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.
