SOLVING INVERSE PROBLEMS FOR THE HAMMERSTEIN ...

International Journal of Pure and Applied Mathematics 

————————————————————————– 

Volume 60 No. 4 2010, 393-408 

SOLVING INVERSE PROBLEMS FOR THE HAMMERSTEIN 

INTEGRAL EQUATION AND ITS RANDOM ANALOG 

USING THE “COLLAGE METHOD” FOR FIXED POINTS 

H.E. Kunze 1 § , D. La Torre 2 , K.M. Levere 3 , E.R. Vrscay 4 

1,3 Department of Mathematics and Statistics 

College of Physical and Engineering Science 

University of Guelph 

50, Stone Road East, Guelph, Ontario, N1G 2W1, CANADA 

1 e-mail: hkunze@uoguelph.ca 

3 e-mail: klevere@uoguelph.ca 

2 Department of Economics, Business and Statistics 

University of Milan 

Milan, ITALY 

e-mail: davide.latorre@unimi.it 

4 Department of Applied Mathematics 

University of Waterloo 

Waterloo, Ontario, N2L 3G1, CANADA 

e-mail: ervrscay@math.uwaterloo.ca 

Abstract: Many inverse problems in applied mathematics can be formulated 

as the approximation of a target element u in a complete metric space (X,dX) 

by the fixed point ¯x of an appropriate contraction mapping T : X ↦→ X. The 

method of collage coding seeks to solve this problem by finding a contraction 

mapping T that minimizes the so-called collage distance d(x,Tx). In this paper, 

we develop a collage coding framework for inverse problems involving deterministic 

or random Hammerstein integral operators. Such operators are used to 

model image blurring. We illustrate the method with examples. 

AMS Subject Classification: 45Q05, 45R05, 60H25 

Key Words: Hammerstein integral equations, inverse problems, fixed point 

equations, random fixed point equations, collage theorem 

Received: March 6, 2010 c○ 2010 Academic Publications 

§ Correspondence author

394 H.E. Kunze, D. La Torre, K.M. Levere, E.R. Vrscay 

1. Introduction 

In this paper, we examine the Hammerstein integral equation in the context of 

Banach’s celebrated contraction mapping theorem. The Hammerstein integral 

operator is 

(Tu)(x) = 

� 1 

0 

K(x,y,u(y))dy + l(x). 

In imaging applications, u represents an input signal (or image) and the kernel 

K describes how this signal has been blurred. Of course, the kernel can take 

on many forms, as an image might be blurred for a wide variety of reasons 

ranging from effects of motion capture to focus problems to diffraction effects 

of the medium in which a photograph was taken. The term l(x) represents the 

contribution of additive noise to the recorded image. The result of applying the 

Hammerstein integral operator, Tu, is a blurred and noised image. 

Normally, Banach’s Theorem is used to establish the existence and uniqueness 

of solutions to the related fixed point equation u = Tu. Our objective, 

however, is to show that it can be used to solve the associated inverse problem: 

given a recorded signal as well as l, denoise and deblur the image by 

recovering the parameters in K (and possibly its functional form). 

We also show that the random Hammerstein integral and its related inverse 

problem can be treated with a random version of Banach’s Theorem. 

As we have shown in previous works by Forte and Vrscay [7], [8], La Torre 

and Iacus [9], [10], and Kunze et al [12], [13], [14], [16], [17], a number of inverse 

problems in applied mathematics can be formulated as the approximation of 

a target element x in a complete metric space (X,dX) by the fixed point ¯x of 

an appropriate contraction mapping T : X ↦→ X. In practice, one considers a 

family of such contraction mappings Tλ, λ ∈ R n , with respective fixed points 

¯xλ, and seeks to minimize the approximation error � x − ¯xλ �. However, it is 

generally impractical to search directly for contraction maps Tλ that minimize 

this approximation error. Instead, we resort to the following simple consequence 

of Banach’s Fixed Point Theorem, referred to as the collage theorem in fractal 

imaging literature (see Barsley et al [3]): 

dX(x, ¯x) ≤ 1 

1 − c dX(x,Tx), for all x ∈ X, (1) 

where c is the contraction factor of T. One now seeks a contraction mapping 

Tλ that minimizes the so-called collage error dX(x,Tx). In other words, we 

look for a contraction mapping T that sends the target x as close as possible 

to itself. This is the essence of collage coding that has provided the basis of

SOLVING INVERSE PROBLEMS FOR THE HAMMERSTEIN... 395 

most, if not all, fractal image coding and compression methods (see Fischer [6], 

Forte and Vrscay [7], [8], and Lu [18]). Indeed, many problems in the parameter 

estimation literature (see, for example, Anderson [1] and Milstein [19]) can be 

formulated in such a collage coding framework [17], [14], [13], [12]. 

The structure of the paper is as follows. In Section 2, we state some relevant 

mathematical results regarding contraction mappings and their fixed points, 

and the inverse problem of approximation by such fixed points. In Sections 

3 and 4, we examine Hammerstein-type integral equations and also present a 

collage coding method to solve inverse problems. After discussing the deterministic 

case in these two sections, we present a parallel discussion in Sections 

4-6, ending in the latter section with examples of the solution of the inverse 

problem for the random Hammerstein integral equation via the collage theorem 

for random fixed point equations. 

2. Fixed Point Equations and the “Collage Theorem” 

For the benefit of the reader, we mention some important mathematical results. 

Definition 1. Let (X,dX) be a metric space. A mapping T : X ↦→ X is 

said to be Lipschitz on X if there exists a positive constant K > 0 such that 

dX(Tx,Ty) ≤ KdX(x,y), (2) 

for all x,y ∈ X. If the inequality is satisfied for K < 1, then T is said to be 

contractive. In this case we define its contraction factor c to be the infimum of 

all K values. 

Theorem 2.1. (Banach Fixed Point Theorem) Let (X,dX) be a complete 

metric space. Also let T : X ↦→ X be contractive. Then there exists a unique 

¯x ∈ X such that ¯x = T ¯x. Moreover, for any x ∈ X, dX(T n x, ¯x) ↦→ 0 as n ↦→ ∞. 

A simple triangle inequality argument along with Banach’s Theorem yields 

the following result. 

Theorem 2.2. (“Collage Theorem”, see Barnsley et al [2], [3]) Let (X,dX) 

be a complete metric space and T : X ↦→ X a contraction mapping with 

contraction factor c ∈ [0,1). Then for any x ∈ X, 

dX(x, ¯x) ≤ 1 

1 − c dX(x,Tx), (3) 

where ¯x is the fixed point of T. 

Theorem 2.3. (Continuity of Fixed Points, see Centore and Vrscay [5])


Let (X,dX) be a complete metric space and T1,T2 be two contractive mappings 

with fixed points ¯x1 and ¯x2. Then 

1 

dX(¯x1, ¯x2) ≤ 

1 − min{c1,c2} dsup(T1,T2), (4) 

where 

dsup(T1,T2) = sup dX(T1(x),T2(x)) (5) 

x∈X 

and ci is the contractivity factor of Ti. 

Another manipulation of the triangle inequality involving x, Tx and ¯x yields 

the following interesting result. 

Theorem 2.4. (“Anti-Collage Theorem”, see Vrscay and Saupe [20]) 

Assume the conditions of the collage theorem above. Then for any x ∈ X, 

dX(x, ¯x) ≥ 1 

1 + c dX(x,Tx). (6) 

3. The Hammerstein Integral Equation 

We are now interested in finding solutions of the fixed point equation u = Tu 

where T is a Hammerstein integral operator having the form 

(Tu)(x) = 

� 1 

0 

K(x,y,u(y))dy + l(x). (7) 

Here, l : [0,1] ↦→ R is a given function and K : R × R × R ↦→ R satisfies 

|K(α1,β,γ) − K(α2,β,γ)| ≤ C1(β,γ)|α1 − α2| (8) 

and |K(α,β,γ1) − K(α,β,γ2)| ≤ C2(α,β)|γ1 − γ2|. (9) 

Furthermore suppose that for u ∈ L 2 ([0,1]) the function ξ(y) := C1(y,u(y)) 

belongs to L 2 ([0,1]). These next results are quite simple, and we mention them 

for the benefit of the reader. The first result states the existence and uniqueness 

of the solution in the space C([0,1]). 

Theorem 3.1. Let l : [0,1] ↦→ R be a continuous function, then: 

(i) (Tu)(x) is a continuous function. 

(ii) if �C2�∞ := sup α,β∈[0,1] |C2(α,β)| < 1 then T : C([0,1]) ↦→ C([0,1]) is a 

contraction.


Proof. Given a continuous function u ∈ C([0,1]), we have 

|(Tu)(x) − (Tu)(z)| ≤ 

� 1 

0 

≤ |x − z| 

≤ |x − z| 

|K(x,y,u(y)) − K(z,y,u(y))|dy + |l(x) − l(z)| 

� 1 

0 

�� 1 

0 

|C1(y,u(y))|dy + |l(x) − l(z)| 

|C1(y,u(y))| 2 � 

dy 

1 

2 

+ |l(x) − l(z)| (10) 

and this shows the continuity of the function (Tu)(x). To prove contractivity 

we have 

d∞(Tu,Tv) = sup 

≤ sup 

x∈[0,1] 

� 1 

0 

x∈[0,1] 

|(Tu)(x) − (Tv)(x)| 

|K(x,y,u(y))−K(x,y,v(y))|dy ≤ sup 

x∈[0,1] 

� 1 

Consider now the operator T on the space L 2 ([0,1]). 

0 

|C2(x,y)||u(y)−v(y)|dy 

≤ �C2�∞d∞(u,v). � 

Theorem 3.2. Suppose l ∈ C([0,1]) and C2 ∈ L 2 ([0,1] × [0,1]) with 

�C2�2 < 1. Then T : L 2 ([0,1]) ↦→ L 2 ([0,1]) is a contraction. 

Proof. We first prove that Tu ∈ L2 ([0,1]). For each fixed x1,x2 ∈ [0,1], we 

compute 

|Tu(x1) − Tu(x2)| ≤ 

� 1 

0 

≤ |x1 − x2| 

≤ |x1 − x2| 

|K(x1,y,u(y)) − K(x2,y,u(y))|dy + |l(x1) − l(x2)| 

� 1 

0 

�� 1 

0 

|C1(y,u(y))|dy + |l(x1) − l(x2)| 

|C1(y,u(y))| 2 � 

dy 

1 

2 

+ |l(x1) − l(x2)|. 

So the function (Tu)(x) is bounded on [0,1] and then it belongs to L2 ([0,1]). 

To prove contractivity we have 

�Tu − Tv� 2 2 = 

≤ 

� 1 

0 

� 

� 

� 

� 

� 1 

0 

≤ 

� 1 

0 

|(Tu)(x) − (Tv)(x)| 2 dx 

K(x,y,u(y))dy + l(x) − 

� 1 �� 1 

0 

0 

� 1 

0 

� 

� 

K(x,y,v(y))dy − l(x) � 

� 

|K(x,y,u(y)) − K(x,y,v(y))|dy 

�2 

dx 

2 

dx


≤ 

� 1 �� 1 

0 

0 

|C2(x,y)||u(y) − v(y)|dy 

≤ 

� 1 

0 

�2 

dx 

�C2� 2 2�u − v�22 dx = �C2� 2 2�u − v�22 . � 

Corollary 3.1. If C2 ∈ L 2 ([0,1] × [0,1]) and �C2�2 < 1 then the fixed 

point equation Tu = u has a unique solution ū belonging to L 2 ([0,1]) and for 

all u0 ∈ L 2 ([0,1]) the sequence un = Tun−1 converges to ū. If �C2�∞ < 1 then 

ū is continuous. 

4. Inverse Problem for the Hammerstein Integral Equation 

For each fixed u ∈ R we have K(x,y, ·) ∈ L2 ([0,1] × [0,1]), and then 

∞� 

K(x,y,u) = ai,j(u)φj(x)φi(y), (11) 

i,j=1 

where φi(y) is an orthonormal basis in L 2 ([0,1]). For u ∈ C([0,1]), we suppose 

that the functions ai,j(u) satisfy: 

(i) ai,j(u) ∈ L2 ([umin,umax]) where umin = min 

0≤x≤1 u(x) and umax = max 

0≤x≤1 u(x), 

so that 

∞� 

ai,j(u) = ai,j,kψk(u), 

k=1 

where ψk(u) is a basis for L2 ([umin,umax]). 

� 

� ∞� 

� 

� 

� 

� 

(ii) Let |ai,j(u) − ai,j(v)| = � ai,j,k(ψk(u) − ψk(v)) � 

� 

� 

k=1 

≤ bi,j|u − v|. Then 

� 

� 

� 

� 

�� 

� 

|K(x,y,u) − K(x,y,v)| = � 

� (ai,j(u) − ai,j(v))φj(x)φi(y) � 

� 

� i,j 

� 

� 

� 

� 

�� 

� 

� 

≤ � 

� bi,jφj(x)φi(y) � 

� |u − v|, 

� i,j � 

� 

� 

� 

�� 

� 

� 

so C2(x,y) = � 

� bi,jφj(x)φi(y) � 

� in (9). 

� 

� 

i,j 

We truncate each sum at upper limit n. Given a target solution u ∈ C([0,1])


and the function l(x), the inverse problem consists of finding coefficients ai,j,k 

in the expansion of K so that u is approximately admitted as the fixed point 

of the corresponding Hammerstein integral operator. Our collage distance is 

�u − Tu� 2 2 = 

� � 1 � � 

� 1 

2 

� 

� 

�u(x) − K(x,y,u(y))dy − l(x) � 

� dx 

0 0 

� 

� � 1 � 

= � 

� 

0 � − 

� 

n� 

n� 

� 1 n� 

�2 

� 

ujφj(x) + ljφj(x) + ai,j,kψk(u(y))φi(y)φj(x)dy � 

� dx 

j=1 j=1 

0 

i,j,k=1 

� 

� ⎛ 

⎞� 

� � 1 � n� 

n� 

� 1 n� 

�2 

� 

= � 

� φj(x) ⎝−uj + lj + ai,j,k ck,lφl(y)φi(y)dy ⎠� 

� dx 

0 �j=1 

i,k=1 

0 

l=1 

� 

� ⎛ 

⎞� 

� � 1 � n� 

n� 

�2 

� 

= � 

� φj(x) ⎝−uj + lj + ai,j,kck,l⎠� 

� dx 

0 �j=1 

i,k,l=1 � 

⎡ 

� 1 n� 

≤ ⎣ φ 2 ⎤ ⎡ ⎛ 

⎞2⎤ 

n� 

n� 

j(x) ⎦ ⎣ ⎝−uj + lj + ⎠ ⎦ dx 

0 

⎡ 

= ⎣ 

� 1 

0 

j=1 

j=1 

j=1 

n� 

φ 2 ⎤ ⎡ ⎛ 

n� 

j(x)dx⎦ 

⎣ ⎝−uj + lj + 

j=1 

= n 

i,k,l=1 

ai,j,kck,l 

n� 

i,k,l=1 

⎛ 

n� 

⎝−uj + lj + 

j=1 

ai,j,kck,l 

n� 

i,k,l=1 

⎞2⎤ 

⎠ 

⎦ 

ai,j,kck,l 

where ck,l = ck,l(u1,...,un) have known values since ψk are chosen and u = u(y) 

is our known target function. Regarding the constraints we have that: 

� 1 � 1 

(C2(x,y)) 2 ⎛ 

� 1 � 1 

dxdy = ⎝ � 

⎞2 

bi,jφj(x)φi(y) ⎠ dxdy 

0 

0 

= 

0 0 

� 1 � 1 

0 

+2 

= � 

i,j 

0 i,j 

� 1 � 1 

0 

b 2 i,j 

i,j 

� 

(bi,jφj(x)φi(y)) 2 dxdy 

0 

� 

i,j,h,k 

� 1 

+ 2 � 

i,j,h,k 

φj(x)φi(y)φh(x)φk(y)dxdy 

0 

φj(x)φh(x)dx 

� 1 

0 

⎞ 

⎠ 

2 

φi(y)φk(y)dy 

,


= � 

b 2 i,j . 

So now the inverse problem becomes: 

i,j 

min �u − Tu� 

A 2 , (12) 

where A = { � 

i,j b2i,j ≤ CA < 1}. 

Example. We set l(x) = x and set Ktrue(x,y,u) = ktrue(x,y)u. In this 

case, ai,j(u) = ai,ju, so that 

∞� 

Ktrue(x,y,u) = ai,jφj(x)φi(y), 

�u − Tu� 2 2 ≤ 

and bi,j = ai,j. We define 

Let 

⎧ 

⎪⎨ 

ktrue(x,y) = 

⎪⎩ 

i,j=1 

n� 

� 

−uj + lj + 

i=1 

φ(·) ∈ {φs(·),φc(·)} = 

n� 

i=0 

ai,juj 

� 2 

π(−1 − x + y), x − y < − 1 

2 , 

π(x − y), − 1 1 

≤ x − y ≤ 

2 2 , 

1 

π(1 − x + y), < x − y ≤, 

2 

�√ √ � 

2 sin(π·), 2 cos(π·) . (13) 

In this rather limited finite basis, the representation of ktrue(x,y) is 

kbasis(x,y) = 2 

π φs(x)φc(y) − 2 

π φc(x)φs(y) = 4 

sin(π(x − y)). 

π 

We build the Hammerstein integral operator 

(Ttrueu) (x) = 

� 1 

0 

ktrue(x,y)u(y)dy + l(x), 

and construct an approximation of its fixed point ūtrue(x) by iterating the map 

20 times on the initial function u0(x) = 0. We set utarget(x) = T 20u0(x). See 

Figure 1. Thus, our example inverse problem is: Given utarget(x) and l(x), find 

a representation of the kernel k(x,y) in our finite basis such that the corresponding 

Hammerstein integral operator admits utarget(x) as an approximate 

solution. Following our development, we minimize the constrained squared L2 collage distance 

∆ 2 = �u − Tu� 2 ⎛ 

2 + λ⎝ 

� 

⎞ 

− CA⎠, 

i,j 

a 2 i,j 

,


Figure 1: The graph of z = ktrue(x,y), z = kbasis(x,y), and y = 

utarget(x) 

where CA ∈ (0,1) is a chosen fixed value. We denote the kernel that minimizes 

∆ 2 by kcollage(x,y), and we observe that the desired result is kcollage(x,y) = 

kbasis(x,y). To achieve this result, we would have to set 

CA = 2 

� 2 

π 

� 2 

≈ 0.81057. 

Instead, we systematically set CA = i 

10 , i = 1,2,... ,9, and minimize each 

corresponding ∆2 . Perhaps, one might anticipate that the minimum of the 

nine minimal collage distances we find will occur when CA = 0.8. 

In Table 1, we present the results obtained when we restrict our basis fur-


ther, so that it consists of precisely the two functions ψ1(x,y) = 2sin(πx)cos(πy) 

and ψ2(x,y) = 2cos(πx)sin(πy), which appear in ktrue(x,y). The pleasing re- 

CA minimal ∆ 2 a1 a2 

0.1 0.17153 0.29898 −0.10301 

0.2 0.12759 0.40955 −0.17964 

0.3 0.09614 0.48283 −0.25860 

0.4 0.07136 0.53421 −0.33856 

0.5 0.05088 0.57113 −0.41690 

0.6 0.03356 0.59838 −0.49187 

0.7 0.01929 0.61908 −0.56280 

0.8 0.01120 0.63526 −0.62964 

0.9 0.01613 0.64823 −0.69267 

Table 1: Collage coding results with kcollage(x,y) = a1φ1(x,y) + 

a2φ2(x,y) and a2 1 + a22 = CA 

sult is that the minimum of the minimal collage distances occurs when CA = 0.8. 

Indeed, 

a1 and − a2 ≈ 2 

≈ 0.63661, 

π 

the true best result. 

Note that when we minimize ∆ 2 , treating CA as an additional variable, we 

obtain λ = 0, a linear system in the coefficients a1 and a2, and a constraint 

equation, as opposed to an inequality. Still, upon solving the linear system for 

a1 and a2, we can check whether these minimizing values lead to a value of 

CA < 1. For this example, we obtain a1 = 0.63644 and a2 = 0.63675, giving 

CA = 0.81051. 

When we add the basis function φs(2x)φs(y) with coefficient a3, we obtain 

a1 = 0.63675, a2 = −0.55321, and a3 = 0.04580, with CA = 0.71359. The 

quality of the approximation diminishes in part because in this example �C2�2 = 

0.90690, meaning the operator is not very contractive. 

If we modify κtrue, replacing each π by π 

8 , then the new value of �C2�2 is the 

smaller 0.113362. We obtain a1 = 0.07957, a2 = −0.07495, and a3 = 0.00722, 

compared to the desired values a1 = −a2 = 1 

4π ≈ 0.07958 and a3 = 0.


5. Random Fixed Point Equations and the 

“Random Collage Theorem” 

Again let (X,dX) denote a complete metric space. If (X,dX) is separable, 

then it is said to be a Polish space. We also let (Ω, F,µ) be a probability 

space. A mapping T : Ω × X ↦→ X is called a random operator if for any 

x ∈ X the function T(·,x) is measurable. The random operator T is said to 

be continuous/Lipschitz/contractive if, for a.e. ω ∈ Ω, we have that T(ω, ·) is 

continuous/Lipschitz/contractive (see Bharucha and Reid [4]). 

A measurable mapping x : Ω ↦→ X is called a random fixed point of the 

random operator T if x is a solution of the equation 

T(ω,x(ω)) = x(ω), a.e. ω ∈ Ω. (14) 

Consider the space Y of all measurable functions x : Ω ↦→ X. If we define the operator 

˜ T : Y ↦→ Y as ( ˜ Ty)(ω) = T(ω,x(ω)), then the solutions of this fixed point 

equation on Y are the solutions of the random fixed point equation T(ω,x(ω)) = 

x(ω). Suppose that the metric dX is bounded, that is, dX(x1,x2) ≤ K for all 

x1,x2 ∈ X. Then the function ψ(ω) = dX(x1(ω),x2(ω)) : Ω ↦→ R is an element 

of L1 (Ω) for all x1,x2 ∈ Ω. Define the following function over the space Y × Y : 

� 

dY (x1,x2) = dX(x1(ω),x2(ω))dPω = E(dX(x1,x2)). (15) 

Ω 

In Kunze et al [15], we proved the following two results. 

Theorem 5.1. The space (Y,dY ) is a complete metric space. 

Theorem 5.2. Suppose that: 

(i) for all x ∈ Y the function ξ : Ω ↦→ X1 defined by ξ(ω) := T(ω,x(ω)), 

∀ ω ∈ Ω belongs to Y , 

(ii) dY ( ˜ Tx1, ˜ Tx2) ≤ cdY (x1,x2) with c < 1. 

Then there exists a unique solution of ˜ T ¯x = ¯x, that is, T(ω, ¯x(ω)) = ¯x(ω) 

for a.e. ω ∈ Ω. 

Hypothesis (i) can be avoided if X is a Polish space, in which case the 

following result (see Itoh [11]) holds. 

Theorem 5.3. Let X be a Polish space, that is, a separable complete 

metric space, and T : Ω × X ↦→ X be a mapping such that for each ω ∈ Ω 

the function T(ω,.) is c(ω)-Lipschitz and for each x ∈ X the function T(.,x) 

is measurable. Let x : Ω ↦→ X be a measurable mapping; then the mapping 

ξ : Ω ↦→ X defined by ξ(ω) = T(ω,x(ω)) is measurable.


Corollary 5.1. Let T : Ω × X ↦→ X be a mapping such that for each 

ω ∈ Ω the function T(ω,.) is a c(ω)-contraction. Suppose that for each x ∈ X 

the function T(.,x) is measurable. Then there exists a unique solution of the 

equation ˜ T ¯x = ¯x that is T(ω, ¯x(ω)) = ¯x(ω) for a.e. ω ∈ Ω. 

As a consequence of the collage and continuity theorems, we have the following. 

Corollary 5.2. Suppose that the hypotheses of Theorem 5.2 are satisfied. 

Then for any x ∈ Y , 

1 

1 + c dY (x, ˜ Tx) ≤ dY (x, ¯x) ≤ 1 

1 − c dY (x, ˜ Tx), (16) 

where ¯x is the fixed point of ˜ T, that is, ¯x(ω) := T(ω, ¯x(ω)), ω ∈ Ω. 

6. Inverse Problem for the Random Hammerstein Integral Equation 

We now consider the case when the kernel K and the function l also depend on 

ω ∈ Ω, where (Ω, F,p) is a probability space. Suppose that for a.e. ω ∈ Ω, we 

have 

P (w : |K(ω,α1,β,γ) − K(ω,α2,β,γ) ≤ C1(β,γ)|α1 − α2|) = 1, (17) 

and P (w : |K(ω,α,β,γ1) − K(ω,α,β,γ2) ≤ C2(α,β)|γ1 − γ2|) = 1, (18) 

and that for u(y) ∈ L2 ([0,1]) we have C1(u,u(y)) ∈ L2 ([0,1]). Furthermore, 

assume that l(ω, ·) ∈ C([0,1]) and that �C2�2 < 1 for a.e. ω ∈ Ω. Then for 

a.e. ω ∈ Ω, there is a unique function ū(ω,x) ∈ Ω × L2 ([0,1]) such that 

ū(ω,x) = T(ω,x,ū(ω,x)) = 

� 1 

0 

K(ω,x,y,ū(ω,y))dy + l(ω,x). (19) 

The inverse problem can be formulated as: Given a function u : Ω ↦→ L 2 ([0,1]) 

and a family of operators ˜ Tλ : Y ↦→ Y , where Y = {all measurable functions x : 

Ω ↦→ X}, find λ such that u is the solution of random fixed point equation 

˜Tλu = u, (20) 

or, equivalently, 

Tλ(ω,ū(ω)) = ū(ω). (21) 

Example. Extending our deterministic example, we set l(x,ω) = a(ω)x, 

where a(ω) is a random variable with known mean µa and variance σ2 a , and 

define 

kbasis(x,y) = ksc(ω)φs(x)φc(y) − kcs(ω)φc(x)φs(y),


where ksc(ω) and kcs(ω) are random variables with unknown means µsc and µcs 

and variances σ 2 sc and σ 2 cs, respectively. Given data from realizations u(ωj,x), 

j = 1,... ,N, of the random variable u(ω,x), the fixed point of the corresponding 

T, along with l(x,ω), we wish to recover the means and variances of the 

kernel parameters. 

Each of the realizations u(ωj,x), j = 1,... ,N, of the random variable 

u(ω,x) is the solution of a fixed point equation 

u(ωj,x) = 

� 1 

0 

(ksc(ωj)φs(x)φc(y) − kcs(ωj)φc(x)φs(y)) u(ωj,y)dy + a(ωj)x. 

Now, for each of our target functions, u(ωj,x), we can find the constant 

values ksc(ωj) and kcs(ωj) via the collage coding method for the deterministic 

Hammerstein integral equation, as in Section 4. Upon treating each of our 

realizations, we will have found values for ksc(ωj) and kcs(ωj) for j = 1,... ,N. 

We then construct the approximations 

N� 

N� 

µsc ≈ µsc,N = 1 

N 

j=1 

ksc(ωj) and µcs ≈ µcs,N = 1 

N 

j=1 

kcs(ωj). (22) 

Note that the results obtained from collage coding each realization are independent. 

Using these approximations of the means, we can also calculate the 

variances: 

σ 2 sc ≈ σ 2 sc,N = 

and σ 2 cs ≈ σ2 cs,N = 

1 

N − 1 

1 

N − 1 

N� 

(ksc(ωj) − µsc,N) 2 

j=1 

N� 

(kcs(ωj) − µcs,N) 2 . 

Some results are presented in Table 2. The first three columns present the 

distributions of a, ksc, and kcs used to generate the realizations. The final two 

columns present the approximations of the means and variances obtained by 

our collage method for the various values of N specified in the middle column. 

In all cases, we have treated the constraint value CA as a parameter when 

minimizing each collage objective function. 

Our method correctly identifies the distribution of the random variable and 

finds good approximations of the mean and variance. As expected, a larger 

sample size produces better estimates of the mean and variance. 

j=1


True Values Collage Method Approximations 

a ksc kcs N ksc kcs 

N(1, 0.01) N (0.6, 0.0004) N (0.6,0.0004) 10 (0.62681,0.00067) (-0.62649,0.00066) 

N(1, 0.01) 

exp(1) 

N (0.6, 0.0004) 

exp 

N (0.6,0.0004) 100 (0.63655,0.00065) (-0.63613,0.00063) 

` ´ 

π 

exp ` ´ 

π 

10 (0.46817,0.41222) (-0.46817,0.41222) 

2 

exp(1) exp ` ´ π 

2 

2 

exp ` ´ π 

2 

100 (0.65473,0.42717) (-0.64155,0.37227) 

Table 2: Results for the random Hammerstein integral operator inverse 

problem, approximations are in the form (mean, variance) 

Acknowledgments 

This work was developed during a research visit by D. La Torre to the Department 

of Applied Mathematics of the University of Waterloo. D. La Torre 

thanks ERV for this opportunity. For D. La Torre this work has been supported 

by COFIN Research Project 2004. This work has also been supported in part 

by research grants (H.E. Kunze and E.R. Vrscay) and a Canadian Graduate 

Scholarship (K.M. Levere) from the Natural Sciences and Engineering Research 

Council of Canada (NSERC), which are hereby gratefully acknowledged. 

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