On dichotomy and well conditioning in BVP

On dichotomy and well conditioning in BVP

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. https://doi.org/10.1137/0724008

DOI:

10.1137/0724008

Document status and date: Published: 01/01/1987 Document Version:

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ON DICHOTOMY AND WELL CONDITIONING IN

BVP*

F. R. DE HOOGf AND R. M. M. MATTHEIJ*

Abstract. Weinvestigatetherelationships betweenthestability bounds oftheproblemontheone hand and thegrowthbehaviourof thefundamentalsolution ontheotherhand. Itisshown that if thesestability bounds are moderate (i.e. ifthe problem iswell conditioned) thenthe homogeneous solution space is dichotomic,which means that it canbesplit intoasubspace of nondecreasingand acomplementarysubspace of nonincreasing modes. This is done by carefully examiningthe Green’s functions. Ifthese exhibit an exponentialbehaviourthen thesolutionspaceisalso exponentiallydichotomic.Ontheotherhand,wealso show that (exponential) dichotomy implies moderate stability constants,i.e.wellconditioning.Fromthis it followsthat both concepts are more orless equivalent.

Keywords, boundaryvalueproblems, conditioning, dichotomy, stability, Green’sfunctions AMS(MOS)subject classifications. 65L07,65L10,34B27

1. Introduction. Letusconsiderthe first-orderlinearsystem ofordinary differential equations

(1.1)

y:=y’-Ay=f, 0<t<l,

where

A[Lp(O,

1)]

"" andf[Lp(O,

1)]"

forsomep satisfying 1-<p-<oo.

We

seeka solution y subjectto the following boundaryconditions

(BC)

(1.2)

3y :=

Boy(0)

+

Bly(1 b

where

Bo,

_B1

Rnxn

and b

R".

Itiswell known that the boundaryvalueproblem

(BVP)

(1.1),

(1.2)

has aunique

solution ifand only ifY3Yisnonsingular where Yisany fundamental solutionof

(see

for example Keller

[4]).

In thiscase, we canformally write the solutiony as

(1.3)

y(t)=

Y(t)[BoY(O)+

B1Y(1)]-lb+

G(t, s)f(s)

ds,

wherewe have defined the Green’s

_function

G as

(1.4)

G(t,s)={

Y(t)[BY(O)+BIY(1)]-IBY(O)Y-’(s)’

t>s,

-Y(t)[BoY(O)+B1Y(1)]-B1Y(1)Y-(s),

t<s.

Thus, in principle,a knowledge of the fundamental solutionenables one to calculate theGreen’s functionand 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 willbe usingorthogonaltransformationswetake

I"

tobethe Euclideannorm on

".

Ofcourse, as all norms on

"

are equivalent, the use ofanothernormchangesonly thenumericalvalueoftheconstantsinthe subsequent results. Let

(1.5)

Ilull,:--

lu(s)l"

ds l

<-p<-oo,

*Receivedbythe editorsMarch 2, 1984,and in revised formSeptember 14, 1985.

tDivisionofMathematicsandStatistics,CSIRO,P.O. Box1965,Canberra,2601 Australia. MathematischInstituut,Katholieke UniversiteitToernooiveld, Nijmegen, the Netherlands.

89

and

Ilulloo

:-sup

its limiting value as p-->

.

Then, we find from

(1.3) (cf. (1.1), (1.2))

that,

1 1

(1.6)

Ilyll

_:-IIYlI

<=fll3y] +llyll,

----

1, p q

(1.7)

(1.8)

Y[BoY(O)+ B,

Y(1)]-’II

The most appropriate norm in

(1.6)

depends on the problem under consideration.

However,

intheinterestof clarity,weshallhenceforth consideronlythe caseof p 1.

Nevertheless,many of the arguments usedcanbe generalizedtothecaseofanarbitrary p.

When p= 1,

(1.6)

and

(1.8)

reduce to

(1.9)

Ily

_IIo-

-</3lyl

/

II-y

II,,

(1.10)

/3

YEY]-’lloo,

(1.11)

a=supIG(t,s)

_I.

In addition,iftheboundary 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)[

andhence

Thus,in this case, the stabilityconstant a gives ameasure for the sensitivity of

(1.1)

and

(1.2)

to changesinthe data.As has been shownin de

Boor,

de

Hoog

and Keller 1

],

the stabilityconstantsof manynumericalschemes approach those ofthe continuous

problem asthemeshsize goestozero

(see

also Mattheij

[8]).

Atanyrate,ifa is large then one may expectto have difficultyinobtaining an accuratenumerical solution to

(1.1),

(1.2).

Itis clear from

(1.10)

and

(1.11)

that both the fundamental solution

(and

hence

thestructureof

(1.1))

and theboundaryconditions

(1.2)

willdeterminethe 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 equationsothata

and/3

arelarge. Consequently,if

(1.1)

cansupportawell conditioned problem, the conditioning (i.e.the magnitude ofa and

fl)

is intimately relatedto the

choiceof theboundaryconditions

(see

also Lentini, Osborne and Russell

[5],

Mattheij

[9]).

Itis forthis reasonthatinitial valuetechniques suchas simple shooting mayfail to give satisfactory results for some well conditioned boundary value problems. The factthatthe associated initialvalueproblem may bepoorlyconditioned can easily be demonstrated for equations of theform

(1.1)

thathave both increasing and decreasing componentsinthe fundamentalsolution.In fact,manyproblems that occurinpractice

have such fundamental solutions andit has become almosttraditional to assumethat

the solutionspace

can be split into a growing and a decaying part (i.e. there exists a dichotomy). This

idea isimportantnotonlyinthestudyofthe stability ofnumericalschemes(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 weakerversionof this notion.

DEFINITION1.12. Wesay that 5 is dichotomic ifthereexists asplitting 6e ,-1(

_0’092

and aconstant t suchthat

Equivalently, for every fundamental matrix

Y,

there exists a projection matrix

P

""

such that

(a)

(1.13)

(b)

6e

:=

_{

_YPcIc

e

"},

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 ofview we are only interested in bounds for t that

are not toolarge.Forexample,if one considers a classof 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 ofsolutions in

6el

and

_2.

A stronger concept of dichotomy has been used by a number of authors

(see

for example

[2],

[5])

andis defined below.

DEFINITION 1.14. 5 is strongly dichotomic if there exists a constant K and a

projection P

""

such that fora fixedfundamental solution Y.

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.

DEFINITION1.15. 5 isexponentiallydichotomic ifthereexists a constantto,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

ofthe BVP

(1.1),

(1.2).

In fact, the two conceptsare very closely related and it has been shown by Mattheij

[8]

that one may estimate the stability

constant a (definedby

(1.11))

in terms

of/3

(definedby

(1.10))

and t

(the

bound for the dichotomyinDefinition

1.12).

Inthispaperweshow that theconverse is also true.

Specifically, we show that dichotomy constants t in Definitions 1.12 and 1.14 are bounded by t 2or2 or sometimes even t-a. Essentially this means that a well conditioned problem musthave a dichotomic solution space 6e. One may draw some

importantconclusionsfromthis.First, dichotomyis auseful conceptin

BVP,

providing

acounterpart of

Lyapunov

stabilityin initial valueproblems. Second, inconstructing algorithms for solving

BVP,

one may

(or should)

design them such that they utilize

the dichotomy structure (cf.

[5], 10]).

The paperisorganised 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 ofexponential dichotomy. The reverse

result, viz., that dichotomy implieswell conditioning, isconsidered in 5. Finally we

give a number ofnumericalexamples 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])

inthe analysis ofnumerical

schemes for BVP and of the algorithms for their implementation, it is worthwhile investigating how these two concepts differ. We first notethe following.

LEMMA 2.1. Let

_b

and

_’2

be

_defined

by

(1.13a,

b). Then,

<-[r(t)PY-(s)l,

t>=s,

’

(s)

I,(s)

<=IY(t)(I-P)Y-(s)I’

t<=s"

Proof

Let

b

5e,.

Then there is a c

"

such that

Thus, for _->s we have

I (t)l

IY(t)Pcl

I(s)l

Y(s)Pcl

b(t)

Y( t)Pc.

IY(t)PY-’(s)

Y(s)Pcl

y(s)Pcl

--<IY(t)PY-’(s)I.

The second inequality followsin a similar manner. I3

Hence,

strong dichotomy implies dichotomy.The essential differencebetween the

two concepts isthat strong dichotomy also implies adirectional separation between the subspaces

₁

and

₂

andthis is investigated below. Itisknown

(see

[7],

[8])

that the directional separation of the solution subspaces is an important aspect in the conditioning ofa 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]).

DEFNrrION2.2. 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 thenexttheorem.

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

orthogonaltoy the resultistrivial andweassumethat thisis notthecase.Nowdefine

=x,

:P=-(x’y)-y.

Clearly, is orthogonalto

+)3

and hence

(2.4)

cot

Since and

_Se2

there is a

cR"

such that

=

Y(t)Pc

and

=

Y(t)(I-P)c.

Substitution in

(2.4)

nowyields

cot

l( t)

Y(

t)Pcl

Ir(t)cl

IY(t)Pr-l(t) Y(t)cl

Ir(t)cl

IY(t)py-l(t)dl

<_- max --<K.

D

From Theorem 2.3 we deduce that the angle between the subspaces and

2

cannot becomesmaller than some thresholdvalue

cot-(r).

3. Bounds for dichotomy. Inthis section weshow that moderate stabilityconstants

imply a dichotomy with a moderate r bound. Thus far we have not specified the fundamentalsolution Ycompletely.Ausefulchoiceforourpurposesisthe following.

DEFINITION3.1. Let Ybe the fundamentalsolution of

(1.1)

suchthat

Bo

r(0)

+

B, Y(1

I.

Note that we now have

Y

I and this simplifies the notationand the algebra

below.

Wefirst considerthecase ofseparableboundary conditions. THEOREM 3.2. Letthe BC beseparable in the sense that

rank

(Bo)=

n-r, rank

(B)=

r.

en

there is a projection Psuch that

Y(t)PY-’()I

,

>

,

Y(t)(I-P)Y-(s)[a,

t<s, wherea is the stabilityconstant givenby

(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,itfollows that

EBoY(O)E

T

+

EB

Y(1)E

=

I.

On equating the last

(n

r)

rows of theabove equationwe find that

fi

:=

_EBo

Y(O)E

r has thestructure

16

Pll,

P12

0

I,’

Sincerank n-r,

PI

=0and hence

[p]2=/3.

Thus,

p2=

_ET[]2E

ETIE

p

andso P is aprojection.

The resultnow follows from

(1.11)

onnoting that

(t, s)

Y(

t)py-l(s),

Y(t)(I- P)

Y-I(s),

t>s,

t<s.

D

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 ourweakerversionof dichotomy.

FormoregeneralBCthe situation issomewhatmorecomplicated, themainreason

being that these BC donotprovideanaturalprojectionmatrixP. Wethereforeproceed by constructing separableBCsothat the correspondingBVPiswellconditioned.Once

this has been achieved, we can use the results of Theorem 3.2 to obtain bounds for the dichotomy. In orderto constructthe separable boundary conditions, we monitor

the growth ofsolutions over the entire interval. Let the singular value decomposition of theincremental matrix

Y(1)Y-l(0)

be given by

(3.3)

Y(1)

Y-l(O):=

UDV

T,

where

U,

Vare orthogonalmatricesand D is apositive diagonalmatrix withordered elements forwhich we shall use the notation

(3.4)

D=diag(-

--1

dr+l,’’"

d,)

with 0<

dj_<-1,j

1,..., n. In connection with this diagonal matrix definealso

(a)

Dl=diag

(dl,’",

dr, 1,...,

1),

(3.5)

(b)

_D:

diag

(1,

,

1, dr+l,..., d,),

and

(3.6)

Pl=

Wenowdefineseparated boundaryconditions which arespecified by thematrices

(3.7)

/o

:=

_Pl

V

r,

/1

:=

(!

P)

U.

It iseasy toverify that

/o

I7(0) +/, 17(1)

I,

where

(3.8)

I7(t)

:=

_Y(t)

Y-’(O)

VD,

Y(t)

Y-’(1)

UD2.

Wetherefore associate with

Bo,

B1

the corresponding Green’sfunction

{

f(t)o f(o) -’(s),

t>s,

(3.9)

G(t,s)=

I7(t)/}l

I7.(

1

I7,._

(s)

t<s.

From

(3.7)

it is clear that rank

(/o)

r and rank

(/l)

n-r. Thus if we can establishthe stabilityconstants fortheproblem

(1.1)

subjecttothe 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

suchbounds,werequiresomerelationsbetween theGreen’sfunctions

G

and

t

defined by

(1.4)

and

(3.9)

respectively. These are derivedbelow. It should be noted however that the properties are independent of the form of

Bo

and

_B1

constructed above.

PROPERTY 3.11. Forany

fundamental

solution

Z,

(a)

Z(t)=G(t,s)Z(s)-G(t,u)Z(u),

O<-s<t<u<-l,

(b)

(c)

z-l(t)=Z-l(u)G(u,t)-Z-(s)G(s,t),

O<_s<t<u<-l,

(t,

s)=

G(t, s)-

lT(t)[/lG(1, s)+/oG(0,

s)].

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 forZ 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 relationforZ

-

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),

andsimilarly

o

(0)=

P,

VrY(O)

Y-’(O)

PI,

B

Y(1)=

I-P.

Thus,to the BC

(3.10)

we can associatetheGreen’s function

{(tlPl-’(s),

t>s,

(3.12)

G(t,s):=

-"(t)(I-P,)-’(s),

t<s.

We

have then thefollowing lemma.

LEMMA

3.13.

(a)

I(1)P, -’(s)l=16(1,

_s)l<=2,

(b)

f(o)(I-

p,)

f’-’(s)l [,(O,

s)l

<-_2,

(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),

SO

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)

wehave

Y(t)

G(

t,

0) Y(0)

G(

t,

1)

Y(

1

),

I()1--< 19(0)1

+

Wethus obtainthe following

(weak)

result.

THEOREM 3.14.

(a)

f(s)P,

xl

/2

2,

t>s,

(

t)(

I

P,)xl

_<

(b)

9()(t-

n,)xl

+

2

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)

whencethe 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)

1(,,)1+4

,

(b)

(,

’)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 findfrom the previous lemma

THEOREM 3.16.

(a)

(a)

(b,)

(b2)

4. Bounds forexponential dichotomy. Inmanyrealisticproblemsthesolutionspace Se is exponentially dichotomic.

As

we shall see in 5

(cf.

also

[8]),

this implies a certainexponential behaviour of theGreen’sfunctionsaswell.Anticipating thatresult,

we consider here the reverse implications. Specifically, we investigate

BVP,

where

instead of

(1.11)

we have

(a)

[G(t,s)[<-aexp(/z(s-t)),

t>s, >0,

(4.1)

(b)

IG(t,s)l<-aexp(v(t-s)),

t<s, v>0.

Usingsimilartechniquesasinthe preceding section,we canshow that

(4.1)

implies

an exponentially dichotomic solution space. Wehavethe 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 that3,=

(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,

[(t)P1 -(s)]

=<

cr exp

(/(s

t))

+

),.

r/(s),

>

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 theproofofLemma3.15 we then derive, e.g. for

>

s"

](t,

s)l_-

<_{a exp}

_{((s- t))+ (t)n(s),}

giving the bound

(al),

as.inTheorem 3.16.

The other estimates areessentiallysimilar

(for (bl)

and

(b2)

one should use

(**)

inthe proof ofLemma

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 aslightly stronger assumption

we caneven indicateabound for the factor r

(note

that the estimatesinTheorem 4.2 have "pollutionterms"containing the

(t)

and

r/(s) factors).

This is donebelow but for simplicity only for the case/ v.

THEOREM4.3. Let_I u in

(4.1).

Then

(a)

(a2)

where

I(t)P

9-1(s)l

Kexp

(/z(s

t)),

>

s,

I(s)(I-P,)9-(s)]<=K

exp

(k(t- s)),

u=a+4c

u,

/flt-sl<1/2-t<s,

--ce+3cu+2c 3,

iflt-sl>1/2.

Proofi

First let 0

<

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)).

Using Theorem

4.2(al)

gives the required estimate for these values of s,t. A similar

resultfollows from Theorem

4.2(a2)

for0

<

s <-

_1/2.

_{Toshow that these}estimates also

hold for

It-

s[

>

1/2

we first givean estimate for the numbers

di (cf. (3.4))"

From

Property

3.1

l(a),

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

<

eTM

₊

_a

e-d

2,

for all i. Since

_di

<

1 itfollows that

di

<2a

-.

This estimate will now be used to estimate

I(t,

s)[

from

(,)

in the proof ofLemma3.15 more directly:

We

obtain, substituting

Y(t)

as is given in

(3.8),

[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. Furtherrelationships betweendichotomy and conditioning. Intheprevious

sec-tions we have shown that the solution space of

(1.1)

is dichotomic ifthere exists a

boundaryconditionofthe form

(1.2)

such that theBVP

(1.1), (1.2)

is wellconditioned.

Wenowconsiderthe case whenthe solutionspace of

(1.1)

isstronglyorexponentially dichotomic and examinethe 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 thecondition

(5.1)

BoB+B,B=I.

Note that

(5.1)

is equivalent to scaling

Bo,

B

such .that the rows of

[Bo[B1]

are

orthonormal. Such ascaling isuseful as it allows us toobtain abound for

3

in terms

ofa. Recall 1,where we noted the following:

PROPERTY 5.2. If Yisa

_fundamental

solution

_of

(1.1)

satisfying

Y=

Iand

(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.

Wenow show thata strong dichotomy enablesus tochooseseparatedboundary

conditions such that theproblem is well conditioned.

THEOREM 5.3. Let Ybea

_fundamental

solution

_of

(1.1).

LetPbea projectionand bea constantsuch that

[Y(t)PY-(s)[<-_,

t>s,

IY(t)(I-P)Y-’(s)I<=,

t<s.

Then, there exist matrices

Bo,

B

so that

Ilylloo

<--/IY[

+

with

and

y

=/oy(0)

+/ly(1),

Proof.

Considerthe

QR

decomposition

where Re

N

is an upper triangularmatrix and

If we now define

then itis easyto verify that

and

(t)=

Y(t)R

T,

/o

I7"(0)

+/117"(1)

I

(Y(t)PY-(s),

t>s,

G(t,s)=

_y(t)(i_p)y_(s)

t<s.

The result now follows.

Forthe sakeof completeness weshall finally giveageneralization ofaresult that

was discussed in

[5]

for separated BC and in

[8]

fordiscrete problems.Weonlystate ithere for exponential dichotomy.

THEOREM 5.4. Let beexponentiallydichotomicandlet

[Y(t)[<_-/3

(cf. 1.10);

then

IG(t,s)l<-_[ee’-)+,flnole-e+lnleX<’-’)}],

t<s,

IG(t,s)l<-_,[eX<S-’)+{lBole-’+lBleX<’-’)}],

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 resultfor >s is similarly proven.

Thequantityfl,which wascalledtheconditionnumberin

[9]

thereforealsoplays

a

r61e

in estimatesforthe quantity a.

However,

the further away fromthis boundary theless its influence is felt, particularly if

It-sl

isfairly small.

6. Examples. Inthis section we consider twoexamples, thefirstonedemonstrating

a genuine exponential dichotomy and a well conditioned problem atthe same time and thesecond one a lesswellbehaved problem.

Example 6.1. Let theODE

(cf.

[9])

(6.2)

dy_[

1-19cos27rt

l+19sin2rt]

dt-

-l+19sin27rt 1+19cos2rt y and the BC

(6.3)

y(0)

+y(1)

1

(so

Bo

B1

I)

be given. It caneasilybe checked that Zwith

(6.4)

Z(t)=[sinrt-cos

7rt]

[exp

(20(t

7r))

0

]

cos7rt sin7rt 0 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)

G(t,s)=[

s(t)

c(t)

c(s)]

s(s)

t<s,

where

s(t)

sinzrt,

c(t)

cos7ft.

These expressions show that theestimates inTheorem 5.4 arequalitatively sharp.

Note in particular the

O(1)

values of

IG(t,

s)l

if

It-sl

1.

Example 6.6. Considerthe "artificial layerproblem"

(cf. [3], [6], [11])

-3Au

(6.7)

(a)

u"=

(A

+ t2)

2’

[-0.1,

0.1],

where A is a small positive number

t>s,

0.1

(6.7)

(b)

u(0.1) =-u(-0"l)

_=’Av

+0.01

(We

did nottransform the problemtotheinterval

[0,

1]

forreasonsof similaritywith

[3], [6], [11].)

This problem has been used as a test problem for a long time, which makes it even more interestingto examine it on itsconditioning aspects. Inorderto do so we first rewrite it as a linearsystem for the vector[u

u’]r:

(6.8)

u"

-3A

u

(A

+ t2)

0

u’

Afundamental solutionZ of

(6.8)

is given by

(6.9)

Z(t)

u’

v’

where

(6.10)

t2-A

(a)

u(t)

(c)

v(t)

(,

+

t)

’/’

_(x

_{+ t)}

’/’ A

t3+3At

(b)

u’(t)=

(d)

v’(t)=

(A

+ t2)

3/2’

_(A

_{+ t2)}

3/2" See also Figs. 6.1-6.4.

The BC then read

(6.11)

Bo

[10

00]

gl--[

00].

We obtainfor A#0.01,

-1 +1

(6.12)

[BoZ(-O.1)+BZ(O.1)]-=5(A

+0.01)

1/2 _{-0.1 0.1}

A-0.01 A-0.01

As is clear from

(6.12)

the problem is not even well posed for A =0.01, and hence

ill-conditionedforA close to0.01;this was alsonoted by Deuflhard

[3],

whousesthe

inverse of our notion of conditioning and calls this problem very "insensitive" for A 0.01.

FIG.6.1. A 10-3,graphofU.

-0.1 0.0 0.1 FIG.6.2. A 10-3,graphofU’.

-0.1

N//

0.1

FIG.6.3. A 10-3,graphofV.

However,

apart fromthis ill-conditioning, moreorlessarising fromthe

BC,

also

an ill-conditioning occurs for very small h in that the norms of the Green’s function

maybecomelarge.Rather than giving analytical expressions for

G(t, s)

wehave drawn

graphsforthe normof thefirstcolumn (straight line)andsecondcolumn

(dotted

line)

of

G(t, s)

for typical values of and s, see Figs. 6.5-6.8.

As

we clearly see from Figs. 6.1-6.4we donothave exponential dichotomyandonly dichotomywith alarge bound

a. This is also apparentin the Green’s functions which have alarge K bound of the

FIG.6.4. A 10-3, graphofV’.

"2

FIG.6.5. Green’s function, A 10-3, =0.

-0.1

"

0.1s

/

FIG.6.6. Green’s function, A 10-3, 0.05.

O.lt

FIG.6.7. Green’s function, A 10-3,s 0.

21

il

O

\\

_O’lt

-1--

_V

FIG.6.8. Green’s function, A 10-3,s--0.05.

same orderas a. Itis simpletosee from

(6.10)

thatthese constants are

O(A-1/2).

(In

the graphs we have used a fairly "large" A for aesthetic reasons; for smaller A one mayrescale these graphs accordingly.)

REFERENCES

C.DEBOOR,F.DEHOOGANDH. B.KELLER,Thestability_ofone-stepschemes_{forfirst-order}two-point boundaryvalueproblems,thisJournal,20(1983),pp. 1139-1146.

[2] W. A. COPPEL, Dichotomies inStability Theory,Lecture Notesin Mathematics629, Springer-Verlag, NewYork, 1978.

[3] P.DEUFLHARD, Nonlinear equationsolversinboundaryvalueproblemcodes,inLectureNotes Computer Science76, B.Childsetal., eds.,Springer-Verlag, New York, 1979, pp.40-66.

[4] H. B. KELLER, Numerical solutionoftwo-pointboundary value problems, SIAM RegionalConference Series inAppliedMathematics24, Philadelphia,1976.

[5] M.LENTINI,M. R. OSBORNEAND R. D. RUSSELL, The closerelationships between methods_forsolving two-pointboundary value problems,thisJournal, 22(1985),pp. 280-309.

[6] M. LENTINI AND V. PEREYRA, A variable order, variablestep,finite difference method_for nonlinear multipoint boundary-valueproblems,Math.Comp.,28 (1974),pp. 981-1003.

[7] R.M.M. MATI’HEIJ,Characterization_ofdominantand dominated solutions_oflinear recursions,Numer. Math.,35(1980), pp.421-442.

[8]

.,

Estimates_fortheerrors inthe solution_oflinearboundaryvalueproblems duetoperturbations, Computing,27(1981),pp.299-318.

[9] Theconditioning_oflinearboundaryvalueproblems,thisJournal,19(1982),pp. 963-978. [10] Decouplingandstability_ofBVPalgorithms,SIAMRev.,27(1985),pp. 1-44.

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orthonormalization,thisJournal,14(1977),pp. 40-70.