Foundations of Computational Mathematics 0521003490, 9780521003490

Citation preview

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24-29 St Giles, Odord OX I 3LB. United Kingdom 1nc titles below are available from booksellers, or, in ca!li.C of difficulty, from Cambridge University Press.

46

p-adic Analysis: a short course on recent wort., N. KOBLITZ

.59 66 86 SR

Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Topological topics, I.M. JAMES (ed) FPF ring theory, C. FAITH & S. PAGE Polylopes and symmetry, S.A. ROBERTSON Diophantine equations 0'Yer function fields, R.C. MASON Varieties of constructi'Yt mathcmatk:s, D.S. BRIOOES & F. RICHMAN Methods of differential gcomc1ry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time tcchniques for anaJysts and probabilists, L. EGG HE Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDEL YI & WANG SHENGWANG Compa:tification of Siegel moduli schemes, C.·L. CHAI Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) Lectures on the asymptotic theory of ideals, D. REES Representations of algebras, P.J. WEBB (ed) Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups· St Andrews 1985, E. ROBERTSON & C. CAMPBEU. (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M.-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advaoccs in homotopy theory, S. SALAMON, 8. STEER & W. SUTHERLAND (eds) Ge

h

is

0,

N

q u e s t i o n s

e a c h

- »

a t

o n

h

t h a t

c o m p u t i n g

c o m p u t e d

e n d - p o i n t s

h ~ * x *, N u*(x)

i

b e

i n c r e a s i n g

t h e

i n t o

e l e m e n t

W h a t

a s

c r e d i t

{

i n

a t

e a c h c a n

i g n o r e d .

s l o p e

h a r d

o n

I ( v h )

m o n o t o n e

[0,1]

c o n t i n u o u s

s o

c o n v e r g e s

infinite

b e h a v i o u r

a r e affine

d i s c r e t e

uo

i" a m o n g

i n t e g r a n d ) ,

b e

u*h

s u b d i v i d i n g

i n t e g r a l

t h e

i n s t a n c e

t o

f i n i t e - e l e m e n t

m e s h

w h i c h

Vh

R e m a r k a b l y ,

f a c t

T h i s

(2.2)

f o r m

first

n a t u r a l

m i n i m i z e

f u n c t i o n

e x p l i c i t

m i n i m i z e r h

E

3

m i n i m i z e r s

T h u s

0

a

w h e r e

of

S u r e l y

l i m ^ o

u

I (

h )

=

d x = — h - \

x

0 ! (see

B a l l

&

K n o w l e s

[12])

t h a t

if

1




of

I

call

it

i n

( 0 , 1 )

initial t h e t h e n

0, v ( l )

m i n

1

=

=

1 }

c o n s i d e r e d

a b o v e )

t h e n

0.

A i

L a v r e n t i e v T h e

=

f u n c t i o n s

A 3 / 2

A o o

T h e

1)

different

f u n c t i o n

s p a c e s

p h e n o m e n o n

(see

c a l c u l a t i o n

a b o v e

repulsion J ( u '

J

property, ' ' )

- >

00.

c a n

L a v r e n t i e v h a s

t h a t

t h e if

u ^

b e

dif-

[37]

for

f o l l o w i n g G

A s / 2

4

J.M.

I n in

t h e

M a n i a

b u t

py

[ 1 3 , 1 4 ] , for

n o t t h e

elliptic

x , - u , p .

e x a m p l e

s t r i c t l y

e x a m p l e

=

I {(u«))

in

A

=

p

{v

i n t e g r a n d

e

s m a l l

is a

s m o o t h

h a s

d e r i v a t i v e

i n d i c a t e d

m i z e r

is

a

i n t e r v a l u ^

E

h o l d s

i n

m i z e r s ?

b y

[(x4 f[{x*

u ' f u f

: e

- u

-

t h e

v ( - l ) 2

)

2 S

p

is a n

0,

m i n / > i n f /

(2.4),

I

a t t a i n s of

its

t h e

r e p u l s i o n

- >

u *

a.e.

C o n s i d e r

n u m e r i c a l t h e

I ( u )

of

=

of

/

B a l l

/i

c o n v e x

&

in

>

0

fpp

>

for

all

i n

N o t e

2e

>

w*

t h a t

0.

of J

t h e

H e r e ,

in

A i

for t h a t

[ — 1 , 0 ) U ( 0 , 1 ] T h e

b u t

L a v r e n t i e v

(2.4)

e v e r y

e q u a t i o n

h o l d s

->

h o l d

m i n i m i z i n g

1 } .

^ 3 , a n d

a l s o

i n

i n

t h e

s u c h

m i n i -

t h e

w h o l e

f o r m

t h a t

if

00.

s i n g u l a r

for

M i z e l

c a n

/ ( « * ) .

I ( u ^ )

m e t h o d s

p r o b l e m

=

E u l e r - L a g r a n g e

t h e n

of

|x|3"signa\

i n f i m u m

p r o p e r t y

C o m p u t a t i o n

p o s s i b l e

=

is

p r o p e r t y

m i n i m i z e r

~

6

(2.3)

e q u a t i o n

u*(x)

p

dx

satisfies

=

s o l u t i o n

eul]

2

J

u ^

+

a b s o l u t e

w h e r e

b y

> [14]

2

— : r )

fpp(X)U,p)

p r o b l e m

E u l e r - L a g r a n g e

=

r e p u l s i o n

— l , v ( l )

inf

T h e

t h e

s

( u

s h o w n

=

+ e p

=

w a s

w h i c h

f o r m

[ - 1 , 1 ] .

a r e

x

for

t h e

2 . 2

W h a t

4

of t h e a t

w i t h

J_

a s

a n d

g i v e n

0, t h e r e

- f o o

in

is

( x

>

s m o o t h

A 3

=

e

t h o s e

( - l , l )

s o l u t i o n

p h e n o m e n o n

A s

p

W ^

f ( x , u , p )

sufficiently

i.e.

/ ( # , u , p )

H o w e v e r ,

p h e n o m e n o n

i n t e g r a n d s , a n

i n t e g r a n d

c o n v e x .

L a v r e n t i e v

S u c h

t h e

B a l l

m i n i m i z e r s

d e t e c t i n g

s u c h

s i n g u l a r

m i n i -

m i n i m i z i n g

f ( x , u , u

x

) d x

J a i n

A i

w h e r e A u

a , first

f r o m

/3

a r e

{ u e

g i v e n

m e t h o d its

=

W ^ f a b )

=

a , u ( b )

=

/?},

c o n s t a n t s .

p r o p o s e d

d e r i v a t i v e .

: u(a)

T h u s

b y

B a l l

g i v e n

& e

K n o w l e s >

0

w e

[12] c o n s i s t s

in

d e c o u p l i n g

m i n i m i z e

rb I ( u , v )

=

/

f ( x , u , v ) d x

J a a m o n g

p i e c e w i s e

affine

f u n c t i o n s

u

i n

A \

o n

a

u n i f o r m

m e s h

of

size

h ,

C o m p u t a t i o n

a n d

f u n c t i o n s

v

t h e

c o n s t r a i n t

t h a t

a r e

of

p i e c e w i s e

m i n i m i z e r s

c o n s t a n t

o n

5

t h e

s a m e

g r i d ,

s u b j e c t

t o

b

tp(ux

- v ) d x


for all

all

0

is

v

e

p i , P 2

s u i t a b l e t e e i n g

s u i t a b l e

R ,

w h e r e

£

R .

g r o w t h

t h a t

t o

m i n i m i z e r s

,

n

U

/ i , e - > 0 ,

effect

M a n i a affine b u t i n

f ( x , u , p ) / M ( ^

c a l l y

a s

M

e n s u r i n g

in

A \

h




v?(P2))

T h e n

b e

u n d e r g u a r a n -

s h o w n

f u n c t i o n

after

ip(p)

+

p a r t i c u l a r

it

p o s s i b l y

inf

is

t h a t

t h e

I ( u )

=

t h a t

7 ,

c o n -

e x t r a c t i o n

of

a

J ( w * ) .

=

t h e




/ M

w h i c h

m o n o t o n i a s

|p|

~»

0 0 ,

i n t e g r a l

) d x

x

a

p h e n o m e n o n ,

t h a t

e x a m p l e , for

i n s t e a d ,

s u c c e e d s

f o r m

M ,

( x , u , u

A l t h o u g h

a s

f a c t

(2.1) a m o n g

i n t e g r a n d

m i l d

M

r u l e

[12]; i n

x .

t r u n c a t e d

f

i n

of

t r u n c a t i o n

|p|

s u i t a b l e

i n

s t u d i e d

s p e c i a l

z

u

is

t h e

00.

a l s o

t r u n c a t e d

h a s

f u n c t i o n s ,

l a r g e

it

t h i s

m a y

in

t h e

t h e s i n c e

t h e n

m e t h o d

n o t c a s e

b e

is

a

c a n

w o r k s

e a s y

w h e n

i n t e g r a l it

w e

t o

I

m i n i m i z e

t h e o r e t i c a l l y , find

a n

a t t a i n s

(2.3), t h e r e l o c a l

first

a

a b s o l u t e m i n i m u m

is t h e

m i n i m i z e r

it

d a n g e r

of

IM

for

of all

M ) .

i n t e r e s t i n g

u s e

for

t r a p e z o i d a l

w h e n

L a v r e n t i e v

m i n i m i z e r

sufficiently

o n

(i)

C((p(pi)

=

a t t a i n e d )

=

w h e n e v e r

d r a w b a c k s IM


u , p )

- > i n

A \

r e p l a c e d =

^ , P )

5

o n e

g i v e n of

t h e

b o u n d a r y

t r a c t i o n .

d e r i v a t i o n

l e a d s

y

specified

a p p l i e d

r e q u i r e m e n t

t h i s

( 3 . 2 )

m i n i m i z a t i o n

t o

c o n s e q u e n c e

(3.3)

a r e a

m e a n i n g f u l ,

a n o t h e r

=

t

c o n d i t i o n

w h e r e

p a r t i c u l a r

g u a r a n t e e

(see,

m i n i m i z a t i o n

G a r r o n i

c o m p l e t e

c o m p l e x i t y

i n t e g r a t e d .

N o n l i n e a r

p o s i t i v e

d e t

T o

s p a c e

of

&

p h e n o m e n o n .

c o r r e s p o n d i n g

ft.

m o d e l

h y p o t h e s i s

c o m p u t a t i o n

f a m i l i a r

c o n d i t i o n

m a p p i n g .

o n

b e

a m

p h e n o m e n a

t h e

t h e

I

M a s o

b o u n d e d

a t

f u n c t i o n

f u n c t i o n

i n t r o d u c t i o n ,