implies
z-:*=(J-GF)z-(l-GF)z*
while (3.7) implies, with
Cl*=:*,
T-!*
=U-FG)
i-(1-FG)
I*.
(3.8)
(3.9)
The Defect Correction Approach
5
This contractivity is,
of
course, closely related to the approximate inverse property
of
G,
cf.
(3.4)
and
(3.5) resp.
The
element~
which
we
have gained through defect correction may be used in two
ways:
- We
may
interpret z
-i
as
an
estimate
of
the error
i-z*
of the original
approximation
z,
- we may subject
~
as
our
new approximation to another defect correction step.
The
iterative use
of
the basic defect correction procedures
(3.6)
or
(3.7) leads to the
Iterative Defect
Dorrection
(IDeC) algorithms
of
Stetter [51]:
(A)
zi+
1
:=z;-GFz;+Gy,
(3.10)
(B)
li+
1
:=l;-FGl;+y,
with z;=Gl;:
for injective
G,
(3
.11)
turns
into
zi+t:
= G
[(F-F)z;+
y].
Usual starting values for these iterations are z
0
= G y and /
0
=
y.
(3.11)
(3.11
a)
The
contractivity
of
the operators J - G
For
J - F G resp. implies the convergence of
these iterations,
cf. (3.8)
and
(3.9):
The
z;
of
(3.10) converge to
z*
while the
i;
of (3.11)
converge
to/*,
which implies the convergence of
z;:
= G
I;
to
z*.
(Restrictions arising
from an implementation in a finite computer arithmetic have been disregarded.)
If
the
approximate
inverse G
is
an
affine mapping, i.e. if
Gyl
-GJi=G'(yl
-Ji),
Y1,JiED,
(3.12)
with a fixed linear
operator
G',
the two versions merge into the familiar linear version
of
the basic defect correction step
2=z-G'
d(Z)=[J-G'
FJz+G'
y (3.13)
which leads to
the
linear
IDeC
algorithm
zi+
1
:
=z;-G'
d(z;)=
[1-G'
F]
z;+ G'
y.
(3.14)
Now the contractivity
of
I -
G'
F (or equivalently
of
I - F
G')
becomes the condition
for convergence to
z*.
Note
that
in (3.14) there
is
no
need
for
F
of(3.l)
to be linear.
In (3.14),
G'yeD
is a fixed element which has to be computed only once and
is
usually taken as
starting
approximation
::
0
.
If y = 0, this term vanishes.
Often, the
approximate
inverse G
is
Frechet-differentiable, i.e.
G
(y+
LI
y)-
G
y;::::
G'
(y) Lly.
With this linearization, (3.6) yields the new approximation
~==z-G'(y)d(z)
while (3.7) yields Gl=G(T-d(Z))
and
2
=z+
G (T-d
(Z))-
GT
;::::z-
G'
(T)
d
(i),
cf.
also Fig. 2,
(A)
and
(B).
(3.15)
(3.16a)
(3.16
b)