fulltext

Maximum-likelihood method for numerical inversion

of Mellin transform

M. IQBAL(*)

Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia

(ricevuto il 18 Marzo 1996; approvato il 23 Luglio 1996)

Summary. — A method is described for inverting the Mellin transform which uses

an expansion in Laguerre polynomials and converts the Mellin transform to Laplace transform, then the maximum-likelihood regularization method is used to recover the original function of the Mellin transform. The performance of the method is illustrated by the inversion of the test functions available in the literature(J. Inst. Math. Appl., 20 (1977) 73; Math. Comput., 53 (1989) 589). Effectiveness of the method is shown by results obtained through demonstration by means of tables and diagrams.

PACS 02.60.Nm – Integral and integrodifferential equations.

1. – Introduction

The Mellin transform defined by

M(f) 4p(s) 4



0 Q

ts 21_{f(t) dt}

(1.1)

is one of the most important integral transforms. This arises in a natural way in the solution of boundary value problems concerning an infinite wedge.

If p(s) and f(t) satisfy appropriate conditions (see Snedon [1]), one can prove that the relationship p(s) 4Mf(t) defines a one-one correspondence between p(s) and f(t).

(*) E-mail address: miqbalHdpc.kfupm.edu.sa

This problem (1.1), formally solved by the inversion formula f(t) 4 1 2 pi



x 2iQ x 1iQ t2s_{p(s) ds , a ExEb ,} (1.2)

cannot be solved analytically in most cases. Therefore, in practice, either the asymptotic expansions are employed or numerical methods are used. In the latter case, one of the methods is numerical quadrature of (1.2) having the obvious disadvantage that one numerical quadrature is required per value of t, thus turning out to be time-consuming when the value of f is required at many points t.

Since the Mellin transform is essentially the two-sided Laplace transform, one can reduce the inversion of the Mellin transform to inversion of Laplace transforms.

In the present paper we study that eq. (1.1) has the form of a Fredholm integral equation of the first kind and it is well known that the problem of solving such equations is basically ill-conditioned. Many physicists have discovered after much wasted effort that it is essential to understand this feature before attempting to compute solutions.

The ill-posedness of the Laplace transform inversion, in the case where f  L2_(R 1),

can be investigated by means of the Mellin transform [2]. In practice, however, the case of an infinite set of equidistant points was investigated by Papoulis [3]. Several other methods [2, 4-14] have been proposed and a detailed review and comparison is given in Davis [11] and Talbot [15].

The previous methods do not include regularization techniques. Regularization methods have been discussed by Brianzi [7], Essah and Delves [12], Ang [4], Gelfgat [13] and Wahba [16].

In particular, the theory is used to tackle the Laplace transform inversion in a well-conditioned (regularized) manner. This difficult numerical problem, which is frequently encountered by physicists and engineers, is still the subject of much attention in the literature. Finally, we include some specific numerical examples to illustrate and demonstrate clearly the need to consider information content in order to avoid obtaining meaningless results.

2. – Conversion to the Laplace transform

We will consider an approach to reduce the inversion of the Mellin transform to inversion of Laplace transform. This approach is based on the series expansion of the original function f(t). It has been proposed in [17].

For the numerical inversion of the Mellin transform, we consider the following expansion of Laguerre polynomials [18]:

f(t) 4 1 2 r e 2rtO2

!

j 40 Q Aj(r) lj

g

1 2 rt

h

, (2.1) where Aj(r) 4

!

m 40 j

u

j m

v

g

2 1 2 r

h

m bm (2.2)

with

bm4

p(m 11)

m! , r being a free parameter .

(2.3)

We consider the function g(s) which is represented by

g(s) 4

!

m 40 Q (21)m_b msm, s  C with Re (s) D0 , (2.4) where bmis given by (2.3).

By using eq. (1.1) in (2.4), one can easily see that g(s) is a Laplace transform. In fact if we set L ]f; s( 4



0 Q e2st_{f(t) dt ,} it follows that g(s) 4 L ]f; 2s( . (2.5)

The integral in (2.5) may not exist; however, we may always provide values of a such that the Laplace transform related to p(s 1a) is

g(s) 4 L ]f(t) ta

; 2s( which does exist.

Now, we have to consider



0

Q

f(t) e2st_{dt 4g(s) .}

(2.6)

3. – Solving Laplace transform (2.6) by the regularization method

We are interested in solving eq. (2.6). We make the following substitution. Let

s 4bx and _{t 4b}2y_{, b D1.} (3.1) Then (2.6) becomes g(bx ) 4



2Q Q log b e2bx 2y_{Q f(b}2y_{) b}2y_{dy .} (3.2)

Multiplying both sides by bx_{, we obtain the convolution equation}



2Q Q

K(x 2y) F(y) dy4G(x) ,

where

.

/

´

G(x) 4bx_g(bx ) 4sg(s) , K(x) 4 log b e2bx_bx 4 log bs e2s, F(y) 4f(b2y_{) 4f(t) .} (3.4)

In order that we can apply our deconvolution method to eq. (3.3), G(x) has usually compact support, i.e. G(x) K0 as xK6Q.

In Tikhonov [19] regularization, the approximate solution Fl in (3.3) is defined as C(F ; l) 4 ] VKF2GV2

21 lV(F)( ,

(3.5)

which is minimized over the subspace Hp

L₂and l D0 is a regularization parameter. Here V is some nonnegative “stabilizing” functional which controls the sensitivity of the regularized solution Flto perturbation in G.

We shall restrict our attention to p-th order ( p 42) regularization of the form

C(F ; l) 4VKF2GV2

21 l V F(p)V22

(3.6)

which is minimized over the subspace Hp

L2. Both norms in (3.6) are L₂. F(p)denotes the p-th derivative of F and l the regularization parameter.

4. – p-th order regularization

Consider the smoothing functional C(F ; l) of eq. (3.6) with V(F) 4VF(p)_V2

2. In the

case of convolution equation, (3.3) can be written as

C(F ; l) 4VK(x)

_*

F(x) 2G(x) V2

21 l V F(p)V22,

(4.1)

where

_*

denotes the convolution and both norms in (4.1) are L2. Using Plancherel’s

identity, the convolution theorem for Fourier transforms and the property (F

×

(p)_{) 4} (iv)p_F

_×

_{, we can write (4.1) as}

C(F ; l) 4 1

2 p VK

×

(v) F

×

(v) 2G

×

(v) V

2

21 l V (iv)pF

×

(v) V22;

(4.2)

C(F ; l) is minimized with respect to F when

F

×

_{(v) 4} K

×

G

×

NK

×

N21 lv2 p 4 z(v ; l) G

×

(v) K

×

(v) , (4.3) where z(v ; l) 4 NK

×

N 2 NK

×

N21 lv2 p . (4.4)

z(v ; l) is called the p-th order filter or stabilizer; therefore, (4.3) can be written as Fl(y) 4 1 2 p



2Q Q z(v ; l) G

×

(v) K

×

(v) exp [ivy] dv ; (4.5) Fl(y) in (4.5) is approximated by Fn , l(x) 4

!

q 40 N 21 _G

×

N , q K

×

_{N , q} Zq ; l, where zq ; l4 NK

×

N , qN 2 NK

×

N , qN 2 1 lv2 pq (4.6)

is a filter function dependent on a parameter l and vAq4 2 pq (cut-off frequency),

vq4

.

`

/

`

´

v A_q, v A N 2q, 0 GqE 1 2 N , 1 2 N GqGN21 .

The optimal l in (4.4) is still to be determined and N is the number of data points.

5. – The filter in a stochastic setting

Several authors [7, 12, 13] have treated filters in various frameworks. In this section we relate the p-th order convolution filter (4.6) to certain spectral densities which play a role in the ML optimization of l in the next section. Data function Gn and the

underlying function UnTNsuch that

Gn4 Un(xn) 1enfUn1 en.

(5.1)

We identify both ]Un( and ]en( with independent stationary stochastic processes.

Since in general the expectation E

(g(x

j)

)

is not zero. In the limit N KQ, hK0, for any

discrete process xnwe may write (see, for example, [8]) xn4



0 1

exp [ 2 pivn] dSx(v) ,

(5.2)

where Sx(v) is a stochastic process defined on [ 0 , 1 ). The essential property of Sx we

require is

Lemma 5.1. The variance of any integral

s

u(v) dSx(v) is given by

s

Nu(v) N2dGx(v), where dGx(v) 4E

(

dSx(v) N

)

.

Gx(v) may be interpreted as a spectral distribution function, and accordingly we

Now consider FNTN with F 4 (Fn) f

(F

N(xn)

)

defined by (KF)n4 Un, n 40, 1, R, N21, with K given by

K 4c diag (K

×

N , q) c H_,

(5.3)

where c is the unitary matrix with elements

crs4 1 kN exp

k

2 p N irs

l

, r , s 40, 1, R, N21 . (5.4) From (5.2) we have Fn4

!

m 40 N 21 ](K21)mn(



0 1 exp [ 2 pivm] dsu(v) 4



0 1 [K

×

_N(v) ]21_{exp [ 2 pivm] dS} u(v) , (5.5) where K

×

_{N(v) 4} 1 N n 40

!

N 21 Knexp [ 2 pivn] dSu(v) . (5.6)

Assume that Fn is estimated by

!

N 21m 40lmGn 2m, where ]lm( is a filter which we shall

relate to zq ; land ]Gn( is periodically continued for n  [ 0 , N). Then the error Fn2

!

N 21 m 40 lmGn 2m (5.7) is given by (5.8)



0 1 exp [ 2 pivn]

(

[K

×

_N(v) ]21 2 l

×

N(v)

)

dSU(v) 2



0 1 exp [ 2 pivn] l

×

_N(v) dSe(v) ,

where l

×

_N(v) is defined as in (5.6). From lemma 5.1, the variance of this error is clearly



0 1 N[K

×

N(v) ] 21 2 l

×

N(v) N 2_P u(v) dv 1



0 1 Nl

×

N(v) N 2_P e(v) dv (5.9)

which is minimized when

l

×

_N(v) K

×

_{N(v) 4} Pu(v)

Pu(v) 1Pe(v)

. (5.10)

Since the Fourier coefficients of the filtered solution must satisfy

F

×

_{N , q ; l4 l}

×

_{N , q}G

×

_{N , q4 z}q ; lG

×

N , q[K

×

N , q]21,

we find

Using (5.10), we have zq ; l4 Pu(qh) Pu(qh) 1Pe(qh) . (5.11)

6. – Optimization by maximum likelihood (ML)

We now simply relate the filter (5.11) to the p-th order filter (4.6). Assuming that the errors are uncorrelated, Pe(v) has the form

Pe(v) 4s24 const ,

(6.1)

where s2_{is the unknown variance of the noise in the data. Choosing}

Pu(v) 4 s2_NK

×

_{N(v) N}2 lvA2 p , (6.2) where vA 4

.

`

/

`

´

2 Npv , 2 Np( 1 2v) , 0 GvE 1 2 , 1 2 G v E 1 ,

then yields (4.6) from (5.11). Moreover, the spectral density for ]Gn( is then PG(v) 4Pu(v) 1Pe(v) 4s2

y

1 1 NK

×

N(v) N 2 lvA2 p

z

, whence PG(qh) 4s2( 1 2zq ; l)21. (6.3)

The statistical likelihood of any suggested values of s2_{and l may now be estimated}

from the data. Following Whittle [20], the logarithm of the likelihood function PG is

given approximately by const 2 1 2 q 40

!

N 21 [log PG(qh) 1I(qh)OPG(qh) ] , (6.4) where I(v) 4

N

!

n 40 N 21 Gnexp [22pivn]

N

2

is the periodogram of the data, with

I(qh) 4NG

×

N , qN

We now maximize (6.4) with respect to s2_{and l. The partial maximum with respect to}

s2 may be found exactly (in terms of l) with the maximizing value of s2 given by

s2 4 1 N q 41

!

N 21 NG

×

N , qN 2 ( 1 2z2 q ; l) . (6.5)

The maximum with respect to l may then be found [16] by minimizing

VML(l) 4 1 2 N log

k

q 41

!

N 21 NG

×

N , qN( 1 2 zq ; l)

l

2 1 2 q 41

!

N 21 log ( 1 2zq , l) . (6.6)

Thus the optimal regularization parameter is given by the minimizer of a simple function of l, depending on the known Fourier coefficients G

×

_{N , q} and K

×

_{N , q}. The expression in (6.6) is minimized using the quadratic interpolation technique to obtain a minimum. We have experienced more than one minimum in all the test examples.

In order to solve (3.3) we need to choose two numbers xmin and xmax such that

NG(x) N E e whenever x E xmin and x Dxmax. In what follows we choose e 41024

(

maxNG(x)N

)

. We find xmin and xmax as the smallest and largest solutions of the

nonlinear equation G(x) 4e, we may then pose the deconvolution problem (3.3) on the interval [ 0 , T ], where T 4xmax2 xmin. Since the size of the essential support of G(x)

depends upon “b”, we write T 4Tb. For a fixed number N of equidistant data points

]xn(, we have spacing h 4 TbON.

We have minimized (6.6) with respect to l for values of b F1 and computed the LQ

error of the resulting solution with the values of the true solution. We found that, minimum value of V(l), for optimal l, the L_Q error of the regularized solution is the least.

7. – Numerical examples

In this section we tabulate the results of the above method applied to test examples [4, 7, 2, 17]. All data functions have the property g(s) 4O(s21_{) and since it is}

a severely ill-posed problem, therefore, no noise is added apart from machine rounding error. In all cases we have taken N 4256 data points.

TABLEI. T h b a l V(l) L_Qnorm Fig. 12.50 0.04883 10.0 1.0 0 .9809 31028 _{0 .7772 3 10}3 _{0.006 1} TABLEII. T h b a b l V(l) L_Qnorm Fig. 9.0 0.03516 5.0 5.0 2.2 0 .11 310210 _{0 .83955 310}3 _{0.01 3}

TABLEIII. T h b a b l V(l) L_Qnorm Fig. 12.01 0.04727 10.0 1.0 1.0 0 .21 310210 _{0 .1038 310}2 _{0.001 5}

1.00

0.80

0.60

0.40

0.20

0.00

0

1

2

3

4

5

6

7

8

9

10

11

t

f(t)

Fig. 1. – Mellin transform by ML.sNumerical solution for a 410.0, –m– true solution.

f(t)

t

1.0

0.5

0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t f(t) 0.160 0.140 0.120 0.100 0.080 0.060 0.040 0.020 0.000 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Fig. 3. – Mellin transform by ML.sNumerical solution for a 45.0, –m– true solution.

f(t) t 0.15 0.10 0.05 0.00 0.0 0.5 1.0 1.5 2.0

f(t)

t

0.350

0.300

0.250

0.200

0.150

0.100

0.050

0.000

0

2

4

6

8

10

12

14

16

Fig. 5. – Mellin transform by ML.sNumerical solution for a 410.0, –m– true solution.

t f(t) 0 10 20 30 40 0.4 0.2 0.0

Fig. 6. – J. G. M. Whirter and E. R. Pike. pN(v) as a function of v. b 44p and noise A1023:

Example 1. This example has been taken from ref. [17], case 5, p. 79: f(t) 4e2at_{, a 41.0 ,}

g(s) 4 1 s 1a ,

P(s) 4a2s_{G(s) , Re s D0 .}

The numerical calculations are given in table I and fig. 1. The optimal solution is compared with Theocaris solution in fig. 2.

Example 2. This example has been taken from ref. [17] case 4, p. 79: f(t) 4e2at_{sin bt , a 45.0 , b 42.2 ,}

g(s) 4 U sin V

s2_{1 2 Us cos V 1 U}2 , U 4 (a

2

1 b2)21 O2_{, V 4tan}21_{(bOa) ,}

P(s) 4 (a21 b2)21 O2G(s) sin

(s tan

21(bOa)

)

.

The numerical calculations are given in table II and fig. 3. The optimal solution is compared with Theocaris solution in fig. 4.

Example 3. This example has been taken from refs. [4, 7, 2]: f(t) 4tae2bt _{for a 41.0 , b41.0 ,}

g(s) 4 G(a)

(s 1b)a 11 , P(s) 4b2(s 1 a)_{G(s 1a) .}

The numerical calculations are given in table III and fig. 5. The optimal solution is compared with McWhirter’s solution in fig. 6.

8. – Conclusion

Our method worked well over all the three test examples. The results obtained are shown in fig. 1, 3, 5. Theocaris and McWhirter’s solutions are also presented in fig. 2, 4, 6 for comparison purposes. Our results are better and are shown over a wider range of t than Theocaris and McWhirter’s results.

* * *

The author acknowledges the excellent research and computer facilities availed at King Fahd University of Petroleum and Minerals, Dhahran during the preparation of this paper.

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