On dichotomy and well conditioning in BVP
Hoog, de, F.R.; Mattheij, R.M.M.
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SIAM Journal on Numerical Analysis
DOI:
10.1137/0724008
Published: 01/01/1987
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Citation for published version (APA):
Hoog, de, F. R., & Mattheij, R. M. M. (1987). On dichotomy and well conditioning in BVP. SIAM Journal on
Numerical Analysis, 24(1), 89-105. DOI: 10.1137/0724008
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SIAM J. NUMER. ANAL.
Vol. 24, No. 1, February 1987
1987 Society for Industrial and Applied Mathematics
OO8
ON DICHOTOMY AND WELL CONDITIONING IN BVP*
F. R. DE HOOGf
AND
R. M. M. MATTHEIJ*
Abstract. We investigate the relationships between the stability bounds of the problem on the one hand
and the growth behaviour of the fundamental solution on the other hand. It is shown that if these stability
bounds are moderate (i.e. if the problem is well conditioned) then the homogeneous solution space is
dichotomic, which means that it can be split into a subspace of nondecreasing and a complementary subspace
of nonincreasing modes. This is done by carefully examining the Green’s functions. If these exhibit an
exponential behaviour then the solution space is also exponentially dichotomic. On the other hand, we also
show that (exponential) dichotomy implies moderate stability constants, i.e. well conditioning. From this it
follows that both concepts are more or less equivalent.
Key words, boundary value problems, conditioning, dichotomy, stability, Green’s functions
AMS(MOS) subject classifications. 65L07, 65L10, 34B27
1. Introduction. Let us consider the first-order linear system of ordinary differential
equations
(1.1)
y:=y’-Ay=f,
0<t<l,
where A[Lp(O, 1)] "" andf[Lp(O, 1)]" for some p satisfying 1-<p-<oo. We seek a
solution y subject to the following boundary conditions (BC)
3y := Boy(0) + Bly(1
(1.2)
b
nxn
where Bo, B1 R
and b R".
It is well known that the boundary value problem (BVP) (1.1), (1.2) has a unique
solution if and only if Y3 Y is nonsingular where Y is any fundamental solution of
(see for example Keller [4]). In this case, we can formally write the solution y as
(1.3)
y(t)= Y(t)[BoY(O)+ B1Y(1)]-lb+
G(t, s)f(s) ds,
where we have defined the Green’s function G as
(1.4)
Y(t)[BY(O)+BIY(1)]-IBY(O)Y-’(s)’
G(t,s)={ -Y(t)[BoY(O)+B1Y(1)]-B1Y(1)Y-(s),
t>s,
t<s.
Thus, in principle, a knowledge of the fundamental solution enables one to calculate
the Green’s function and whence the solution y given by (1.3).
We shall now demonstrate how equation (1.3) can be used to examine the
conditioning of (1.1), (1.2). Since we will be using orthogonal transformations we take
to be the Euclidean norm on ". Of course, as all norms on
are equivalent, the
use of another norm changes only the numerical value of the constants in the subsequent
results. Let
I"
"
(1.5)
Ilull,:--
lu(s)l" ds
l <-p<-oo,
* Received by the editors March 2, 1984, and in revised form September 14, 1985.
t Division of Mathematics and Statistics, CSIRO, P.O. Box 1965, Canberra, 2601 Australia.
Mathematisch Instituut, Katholieke Universiteit Toernooiveld, Nijmegen, the Netherlands.
89
90
F. R. DE HOOG AND R. M. M. MATTHEIJ
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and
its limiting value as p-->
(1.6)
(1.7)
.
Ilulloo :- sup
--
Then, we find from (1.3) (cf. (1.1), (1.2)) that,
Ilyll :-IIYlI <=fll3y] +llyll,
Y[BoY(O)+ B, Y(1)]-’II
1
p
1
q
1,
(1.8)
The most appropriate norm in (1.6) depends on the problem under consideration.
However, in the interest of clarity, we shall henceforth consider only the case of p 1.
Nevertheless, many of the arguments used can be generalized to the case of an arbitrary
p.
When p= 1, (1.6) and (1.8) reduce to
(1.9)
(1.10)
(1.11)
-</3lyl / II-y II,,
Ily IIo-
YEY]-’lloo,
a=supIG(t,s) I.
/3
In addition, if the boundary condition (1.2) is scaled so that
B[Bo+BBI=I,
then
[Y(t)[Y]-’12=]Gr(t, O)G(t, O)+G(t, 1)G(t, 1)[
and hence
Thus, in this case, the stability constant a gives a measure for the sensitivity of (1.1)
and (1.2) to changes in the data. As has been shown in de Boor, de Hoog and Keller
1 ], the stability constants of many numerical schemes approach those of the continuous
problem as the mesh size goes to zero (see also Mattheij [8]). At any rate, if a is large
then one may expect to have difficulty in obtaining an accurate numerical solution to
(1.1), (1.2).
It is clear from (1.10) and (1.11) that both the fundamental solution (and hence
the structure of (1.1)) and the boundary conditions (1.2) will determine the magnitude
of the stability constants a and ft. Thus while it is possible to construct differential
equations (1.1) for which no boundary conditions exist such that a and /3 are of
moderate size, it is also possible to find boundary conditions for any differential
equation so that a and/3 are large. Consequently, if (1.1) can support a well conditioned
problem, the conditioning (i.e. the magnitude of a and fl) is intimately related to the
choice of the boundary conditions (see also Lentini, Osborne and Russell [5], Mattheij
[9]). It is for this reason that initial value techniques such as simple shooting may fail
to give satisfactory results for some well conditioned boundary value problems. The
fact that the associated initial value problem may be poorly conditioned can easily be
demonstrated for equations of the form (1.1) that have both increasing and decreasing
components in the fundamental solution. In fact, many problems that occur in practice
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DICHOTOMY AND WELL CONDITIONING IN BVP
91
have such fundamental solutions and it has become almost traditional to assume that
the solution space
can be split into a growing and a decaying part (i.e. there exists a dichotomy). This
idea is important not only in the study of the stability of numerical schemes (cf. [6], [8])
but also in the analysis of algorithms to solve them (cf. [8], [10]). An outstanding
reference to dichotomy is Coppel’s book [2] although we shall also use a slightly
weaker version of this notion.
DEFINITION 1.12. We say that 5 is dichotomic if there exists a splitting 6e ,-1 ( 0’092
and a constant t such that
P
Equivalently, for every fundamental matrix Y, there exists a projection matrix
such that
""
(1.13)
(a) 6e := { YPcIc e "},
(b) SC2 := { Y( I P)cIc a"},
for which the above holds. Of course, for a finite interval such a dichotomy always
exists but from a practical point of view we are only interested in bounds for t that
are not too large. For example, if one considers a class of singular perturbation problems
depending on a parameter e, then an acceptable bound for t would be independent
of e.
The concept of dichotomy introduced above bounds the growth of solutions in
6el and 2. A stronger concept of dichotomy has been used by a number of authors
(see for example [2], [5]) and is defined below.
DEFINITION 1.14. 5 is strongly dichotomic if there exists a constant K and a
such that for a fixed fundamental solution Y.
projection P
""
IY(t)PY-(s)l<-_t, t>-s,
r(t)(I P) r-’ (s)l--<
,
_-< s.
More generally, one may even allow an exponential behaviour of the solutions,
which leads to the following.
DEFINITION 1.15. 5 is exponentially dichotomic if there exists a constant to, positive
constants A, and a projection P
such that
""
IY(t)PY-l(s)l<-_tcexp(A(s-t)), t>--s, A>O,
IY(t)(I-P)Y-’(s)l<--exp((t-s)), t<--s, >0.
It is of interest to know how the concept of dichotomy relates to the stability
constants a and/3 of the BVP (1.1), (1.2). In fact, the two concepts are very closely
related and it has been shown by Mattheij [8] that one may estimate the stability
constant a (defined by (1.11)) in terms of/3 (defined by (1.10)) and t (the bound for
the dichotomy in Definition 1.12). In this paper we show that the converse is also true.
Specifically, we show that dichotomy constants t in Definitions 1.12 and 1.14 are
bounded by t 2or 2 or sometimes even t- a. Essentially this means that a well
conditioned problem must have a dichotomic solution space 6e. One may draw some
important conclusions from this. First, dichotomy is a useful concept in BVP, providing
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92
F. R. DE HOOG AND R. M. M. MATTHEIJ
a counterpart of Lyapunov stability in initial value problems. Second, in constructing
algorithms for solving BVP, one may (or should) design them such that they utilize
the dichotomy structure (cf. [5], 10]).
The paper is organised as follows. In 2 we introduce the fundamental concepts
of dichotomy. Then in 3 we give dichotomy bounds from bounds for the Green’s
functions. In 4 this is extended to the case of exponential dichotomy. The reverse
result, viz., that dichotomy implies well conditioning, is considered in 5. Finally we
give a number of numerical examples to demonstrate the concepts in 6.
2. Dichotomy and strong dichotomy. As the concepts of dichotomy and strong
dichotomy have both been used (see for example [5], [6]) in the analysis of numerical
schemes for BVP and of the algorithms for their implementation, it is worthwhile
investigating how these two concepts differ. We first note the following.
LEMMA 2.1. Let b and ’2 be defined by (1.13a, b). Then,
’
(s)
<-[r(t)PY-(s)l,
t>=s,
I,(s) <=IY(t)(I-P)Y-(s)I’
Proof Let b 5e,. Then there is a
c
b(t)
"Y(
t<=s"
such that
t)Pc.
Thus, for _-> s we have
I (t)l IY(t)Pcl IY(t)PY-’(s) Y(s)Pcl
--<IY(t)PY-’(s)I.
y(s)Pcl
I(s)l Y(s)Pcl
The second inequality follows in a similar manner. I3
Hence, strong dichotomy implies dichotomy. The essential difference between the
two concepts is that strong dichotomy also implies a directional separation between
the subspaces 1 and 2 and this is investigated below. It is known (see [7], [8]) that
the directional separation of the solution subspaces is an important aspect in the
conditioning of a BVP. For
5’1(t) { Y( t)PcIc "}
and
2( t) { Y( t)( I P)clc "},
it is useful to introduce "angles" as follows (cf. [7]).
DEFNrrION 2.2. Define the angle 0 -< r/(t) -< 7r/2 between 6e(t) and 6t’2(t) as
cos r/(t)= max (lx yl}.
xe ,YeSe
We now have the next theorem.
THEOREM 2.3. Let IY(t)PY-(t)l<=K for some Then
cot n(t) =<
Proof Let x Se(t), y oW2(t with Ixl-lyl-1 such that cos n(t)= Ix’yl. If x is
orthogonal to y the result is trivial and we assume that this is not the case. Now define
=x, :P=-(x’y)-y. Clearly, is orthogonal to +)3 and hence
.
(2.4)
cot
93
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DICHOTOMY AND WELL CONDITIONING IN BVP
and Se2 there is a cR" such that
Since
Substitution in (2.4) now yields
=
Y(t)Pc and
=
Y(t)(I-P)c.
Y( t)Pcl
cot l( t)
Ir(t)cl
IY(t)Pr-l(t) Y(t)cl
Ir(t)cl
<_- max
IY(t)py-l(t)dl
D
--<K.
From Theorem 2.3 we deduce that the angle between the subspaces
cannot become smaller than some threshold value
and
2
cot-(r).
3. Bounds for dichotomy. In this section we show that moderate stability constants
imply a dichotomy with a moderate r bound. Thus far we have not specified the
fundamental solution Y completely. A useful choice for our purposes is the following.
DEFINITION 3.1. Let Y be the fundamental solution of (1.1) such that
Bo r(0) + B, Y(1
I.
Note that we now have Y I and this simplifies the notation and the algebra
below.
We first consider the case of separable boundary conditions.
THEOREM 3.2. Let the BC be separable in the sense that
rank (Bo)= n-r,
en there is a projection P such that
rank (B)= r.
,
>
Y(t)PY-’()I
Y(t)(I-P)Y-(s)[a,
,
t<s,
where a is the stability constant given by (1.11).
Proo We first show that P:= BoY(O) is a projection. Let E be an ohogonal
matrix such that the last n-r rows of EB1 are zero. Since E is ohogonal, it follows
that
EBoY(O)E T + EB Y(1)E
= I.
On equating the last (n r) rows of the above equation we find that
has the structure
16
Since rank
n-r,
fi := EBo Y(O)E r
Pll, P12
0
I,’
PI =0 and hence [p]2=/3. Thus,
p2= ET[]2E
ETIE
p
and so P is a projection.
The result now follows from (1.11) on noting that
(t, s)
Y( t)py-l(s),
Y(t)(I- P) Y-I(s),
t> s,
t<s.
D
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94
F. R. DE HOOG AND R. M. M. MATTHEIJ
From Theorem 3.2, a strong dichotomy exists with K c when the boundary
conditions (BC) are separable. It then follows from Lemma 2.1 that the same result
is true for our weaker version of dichotomy.
For more general BC the situation is somewhat more complicated, the main reason
being that these BC do not provide a natural projection matrix P. We therefore proceed
by constructing separable BC so that the corresponding BVP is well conditioned. Once
this has been achieved, we can use the results of Theorem 3.2 to obtain bounds for
the dichotomy. In order to construct the separable boundary conditions, we monitor
the growth of solutions over the entire interval. Let the singular value decomposition
of the incremental matrix Y(1)Y-l(0) be given by
Y(1) Y-l(O):= UDV T,
where U, V are orthogonal matrices and D is a positive diagonal matrix with ordered
(3.3)
elements for which we shall use the notation
D=diag(-
(3.4)
with 0< dj_<-1,j
(a)
(b)
(3.5)
--1
dr+l,’’"
d,)
1,..., n. In connection with this diagonal matrix define also
Dl=diag (dl,’", dr, 1,..., 1),
1, dr+l,..., d,),
D: diag (1,
,
and
(3.6)
Pl=
We now define separated boundary conditions which are specified by the matrices
/o := Pl V r, /1 := (! P) U.
(3.7)
It is easy to verify that
/o I7(0) +/, 17(1)
I,
where
I7(t) := Y(t) Y-’(O) VD, Y(t) Y-’(1) UD2.
(3.8)
We therefore associate with Bo, B1 the corresponding Green’s function
t> s,
f(t)o f(o) -’(s),
G(t,s)=
(3.9)
I7,._
t<s.
I7(t)/}l I7.( 1
(s)
From (3.7) it is clear that rank (/o) r and rank (/l) n-r. Thus if we can
establish the stability constants for the problem (1.1) subject to the boundary condition
{
(3.10)
/oy(0) +/,y(1)
b,
we can use Theorem 3.2 to establish bounds for the dichotomy. In order to obtain
such bounds, we require some relations between the Green’s functions G and defined
by (1.4) and (3.9) respectively. These are derived below. It should be noted however
that the properties are independent of the form of Bo and B1 constructed above.
PROPERTY 3.11. For any fundamental solution Z,
t
(a)
(b)
(c)
Z(t)=G(t,s)Z(s)-G(t,u)Z(u),
O<-s<t<u<-l,
O<_s<t<u<-l,
z-l(t)=Z-l(u)G(u,t)-Z-(s)G(s,t),
(t, s)= G(t, s)- lT(t)[/lG(1, s)+/oG(0, s)].
DICHOTOMY AND WELL CONDITIONING IN BVP
95
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Proof.
(a)
Y(t)= Y(t)[BoY(O)+ B1Y(1)]
Y(t)BoY(O) Y-l(s) Y(s)+ Y(t)BI Y(1) Y-(u) Y(u)
G(t, s) Y(s)-G(t, u) Y(u).
8y substituting Z(t)= Y( t) Y-(O)Z(O) one immediately sees that a similar relation
also follows for Z instead of Y.
(b)
Y-I(t)=[BoY(O)+B Y(1)]Y-(t)
Y-(u) Y(u)BoY(O) Y-(t)+ Y-l(s) Y(s)B1Y(1) Y-(t)
Y-(u)G(u, t)- r-(s)G(s, t).
Again the relation for Z follows immediately.
(c) (i) For > s we have ((t, s) I7(t)/o (0) -’(s).
For IV-’(s) we can write (cf. (b)) ’-’(s)="-’(1)G(1, s)IT"-(0)G(0, s), so we obtain
(t, S)-- ]’(t)/o r(0) -I(1)G(1, s)- Tr(t)/oG(0, s)
IT(t)[I-/}, I7(1)] IT"-l(1)G(1, s)- ’(t);oG(O, s)
IT(t) IT-’(1)G(1, s)- IT"(t)/,G(1, s)- (t)oG(O, s)
G(t, s)- I’(t)/lG(1, s)- IT"(t)/oG(O s).
(ii) For < s, we obtain
(t, s)=- IT"(t)/, I7"(1)
IT(t)/},O(1, s) + ’(t)[I-o’(O)]Y-’(O)G(O, s)
=-"(t),G(1, s)+G(t, s)- (t)oG(O,s).
From (3.7) and (3.8),
(0)= P, VrY(O) Y-’(O)
PI,
-
o
and similarly
B Y(1)= I-P.
Thus, to the BC (3.10) we can associate the Green’s function
(3.12)
G(t,s):=
{(tlPl-’(s),
-"(t)(I-P,)-’(s),
t>s,
t<s.
We have then the following lemma.
LEMMA 3.13.
(a)
I(1)P, -’(s)l=16(1, s)l<=2,
f(o)(I- p,) f’-’(s)l [,(O, s)l <- 2,
(b)
(c)
IT"(t)l_<- 2.
Proof. (a) From Property 3.11(b) we obtain
IT"-’(s) I7"-1(1)G(1, s)- IT"-’(0)G(0, s),
96
F. R. DE HOOG AND R. M. M. MATTHEIJ
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SO
(1)P, IT"-’(s)l < IG(1, s)l + (1)P, -’(0)[ IO(0,
<= a + UD2P, D-’ VT]oI Ol + [D2Plot <- 2a.
(b)
](O)(I-P)-’(s)]<=]’(O)(I-P,)-’(1)]lG(1, s)]+[G(O,s)l<-2o.
(c) From Property 3.11(a) we have
Y(t) G( t, 0) Y(0) G( t, 1) Y( 1 ),
SO
I()1--< 19(0)1 +
We thus obtain the following (weak) result.
THEOREM 3.14.
(a)
(b)
f(s)P, xl / 2 2,
( t)( I P,)xl <
9()(t- n,)xl + 2
t> s,
Proo (a) From Propey 3.11(a) we have
( t)n,x G( t, s) (s)P,x- G( t, 1) (1)P, -’(s) (s)Px
[G(t, s)- o(t, )o(, s)]Y(s)P,x.
The result now follows from Lemma 3.13(a).
(b)
9(t)(t-n,)xl=16(t,o)f(o)(t-P,)x-6(,s)9(s)(t-P,)xl
[ (, 0) (0, s)
whence the result follows from Lemma 3.13(b).
We can also give a strong dichotomy estimate. For this we need the following
lemma.
LEMMA 3.15.
,
(a)
(b)
1(,,)1+4
(, ’)1 + 2% where
Proo
From Propey 3.11 (c),
O(t, s)= 6(t, s)- 9(t)[P, v6(0, s)+ (-n,)vo(1, s)]
6(t, s)+ Y(t)[noO(0, s)+ ,(, s)].
(,)
(**)
Since we know that I(t)l2a (Lemma 3.13) we obtain (a) from bounding (,).
Moreover, since (1, s)l2a and [(0, s)]2w%similarly find (b) from (**).
Now since the "aificially" separated BC Bo, B enable us to use Theorem 3.2
we immediately find from the previous lemma
THEOREM 3.16.
(a)
(a)
(b,)
(b2)
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DICHOTOMY AND WELL CONDITIONING IN BVP
97
4. Bounds for exponential dichotomy. In many realistic problems the solution space
Se is exponentially dichotomic. As we shall see in 5 (cf. also [8]), this implies a
certain exponential behaviour of the Green’s functions as well. Anticipating that result,
we consider here the reverse implications. Specifically, we investigate BVP, where
instead of (1.11) we have
(a)
(b)
(4.1)
[G(t,s)[<-aexp(/z(s-t)),
IG(t,s)l<-aexp(v(t-s)),
t>s, >0,
t<s,
v>0.
Using similar techniques as in the preceding section, we can show that (4.1) implies
an exponentially dichotomic solution space. We have the following.
THEOREM 4.2. Denote (t):= a[exp (-/.t) +exp (u(t-1))] and
r/(t) := c [exp (/z(s 1)) + exp (- vs)].
Let P1 be defined as in (3.6) and as in (3.8). Finally, recall that 3,= (IBoI+IBI).
Then
(al)
(a)
(bl)
(b_)
I(t)P19-(s)lexp(l(s-t))+(t)n(s), t>s,
IP(t)(I-P)-(s)l<-_ exp(v(t-s))+(t)q(s), t<s,
> s,
[(t)P1 -(s)] =< cr exp (/(s t)) + ),. r/(s),
I(t)(I-P)-(s)l- exp (v(t-s))+ /. r/(s),
t<s.
Proof. From Property 3.11(a) we have (cf. Lemma 3.13(c)):
I(t)l _-< a[exp (-/t) -exp (v(t- 1))].
From (,) in the proof of Lemma 3.15 we then derive, e.g. for > s"
](t, s)l_-< a exp ((s- t))+ (t)n(s),
giving the bound (al), as.in Theorem 3.16.
The other estimates are essentially similar (for (bl) and (b2) one should use (**)
in the proof of Lemma 3.15).
From Theorem 4.2 it appears that / and v more or less play the r61es of the
numbers A and respectively in Definition 1.15. With a slightly stronger assumption
we can even indicate a bound for the factor r (note that the estimates in Theorem 4.2
have "pollution terms" containing the (t) and r/(s) factors). This is done below but
for simplicity only for the case/ v.
THEOREM 4.3. Let I u in (4.1). Then
(a)
(a2)
> s,
I(t)P 9-1(s)l K exp (/z(s t)),
I(s)(I-P,)9-(s)]<=K exp (k(t- s)), t< s,
where
u=a+4c
Proofi
First let 0 <
u, /flt-sl<1/2-
--ce+3cu+2c 3, iflt-sl>1/2.
s _-< 1/2. Then
(t)n(s) 2. [exp (/z(s- t- 1))
+exp (/z(t- s- 1)) + exp (-/z(t + s)) + exp ((t + s- 2))]
-< 4c2 exp
(/z(s- t)).
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98
F. R. DE HOOG AND R. M. M. MATTHEIJ
Using Theorem 4.2(al) gives the required estimate for these values of s, t. A similar
result follows from Theorem 4.2(a2) for 0 < s <- 1/2. To show that these estimates also
hold for
s[ > 1/2 we first give an estimate for the numbers di (cf. (3.4))"
From Property 3.1 l(a),
It-
Y(1) Y-l(0)= G(1, 0)- Y(1) Y-(O)G(O, 1) Y(1) Y-l(0);
hence using (3.3) we see that di < e TM + a e-d2, for all i. Since di < 1 it follows that
s)[ from (,) in the proof
di <2a This estimate will now be used to estimate
of Lemma 3.15 more directly: We obtain, substituting Y(t) as is given in (3.8),
-.
I(t,
[t(t, s)[<-[G(t, s)[+lr(t) r-(O) VDP VTG(O, s)
Y(t) Y-’(1) UD2(I P,) UFG(1,
IO(t, s)l+l[o(t, 0)-G(t, 1) Y(1) Y-’(O)]VDIP VTG(O, s)[
+ [[G(t, 1)-G(t, 0)Y(0) Y-(1)]UD2(I-P1)UTG(1, s)[
_<-[G(t, s)[ + IO(t, 0)1 [G(0, s)] + [G(t, 1) UDEP UTG(O, s)
+ IG( t, 1)1 ]G( 1, s)[ + [G( t, 0) VD(I P) VTG( 1, s)
<--_exp(l(S--t))+3a2exp(tz(s--t))+2a exp(/z(-2+s+t))
ift>s.
The other cases are established similarly. 13
5. Further relationships between dichotomy and conditioning. In the previous sections we have shown that the solution space of (1.1) is dichotomic if there exists a
boundary condition of the form (1.2) such that the BVP (1.1), (1.2) is well conditioned.
We now consider the case when the solution space of (1.1) is strongly or exponentially
dichotomic and examine the existence of BC of the form (1.2) such that (1.1), (1.2)
is well conditioned. However, as the scaling of the matrices Bo and BI in (1.2) is
somewhat arbitrary, we impose the condition
BoB+B,B=I.
scaling Bo, B such .that
(5.1)
the rows of [Bo[B1] are
Note that (5.1) is equivalent to
orthonormal. Such a scaling is useful as it allows us to obtain a bound for 3 in terms
of a. Recall 1, where we noted the following:
PROPERTY 5.2. If Y is a fundamental solution of (1.1) satisfying Y= I and (5.1)
holds, then
(i) ]Y(t)l=lG(t, 1)G(t, 1)+ G(t, 0)G(t, 0)1
(ii) /3 <_-x/ a.
We now show that a strong dichotomy enables us to choose separated boundary
conditions such that the problem is well conditioned.
THEOREM 5.3. Let Y be a fundamental solution of (1.1). Let P be a projection and
be a constant such that
Then,
[Y(t)PY-(s)[<-_, t> s,
IY(t)(I-P)Y-’(s)I<=,
there exist matrices Bo, B so that
Ilylloo <--/IY[ +
t<s.
DICHOTOMY AND WELL CONDITIONING IN BVP
99
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with
y =/oy(0) +/ly(1),
and
Proof.
Consider the QR decomposition
where R e N is an upper triangular matrix and
If we now define
(t)= Y(t)R T,
then it is easy to verify that
/o I7"(0) +/117"(1)
I
and
G(t,s)=
(Y(t)PY-(s),
t>s,
_y(t)(i_p)y_(s)
t<s.
The result now follows.
For the sake of completeness we shall finally give a generalization of a result that
was discussed in [5] for separated BC and in [8] for discrete problems. We only state
it here for exponential dichotomy.
THEOREM 5.4. Let be exponentially dichotomic and let [Y(t)[<_-/3 (cf. 1.10); then
IG(t,s)l<-_[ee’-)+,flnole-e+lnleX<’-’)}],
IG(t,s)l<-_,[eX<S-’)+{lBole-’+lBleX<’-’)}],
t<s,
t> s.
Proof. For < s we have
G( t, s)
Y( t)n Y(1) Y-l(s)
Y(t)B Y(1)PY-I(s) Y(t)BI Y(1)(I-P)Y-l(s)
=-Y(t)B Y(1)PY-(s) Y(t)(I-BoY(O))(I-P) Y-(s)
Y(t)B(1)PY-I(s)+ r(t)(I-P) r-(s)- Y(t)BoY(O)(I-P) r-(s).
Hence
IG( t, s)l <-_ K e<’- + ln01
e
-
+
The result for > s is similarly proven.
The quantity fl, which was called the condition number in [9] therefore also plays
a r61e in estimates for the quantity a. However, the further away from this boundary
the less its influence is felt, particularly if It-sl is fairly small.
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100
F. R. DE HOOG AND R. M. M. MATTHEIJ
6. Examples. In this section we consider two examples, the first one demonstrating
a genuine exponential dichotomy and a well conditioned problem at the same time
and the second one a less well behaved problem.
Example 6.1. Let the ODE (cf. [9])
dy_[
dt-
(6.2)
1-19cos27rt
-l+19sin27rt
l+19sin2rt]
1+19cos2rt
y
and the BC
y(0) +y(1) 1
(6.3)
(so Bo B1 I) be given. It can easily be checked that Z with
0
(20(t 7r))
(6.4)
0
cos 7rt sin 7rt
exp (- 18 t)
is a fundamental solution of (6.2) from which it can be seen that the initial value
problem is poorly conditioned. For the Green’s function we then obtain (e.g. using
that Y(t)-- Z(t)[Z(O)+ Z(1)] -1)
(6.5)
]
Z(t)=[sinrt-cos 7rt] [exp
G(t,s)=[
c(s)]
s(t)
c(t)
s(s)
t> s,
t< s,
where s(t) sin zrt, c(t) cos 7ft.
These expressions show that the estimates in Theorem 5.4 are qualitatively sharp.
Note in particular the O(1) values of IG(t, s)l if It-sl 1.
Example 6.6. Consider the "artificial layer problem" (cf. [3], [6], [11])
(6.7)
(a)
u"=
-3Au
[-0.1, 0.1],
(A + t2) 2’
where A is a small positive number
(6.7)
(b)
u(0.1) =-u(-0"l)
=’Av
0.1
+0.01
(We did not transform the problem to the interval [0, 1] for reasons of similarity with
[3], [6], [11].)
This problem has been used as a test problem for a long time, which makes it
even more interesting to examine it on its conditioning aspects. In order to do so we
first rewrite it as a linear system for the vector [u u’]r:
(6.8)
u"
-3A
(A + t2)
u
0
u’
DICHOTOMY AND WELL CONDITIONING IN BVP
101
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A fundamental solution Z of (6.8) is given by
(6.9)
Z(t)
u’ v’
where
(a) u(t)
(6.10)
(, + t)
(b) u’(t)=
’/’
A
(A + t2) 3/2’
t2-A
(x + t) ’/’
t3+3At
(d) v’(t)=
(A + t2) 3/2"
(c)
v(t)
See also Figs. 6.1-6.4.
The BC then read
Bo
(6.11)
[10 00] gl--[ 00].
We obtain for A # 0.01,
(6.12)
[BoZ(-O.1)+BZ(O.1)]-=5(A +0.01)
1/2
-1
-0.1
A-0.01
+1
0.1
A-0.01
As is clear from (6.12) the problem is not even well posed for A =0.01, and hence
ill-conditioned for A close to 0.01; this was also noted by Deuflhard [3], who uses the
inverse of our notion of conditioning and calls this problem very "insensitive" for
A 0.01.
FIG. 6.1. A
10 -3, graph
of U.
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102
F. R. DE HOOG AND R. M. M. MATTHEIJ
FIG. 6.2. A
-0.1
0.1
0.0
-0.1
10 -3, graph
of U’.
0.1
N//
FIG. 6.3. A
10 -3, graph
of V.
However, apart from this ill-conditioning, more or less arising from the BC, also
an ill-conditioning occurs for very small h in that the norms of the Green’s function
may become large. Rather than giving analytical expressions for G(t, s) we have drawn
graphs for the norm of the first column (straight line) and second column (dotted line)
of G(t, s) for typical values of and s, see Figs. 6.5-6.8. As we clearly see from Figs.
6.1-6.4 we do not have exponential dichotomy and only dichotomy with a large bound
a. This is also apparent in the Green’s functions which have a large K bound of the
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DICHOTOMY AND WELL CONDITIONING IN BVP
FIG. 6.4. A
10 -3, graph
of V’.
"2
FIG. 6.5. Green’s function, A
10 -3, =0.
" 0.1s
-0.1
/
FIG. 6.6. Green’s function, A
10 -3,
0.05.
103
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104
F. R. DE HOOG AND R. M. M. MATTHEIJ
O.lt
FIG. 6.7. Green’s function, A
10 -3, s
0.
21
il
O \\
-1--
O’lt
V
FIG. 6.8. Green’s function, A
10 -3, s--0.05.
same order as a. It is simple to see from (6.10) that these constants are O(A-1/2). (In
the graphs we have used a fairly "large" A for aesthetic reasons; for smaller A one
may rescale these graphs accordingly.)
REFERENCES
C.
BOOR, F. DE HOOG AND H. B. KELLER, The stability of one-step schemes for first-order two-point
boundary value problems, this Journal, 20 (1983), pp. 1139-1146.
DE
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DICHOTOMY AND WELL CONDITIONING IN BVP
105
[2] W. A. COPPEL, Dichotomies in Stability Theory, Lecture Notes in Mathematics 629, Springer-Verlag,
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