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scattering theory are integral equations with Toeplitz and. Hankel kernels respectively. It is shown that these facts can be used to reduce the integral equations to ...
Birkh~userVerlag Basel

Integral E q u a t i o n s Vol. i/i 1978

FAST ALGORITHMS

and O p e r a t o r

Theory

FOR THE Ib~EGRAL EQUATIONS SCATTERING

B. D. O. A n d e r s o n

OF THE INVERSE

PROBLEM t and T. Kailath

The G e l f a n d - L e v i t a n and M a r c h e n k o equations of inverse scattering theory are integral equations with Toeplitz and Hankel kernels respectively. It is shown that these facts can be used to reduce the integral equations to differential equations which can be solved with an order of magnitude less c o m p u t a t i o n than generally envisaged.

i.

In solving

tional

the inverse

scattering

to use one of two approaches~

Levitan chenko

(perhaps

as m o d i f i e d

(cf. the surveys

equations.

if a d i s c r e t i z a t i o n 0(N3).

gorithms

developed

Toeplitz

and Hankels

in

we shall assume there tions,

Both lead to integral

~(')

equations

N

small time grows

[~],

[5]

struc-

can be exploited

to

in the sense that

intervals

as 0(N 2)

These reductions

We shall consider shift

or that due to Mar-

solving them--fast

into

normal,

phase

[2]).

in these

fast algorithms

the number of calculations

2.

that due to Gelfand-

by Krein),

[i],

it is conven-

In this note, we indicate how the special

ture of the kernels develop

problem

is made,

rather than,

are based on special for integral

equations

as

alwith

kernels. the S-wave

case,

are no bound

states.

is given satisfying

the G e l f a n d - L e v i t a n - K r e i n

and for convenience Assuming

the standard

procedure

that a condi-

is as follows.

Define TThis work was supported by the Army Research Office under Contract DAAG29-77-C-00~2, by the Air Force Office of Scientific Research, Air Force Systems Command, under Contract AF44-620-74-C-0068 and the A u s t r a l i a n Research Grants Committee.

132

2

A N D E R S O N et al

oo

F(k) : exp

- ~

~

k' dk'

(2.1)

i dk

(2.2)

0

and H(r) = ~

Let

o

cos kr

Ir(k)12

F2r(') be the s o l u t i o n of the integral

equation

2r

I'2r(U) + H(u) + f 0 Then the p o t e n t i a l

F2r(S)H(s-u)

El], [2]

is

2 [F2r (2r)] + 4 F2r(2r)

V(r) = -2 ~d Defining

a(2r;2r-u) T

a(T;s)

ds = 0, 0 ~ u ( 2r (2.3)

= -F2r(U),

+ [ a(T;u)H(u-t)

one can r e w r i t e

du = H(T-s),

(2.4)

(2.3) as

0 ~ s ~ T

(2.5)

J

0 The T o e p l i t z n a t u r e of the k e r n e l

H(u-t)

can then be ex-

p l o i t e d to show that the i n t e g r a l e q u a t i o n can be r e d u c e d to a d i f f e r e n t i a l [4]. [3],

e q u a t i o n form w h i c h

The r e l e v a n t

differential

+ ~]

a(t;s) = - a ( t ; t - s ) a ( t ; 0 )

To see the p o t e n t i a l examine

advantages

Suppose that

~(k,£)

(2.6)

with

is k n o w n for

(2.6)

of this reduction,

a (naive) way of solving it numerically.

a d i s c r e t i z e d v e r s i o n of

we

Consider

~(j;£) = a(jA;£A).

0 ~ £ ~ k ~ j.

Then

yields ~(j+l,£+l)

while

e q u a t i o n turns out to be

[4] [~

(2.5)

is easier to solve

~(j+l,0)

= ~(j,£)

comes from

~(j+l~0) = H ( ( j + I ) A , 0 )

- ~(;j-£)~(£;0)A

(2.5)

(2.7)

as

j+l - [ e(j+I;£)H(£A~0)A £=i

(2.8)

M o r e o v e r , to get ~(j;£) for any £ ~ j takes p r o p o r t i o n a l .2 .3 to 3 o p e r a t i o n s ; this c o m p a r e s with the order of ] op-

133

ANDERSON erations of

allows upper 3.

needed

Eq.

to find

(2.5).

a(')

3

via a direct

Note that this recursive

us to easily limit

et al

T

accommodate

in

The Marchenko

discretization

solution

increased

values

method of the

(2.5). calculations

of

V(r)

the definitions S(k) = exp[2i6[k)],

= ~ _i~

A0(t)

[I],

[2] start

+~ f [S(k)- l]eikt

with

dk (3.1)

7 [

Next

one defines

A(r,t)

for

0 ~ r ( t

as the solution

of A(r,t)

= A0(r+t)

+ ] A(r,s)A0(s+t]

ds

(3.2)

r

w i t h the result

that V(r) = -2 dA(r~r) dr

The integral

equation

fact ean be exploited [5].

The first

uation

(3.2)

has a Hankel

by a method

~(r,t)

kernel,

patterned

step is to introduce

for a "dual"

(3.3)

after

a second

which that

integral

in eq-

quantity

= A0(r+t)

- i ~(r's)A0(s+t)

ds

(3.4)

r

(This equation

is discussed

(3.4),

using

derive

the following

the Hankel

structure

coupled

below.)

From

of the kernel

(3.2) A0

and

one can

equations:

~--- ~(r t) - ~t A(r,t) Dr '

= [A(r,r)

- ~(r r)]~(r,t)

(3.5)

~-~ ~(r,t)

= [A(r,r)

+ ~(r,r)]A(r,t)

(3.6)

- ~-~ ~ A(r,t)

To see the potentialities simple

discretization

-~(~-i,i)

-~(j,i)

~(j,i)

of these

equations,

we consider

a

as

=

- ~(j,i)

-~(j-l,i)

134

further

+ ~(j,i+l)

+ A[~(j,j)

- ~(j,j)]~(j,i)

+ A[~(j,j)

+ s(],j)]e(j,i)

(3.7)

= - ~(j~i+l)

+ e(j,i)

.

(3.8)

4

If

A N D E R S O N et al

~(k,i)

and

supplies i ~ j.

e(k,i)

~(j-l,i)

are known for all j ( k ¢ i,

and

The q u a n t i t i e s

(3.8)

supplies

~(j-l,j-l)

t a i n e d by d i s c r e t i z i n g

(3.4)

e(j-l,i)

and

~(j-l,j-l)

on the d i a g o n a l

scheme needs to be i n i t i a l i z e d by solving for some large fixed tains values With

r'.

diseretized

lations grow at a rate 4.

are ob-

t = r.

(3.2)

and

into

of N

A(r,s)

for

subintervals,

The (3.4)

Then the scheme r e c u r s i v e l y

at d i s c r e t e points

[r,r']

(3.7)

for all

ob-

r < r'. the calcu-

0(N2).

We d e s c r i b e why the added e q u a t i o n

(3.4)

has a solution.

Let a s u p e r s c r i p t hat denote an o p e r a t o r a s s o c i a t e d w i t h an ^

i n t e g r a l kernel.

Define

A(t,s) = ~(t+s),

then

^

^

~ = H - HA

has ~(t,s) = ~

sin kt

o

i

sin ks dk

i

cos ks dk

ir(k)12

^

and

0 = H + HA

has

®(t,s) = ~

cos kt

0 Evidently,

I +

fr(k)J 2

> 0, I + @ > 0.

Now it is shown in

[2]

^

that there exists a V o l t e r r a o p e r a t o r (I+~l(I+H)_(z+~a). = I

(I-A 0) =

Here

H

for w h i c h

and

( I + N I 6 I + ~ I C I + ~ a) = I + (I+K)HA(I+~ a)

A 0 = A0(t+s)

is r e s t r i o t e d

to

0 ~ t,s < =.

(4.1)

It fol-

lows that ^

I+A 0 = (l+~)(I+8)(I+~ a) > 0 and t h e r e f o r e that 5.

The n u m e r i c a l

(3.4)

is solvable.

schemes used above were meant only to il-

l u s t r a t e the o r d e r - o f - m a g n i t u d e

r e d u c t i o n s p o s s i b l e by ex-

p l o i t i n g the special T o e p l i t z or H a n k e l kernels.

M o r e w o r k needs to be done on the n u m e r i c a l analy-

sis a s p e c t s of these 6.

The above

schemes and equations.

ideas suggest that e x t e n s i o n to m a t r i x p r o b l e m s

could well be p r o f i t a b l e . possible

135

s t r u c t u r e of the

for solving

(Such e x t e n s i o n s

are k n o w n to be

integral equations with Toeplitz

and

ANDERSON et al

5

Hankel kernels.) Also, the role of the dual integral equation C3.4) needs to be further investigated. What about e q u a t i o n s with non-Toeplitz or non-Hankel kernels?

It has

been shown that the above algorithms can be extended to arbitrary kernels, but the amount of computation goes up by a factor related to a suitably defined "index" of non-Toeplitzness or non-Hankelness of the kernel--see of such possibilities

[3], [4].

The role

in more general versions of the inverse

scattering problem remains to be investigated. REFERENCES [i] [2] [3] [43

[53

K. Chadan and P. C. Sabatier, The Inverse Problem of Quantum Scattering Theory~ New York: Springer-Verlag, 1977. L. D. Faddeyev, "The Inverse Problem in the Quantum Theory of Scattering~" J. Math. Phys.~ vol. 4, no. i, pp. 72-i04, January 1963. M. G. Krein, "Continuous Analogs of Propositions on Polynomials Orthogonal on the Unit Circle," Dokl. Akad. Nauk SSSR vol. i05~ no. 4~ pp. 637-640, 1955. T. Kailath~ L. ~jung a n d M. Morf~ "Generalized KreinLevinson Equat~Qns for t~e Efficient Computation of Fredholm Resolvents of Nond~splaeement Kernels, t' in Surveys in Mathematical Analysis: Essays Dedicated to M. G. Krein, New York: Academic Press, 1978. B. Levy, M. Morf and T. Kailath "Fast Algorithm for ~ntegral Equations w~th Toeplitz and/or Hankel Kernels," tO be submitted for publication.

B. D. O. Anderson Dept. of Electrical Engineering The University of Newcastle New So. Wales 2308, Australia

136

Thomas Kailath Department of Electrical Engineering Stanform University Stanford~ CA 94305 U.S.A.