Andreas Kleefeld
email: kleefeld@tu-cottbus.de

Numerical results for linear Fredholm integral equations of the first kind over surfaces in 3D

Keywords: Fredholm integral equations, ill-posed problems, regularization, Tikhonov method, Nyström interpolation, boundary element method Fredholm integral equations, ill-posed problems, regularization, Tikhonov method, Nyström interpolation, boundary element method AMS Subject Classification: 45B05, 65F22, 65N38

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction, and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method -the Tikhonov method -to deal with this ill-conditioning and to obtain accurate numerical results.

Integral equations arise in a variety of applications (cf. [START_REF] Lonseth | Sources and applications of integral equations[END_REF]). Depending on the field, one obtains Fredholm integro-differential equations (cf. [START_REF] Akyüz-Daşcioglu | A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form[END_REF][START_REF] Dehghan | Chebyshev finite difference method for Fredholm integro-differential equation[END_REF]), Volterra integral equations (cf. [START_REF] Li | Modified method for determining an approximate solution of the Fredholm-Volterra integral equations by Taylor's expansion[END_REF]), or Fredholm integral equations (cf. [START_REF] Ghasemi | Numerical solution of linear Fredholm integral equations using sine-cosine wavelets[END_REF]). The latter case can be classified further into linear and nonlinear Fredholm integral equations of the first or second kind. Although a recent topic of research, linear Fredholm integral equations of the first kind are less familiar, since they are often ill-posed. They are usually considered only over an interval (cf. [START_REF] Groetsch | Integral equations of the first kind, inverse problems and regularization: a crash course[END_REF][START_REF] Reichel | Greedy Tikhonov regularization for large linear ill-posed problems[END_REF]).

In this article, we consider Fredholm integral equations of the first kind over surfaces in three dimensions with emphasis on their numerical treatment, since they arise in various applications in physics, computer tomography, and heat conduction problems (cf. [START_REF] Rieder | Keine Probleme mit inversen Problemen, Eine Einführung in Ihre stabile Lösung[END_REF]).

The inverse scattering problem is still under investigation with many unanswered questions (cf. [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF][START_REF] Hohage | Lecture Notes on Inverse Problems[END_REF][START_REF] Hohage | On the Numerical Solution of 3D Inverse Obstacle Scattering Problems[END_REF]). In the direct problem, the scattered field has to be calculated for a given obstacle and incident field. The inverse problem consists of finding the shape of an obstacle for given scattered and incident fields. This question arises in applications like sonar, radar, medical imaging, and geophysical exploration. The main issues are that the solution depends nonlinearly on the boundary and the reconstruction of the scattering wave from its far field is illposed.

There are several approaches to solve this inverse problem (cf. [START_REF] Kress | Numerical methods in inverse obstacle scattering[END_REF]). One method is to split the inverse scattering problem into two parts. In the linear ill-posed part, one must reconstruct the scattered wave from the far field pattern. The nonlinear well-posed part consists of finding the boundary of the object. More specifically, the integral equation of the first kind u ∞ (x) = 1 4π Γ e iκ x,y ϕ(y) dΓ y , x ∈ S 2 = x ∈ R 3 : |x| = 1 has to be solved for the density ϕ provided the far-field pattern u ∞ and a closed surface Γ is given (cf. [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF]). In order to attack this complex inverse scattering problem for arbitrary obstacles, we first consider the ill-posed part of this problem by looking at linear Fredholm integral equations of the first kind over surfaces.

The outline of this article is as follows. Section 2 gives the preliminaries about linear Fredholm integral equations of the first kind with a smooth kernel. Singular kernels are not considered here, since they are already discussed in [START_REF]An Empirical Study of the Numerical Solution of Integral Equations on Surfaces in R 3[END_REF] and [START_REF]The Numerical Solution of Integral Equations of the Second Kind[END_REF]. The next section discusses the boundary element method (BEM) for solving the first kind integral equations over a surface with emphasis on the numerical approximation. An integration scheme is presented which reduces the linear system dramatically. Another approach for more complex surfaces might be to consider meshless methods (see [START_REF] Mirzaei | A meshless based method for solution of integral equations[END_REF] for details and the description of the moving least squares (MLS) method). Section 4 is devoted to the Tikhonov regularization method and its usage. Furthermore, the discretization of the operator equation is given in detail. Section 5 illustrates numerical examples for the direct and inverse problem for several smooth and piecewise smooth surfaces and for different kernel and data functions. With the Nyström interpolation the solution can be obtained at every point. The last section gives a summary and an outlook for future work.

Fredholm integral equation of the first kind

In this section, the linear Fredholm integral equation of the first kind is presented. Fredholm integral equations of the first kind can be classified into two categories. The first has singular kernel functions, but those are not considered here. The second class consists of smooth functions k(x, y) with non-degenerate kernel. According to Hadamard, a problem is well-posed if it has a solution (existence), not more than one solution (uniqueness), and the solution depends continuously on the data (stability). Problems are called ill-posed, if they are not well-posed. Fredholm integral equations of the first kind are examples of such ill-posed problems. That is, small changes in the data cause huge changes in the unknown function f (cf. [START_REF] Groetsch | The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind[END_REF]Section 1.1]). That means, the solution does not depend continuously on the data. Strictly speaking, the solution f is extremely sensitive to small changes in g(y). A method to solve an ill-posed problem approximately is called a regularization

Kf = Γ k(x, y)f (x) dΓ x
is linear where the kernel is assumed to be smooth. For a discussion of existence and uniqueness of a solution f , refer to [START_REF] Groetsch | The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind[END_REF][START_REF]Inverse Problems in the Mathematical Sciences[END_REF][START_REF] Wing | A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding[END_REF]. Note that Fredholm integral equations of the first kind over an interval are already discussed in [START_REF] Delves | Computational Methods for Integral Equations[END_REF][START_REF] Groetsch | The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind[END_REF][START_REF] Kirsch | An Introduction to the Mathematical Theory of Inverse Problems[END_REF][START_REF] Rieder | Keine Probleme mit inversen Problemen, Eine Einführung in Ihre stabile Lösung[END_REF][START_REF] Wing | A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding[END_REF].

Boundary element method for solving integral equations

The boundary element method is discussed extensively in [START_REF]An Empirical Study of the Numerical Solution of Integral Equations on Surfaces in R 3[END_REF] and [START_REF]The Numerical Solution of Integral Equations of the Second Kind[END_REF] and we assume that the reader is familiar with the used notation. We briefly summarize the main points with emphasis on the numerical approximation of the Fredholm integral equation of the first kind over a surface.

Triangulation

Assume that Γ is a piecewise smooth surface in R 3 ; that means, Γ can be written as

Γ = Γ 1 ∪ • • • ∪ Γ J . (1)
Each Γ j is divided into a triangular mesh and the collection of those is denoted by

T n = {∆ k | 1 ≤ k ≤ n} (2)
with mesh size

δ n = max 1≤k≤n diam(∆ k ), diam(∆ k ) = max p,q∈∆ k |p -q|.
We assume, that δ n → 0 as n → ∞. Let the unit simplex in the st-plane be defined as

σ = {(s, t) | 0 ≤ s, t, s + t ≤ 1}.
Let {q 1 , . . . , q 6 } be the three vertices and three midpoints of σ given by q 1 = (0, 0), q 2 = (0, 1),

q 3 = (1, 0) q 4 = 0, 1 2 , q 5 = 1 2 , 1 2 , q 6 = 1 2 , 0 . (3)
For each ∆ k , we assume there is a map

m k : σ 1-1 ----→ onto ∆ k , (4)
which is used for interpolation and integration on ∆ k . Define the node points of ∆ k by v k,j = m k (q j ), j = 1, . . . , 6. The first three are the vertices and the last three are the midpoints of the sides of ∆ k . To obtain a triangulation (2) and the mapping (4), use a parametric representation for each region Γ j of (1). Assume that for each Γ j , there is a map

F j : R j 1-1 ----→ onto Γ j , j = 1, . . . , J, (5)
where R j is a polygonal region in the plane and F j ∈ C 3 (R j ). That is, a triangulation of R j is mapped onto a triangulation Γ j . Let ∆k,j be an element of the triangulation of R j with vertices vk,1 , vk,2 and vk,3 and mesh size δ. Then the map (4) is given by

m k (s, t) = F j (uv k,1 + tv k,2 + sv k,3 ), (s, t) ∈ σ, u = 1 -s -t. (6)
Most smooth and piecewise smooth surfaces can be decomposed as in [START_REF] Akyüz-Daşcioglu | A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form[END_REF]. Conforming triangulations satisfying T1-T3 are considered (cf. [5, p. 188]). The refinement of ∆ k ∈ T n is done by connecting the midpoints of the three sides of ∆k giving four new triangles. This also leads to symmetry in the triangulation and cancellation of errors occurs (cf. [5, p. 173]).

Interpolation and numerical integration

For quadratic interpolation on σ, let u = 1 -s -t and define the Lagrange basis functions l j (s, t), 1 ≤ j ≤ 6, by

l 1 (s, t) = u(2u -1), l 2 (s, t) = t(2t -1), l 3 (s, t) = s(2s -1) l 4 (s, t) = 4tu, l 5 (s, t) = 4st, l 6 (s, t) = 4su.
The corresponding basis functions {l j,k (q)} on ∆ k are

l j,k (m k (s, t)) = l j (s, t), 1 ≤ j ≤ 6, 1 ≤ k ≤ n.
For a function f ∈ C(Γ), define

P n f (q) = 6 j=1 f (v j,k )l j,k (q), q ∈ ∆ k , (7)
the piecewise quadratic isoparametric function interpolating f on the nodes of the mesh

{∆ k } for Γ. For f ∈ C 3 (Γ i ), i = 1, . . . , J , we have (cf. [5, p. 165]) f -P n f ∞ = O( δ3 n ).
To reduce integration over ∆ k to σ, use the map

m k : σ → ∆ k to obtain ∆ k f (q) dΓ q = σ f (m k (s, t))J (s, t) dσ (8)
with the Jacobian For simplicity write the right hand side of (8) as

J (s, t) = ∂m k ∂s × ∂m k ∂t (s, t) . (9)
I(f ) = σ f (s, t) dσ.
To approximate I(f ), replace f with the quadratic polynomial given in [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF] to obtain

σ f (s, t) dσ ≈ 6 i=1 w i f (q i ), w i = σ l i (s, t) dσ, (10)
where the weights w 1 , w 2 , and w 3 are zero and w 4 , w 5 and w 6 are 1/6. The nodes q i are given on page (3). Thus, [START_REF] Ghasemi | Numerical solution of linear Fredholm integral equations using sine-cosine wavelets[END_REF] becomes

σ f (s, t) dσ ≈ 1 6 f 0, 1 2 + f 1 2 , 1 2 + f 1 2 , 0 , (11)
which has degree of precision 2.

Approximating the surface

For most surfaces it is difficult to find ∂m k /∂s and ∂m k /∂t. Therefore, we approximate m k (s, t) in terms of q 1 , . . . , q 6 by

m k (s, t) = 6 j=1 m k (q j )l j (s, t), (s, t) ∈ σ. (12)
Thus, each m k is a polynomial of degree ≤ 2 in (s, t) and we have (cf. [2, p. 33])

max k max (s,t)∈σ |m k (s, t) -m k (s, t)| = O( δ3 n ).

Solving integral equations of the first kind over an approximated surface

Let {q 1 , . . . , q 6 } be the interpolation nodes within σ and let

v k,i = m k (q i ), i = 1, . . . , 6, k = 1, . . . , n, which we collect in the set V n = {v 1 . . . , v n v }.
The previous approximation technique applied to the integral equation

Γ k(P, Q)f (Q) dΓ Q = g(P ), P ∈ Γ ( 13
)
yields the Nyström equation with

n v j=1 w j k(v i , v j )f (v j ) = g(v i ), i = 1, . . . ,
w j = i,k v j =m k (q i ) w i ∂ m k ∂s × ∂ m k ∂t (q i ) , w i = σ l i (s, t) dσ. ( 14
)
The weights w 1 , w 2 , and w 3 are zero and w 4 , w 5 and w 6 are 1/6. The weights w j are listed in [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF]Example 10]. Thus, ( 13) is approximated by the n v × n v linear system Kf = g where K is the n v × n v matrix with the entries

w j k(v i , v j ), f is an n v -vector [f (v 1 ), . . . , f (v n v )] T and g is an n v -vector [g(v 1 ), . . . , g(v n v )] T .
Hence, the obtained linear system is of smaller size.

Regularization methods

In this chapter we introduce a well-known method which produces a stable approximation for linear ill-posed operator equations of the form

Kx = y, (15)
where K : X → Y is a linear and bounded operator between Hilbert spaces X and Y and y ∈ R(X ). The regularization method is able to calculate approximations even if the solution x does not depend continuously on the data y, which means that the inverse of the operator is unbounded, K -1 = ∞. Furthermore, the data are perturbed by noise δ which is assumed to be known, i.e.

y δ -y Y ≤ δ.
The method of interest is the Tikhonov regularization. For more details on regularization theory for equations of the first kind as well as the Tikhonov method, refer to [START_REF] Groetsch | The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind[END_REF][START_REF] Hanke | Conjugate Gradient Type Methods for Ill Posed Problems[END_REF][START_REF] Kirsch | An Introduction to the Mathematical Theory of Inverse Problems[END_REF][START_REF] Rieder | Keine Probleme mit inversen Problemen, Eine Einführung in Ihre stabile Lösung[END_REF].

Tikhonov regularization

We want to solve [START_REF] Hohage | Lecture Notes on Inverse Problems[END_REF] with noisy data y δ . A common way to obtain a solution is to determine the best fit in the sense that we minimize the functional

J (x) = Kx -y δ 2
Y in X . However, the solution does not depend continuously on the data. To ensure stability, we add a penalty term to the previous functional. This gives the Tikhonov functional

J α (x) = Kx -y δ 2 Y + α x 2 Y for x ∈ X
with regularization parameter α > 0. The Tikhonov functional is Fréchet differentiable for every α > 0 with the Fréchet derivative

J α (x)h = 2Re K * (Kx -y δ ) + αx, h , since J α (x + h) -J α (x) -J α (x)h = Kh 2 Y + α h 2 Y holds.
The Tikhonov functional J α (x) has a unique minimum x α,δ in X for all y δ ∈ Y and α > 0. This minimum is also the unique solution of the normal equation

(K * K + αI)x α,δ = K * y δ
where K * denotes the adjoint operator of K. The operator K * K + αI is boundedly invertible; that means, x α,δ depends continuously on y δ .

To see this, let x α,δ be the minimum of the Tikhonov functional J α (x). By the Fréchet differentiability and by the definition of a local minimum, we obtain J (x α,δ )h = 0 for all h ∈ X . Using h = K * (Kx -y δ ) + αx yields

(K * K + αI)x α,δ = K * y δ .
Note that the bounded invertibility of K * K + αI follows from the Lax-Milgram Theorem and from the inequality

Re (K * K + αI)x, x = Kx 2 Y + α x 2 Y ≥ α x 2 Y .
Finally, we note that x α,δ , defined by x α,δ = (K * K + αI) -1 (K * y δ ) minimizes J α .

Obviously, the function λ(t) = J α (x α,δ + th) is a polynomial of degree two with λ ≥ 0 and λ (0) = 0 for all h ∈ X \{0}, and therefore λ(t) ≥ λ(0) for all t ∈ R with equality only for t = 0. In general, the regularization parameter α affects two different trends. If α is chosen to be large, then the errors in y δ will be damped. Otherwise, if α is chosen to be small, then J α will be a good approximation of J , but the error affects the solution. To choose a good regularization parameter, we use the Morozov's discrepancy principle. It says that one should take the largest regularization parameter α = α(δ, y δ ) such that the residual Kx α,δ -y δ Y is less than or equal to τ δ for a fixed τ ≥ 1. That is, we find

α(δ, y δ ) = sup Kx α,δ -y δ Y ≤ τ δ with α > 0
where τ ≥ 1 is fixed. That means, we should not satisfy the operator equation more accurately than the known noise error δ. It can be shown that the function α → Kx α,δ -y δ Y is monotonously increasing, thus we only need to find α such that

τ 1 δ ≤ Kx α,δ -y δ Y ≤ τ 2 δ for 1 ≤ τ 1 < τ 2
holds. We can use the bisection method to find such α. In order to use the previously described regularization methods, we have to discretize the operator K and its adjoint K * (cf. [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF]Section 5.4]). We consider the integral operator K : L 2 (Γ) → L 2 (Γ) given by

Kf (P ) = Γ k(P, Q)f (Q) dΓ Q , P ∈ Γ (16) with kernel k ∈ L 2 (Γ × Γ) and f ∈ L 2 (Γ).
Discretizing [START_REF] Hohage | On the Numerical Solution of 3D Inverse Obstacle Scattering Problems[END_REF] yields

Kf (P ) ≈ n v j=1 w j k(P, v j )f (v j )
with weights w j given in ( 14) and node points V n = {v 1 . . . , v n v }. Using the same nodes, one obtains

Kf (v i ) ≈ n v j=1 w j k(v i , v j )f (v j ), i = 1, . . . , n v . (17)
Thus, we obtain the n v × n v matrix K with entries w j k(v i , v j ).

Next, we discretize the adjoint operator of K. For the surface Γ the integral operator

K * : L 2 (Γ) → L 2 (Γ) is given by K * g(Q) = Γ k(P, Q)g(P ) dΓ P , Q ∈ Γ ( 18
)
with kernel k ∈ L 2 (Γ × Γ) and g ∈ L 2 (Γ). Discretizing (18) yields

K * g(Q) ≈ n v j=1 w j k(v j , Q)g(v j )
with weights w j and node points V n = {v 1 . . . , v n v }. Using the same nodes, one obtains

K * g(v i ) ≈ n v j=1 w j k(v j , v i )g(v j ), i = 1, . . . , n v .
Thus, we obtain the n v × n v matrix K * with entries w j k(v j , v i ). Finally, we calculate K * [Kf (P )] (Q), since we also need it for the regularization

K * h(Q) ≈ n v m=1 w m k(v m , Q)h(v m ) ≈ (17) n v m=1 w m k(v m , Q) n v j=1 w j k(v m , v j )f (v j ) = n v j=1 n v m=1 w m k(v m , Q)w j k(v m , v j )f (v j ).
Using the same nodes, one obtains

n v j=1 n v m=1 w m k(v m , v i )w j k(v m , v j )f (v j ), i = 1, . . . , n v .
We get the n v × n v matrix K * K with entries

n v m=1 w m k(v m , v i )w j k(v m , v j ).
Note that we obtain the same entries when we multiply K * and K obtained from above. To calculate the entries for K * K at the position (i, j), we have to multiply the i th row of the matrix K * with the j th column of K. Thus, each entry of K * K is given by

n v m=1 w m k(v m , v i )w j k(v m , v j ).

Numerical results

In this section, numerical results for several linear Fredholm integral equations of the first kind over different surfaces are presented. The considered integral equation is

Γ k(P, Q)g(Q) dΓ Q = f (P ), P ∈ Γ.

Notations

The components of the points P and Q on a surface are given by P = [p x , p y , p z ] T and Q = [q x , q y , q z ] T , respectively. The errors are measured in the discrete 2 -norm

x 2 = 1 N N i=1 |x i | 2 ,
where N denotes the number of midpoints of the triangulation. Furthermore, we denote the normal distributed random error with mean µ = 0 and standard devi- ation σ on the right hand side function f (P ), P ∈ Γ with δ. We assume that

f δ -f 2 ≤ δ,
where δ is a known noise level. We denote the error between the true solution g T and the calculated solution g C by and the error of the defect with ˆ . The parameters of the stopping rule of the Tikhonov method are τ 1 = 1, τ 2 = 1.1 and the interval for α is [10 -8 , 1]. The number of bisection steps is denoted by B. The discretization of the operators is performed as explained in Section 4.2, and the colored surface represents the solution over the surface, since we cannot plot in 4D.

The direct problem

In this section, the direct problem is presented. We calculate the right hand side function provided the kernel function and the true solution are known. This will show that the integration over a surface works well as shown in Table 1. Note that we observe superconvergence at the collocation nodes. Define the estimated order of convergence (EOC) by

EOC = log 2 (E N /E 4N ) ,
where

E N = g -g N ∞
denotes the maximum error between the calculated solution g N and the exact solution g. The EOC seems to converge to four as n is getting large; that is, we obtain a rate of O δ4 n for smooth surfaces. This result has already been stated in [START_REF] Chien | Piecewise polynomial collocation for integral equations with a smooth kernel on surfaces in three dimensions[END_REF].

Example I

The kernel function

K(P, Q) is 1 and ρ(Q) = q 2
x . The surface is a sphere with radius 1. It is easy to check with MAPLE c that the right hand side function f (P ) is 4π/3.

Example II

The kernel function K(P, Q) is given by ∂ ∂n Q |P -Q| 2 and ρ(Q) = 1. The surface is an ellipsoid with a = 1, b = 0.75 and c = 0.5. The function f (P ) is given by 8πabc.

Example III

The kernel function K(P, Q) and the surface are the same as in Example II. The function ρ(Q) is q 2 z . The function f (P ) is 8πabc

q 2 z δ = 0.1% Ellipsoid a = 1.1, b = c = 1 (px -qx) 10 + (py -qy) 10 + (pz -qz) 10 f 2 (P ) q 2 z δ = 0.1% Cube r = 1 |P × Q| 2 f 3 (P ) q 2 x + q 2 y + q 2 z δ = 0.1% L-block b |P × Q| 2 f 4 (P ) q 2 x + q 2 y + q 2 z δ = 0.1% Tetrahedron r = 1 |P -Q| 2 f 5 (P ) q 2 z δ = 0.1% Paraboloid a = b = c = 1 (px -qx) 3 + (py -qy) 3 + (pz -qz) 3 f 6 (P ) q 2 x δ = 0.1%
a See the Appendix A for the detailed data functions.

b The L-shaped surface can be described by

[0, 1] × {[0, 1] × [0, 2] ∪ [0, 2] × [0, 1]}.

The inverse problem

In this section, the inverse problem is considered. We will see that we can obtain a stable approximation of the function ρ(Q) for different surfaces, kernels, and right hand sides with the Tikhonov method.

Next, we consider a spherical, ellipsoidal, cubical, L-shaped, tetrahedral, and elliptic paraboloidal surface. For the setup for each example refer to Table 2. There one finds the description of the surface, the used kernel, data and solution functions, and the added error. The details of the data functions are listed in Appendix A to not impede the reading flow.

The numerical results for the six different surfaces are listed in Table 3. In this table the number of midpoints of the surface, the number of bisection steps, the regularization parameter, the error of the defect and the absolute error between the calculated and true solution are listed. In addition, the condition number of the discretized kernel matrix is presented. For more surfaces refer to [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF].

As one can see in Table 3 the numerical results are very accurate, although the linear system is severely ill-conditioned. For a graphical presentation of the calculated solutions compared to the true solutions refer to the Figures 1, 2, 3, 4, 5, and 6, respectively.

Nyström interpolation

Note that by using Tikhonov regularization we obtain an integral equation of the second kind z which is to be determined. z which is to be determined. x + q 2 y + q 2 z which is to be determined. That means, we can apply the Nyström interpolation (cf. [START_REF]The Numerical Solution of Integral Equations of the Second Kind[END_REF]Section 5.4.1]) to [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF] to obtain the solution at every point P ∈ Γ. Discretizing [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF] yields x + q 2 y + q 2 z which is to be determined. z which is to be determined. x which is to be determined. which can be solved for any point P on the surface Γ as shown in [START_REF] Kleefeld | Numerical results of linear Fredholm integral equations of the first kind over surfaces[END_REF]Section 6.2.13].

(K * K + αI) ρ = K * f. (19)
αρ(P ) + n v j=1 w j k(v j , P ) n v m=1 w m k(v j , v m )ρ(v m ) = n v j=1 w j k(v j , P )f (v j ) ⇔ ρ(P ) = 1 α n v j=1 w j k(v j , P ) f (v j ) - n v m=1 w m k(v j , v m )ρ(v m ) (20
When we use the spherical example, we have obtained the solution ρ at 768 midpoints with α = 5.8594• 10 -3 and = 0.0037. Using the interpolation [START_REF] Lonseth | Sources and applications of integral equations[END_REF], we are able to calculate the solution at the remaining 258 vertices of the triangulation of the surface Γ. The maximum error between the solution at those vertices compared

with the true solution is given by 0.00371.

Summary and outlook

A program written in Matlab is used to approximate linear Fredholm integral equations of the first kind over surfaces with the Nyström method. It is based on a boundary element package (BIEPACK) written in Fortran which solves Fredholm integral equations of the second kind. The direct problem is solved accurately as illustrated in §5.2.

The constructed examples in §5.3 are extremely ill-posed linear integral equations of the first kind. Using standard methods, one would fail to obtain accurate numerical results. But the Tikhonov method yields accurate results, although the right hand side function is disturbed by δ = 0.1%. The regularization parameter α is calculated automatically by a simple bisection method which uses the interval [10 -8 , 1]. Every example fulfills the stopping rule and yields accurate results. The number of iteration steps of the bisection method varies between 6 and 11. Because the Tikhonov method yields a linear integral equation of the second kind over a surface Γ, we can calculate the solution at every point P ∈ Γ by the Nyström interpolation.

The numerical results of the L-shaped block and the cubical surface examples are very accurate compared to the curved surface examples, although the discretized matrix has a large condition number and we have small errors on the right hand side functions. Also, the numerical results over the curved surfaces are very accurate.

In future work we would like to apply this idea to integral equations obtained by solving the inverse scattering problems (cf. [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF][START_REF] Kress | Numerical methods in inverse obstacle scattering[END_REF]).

F

  Numerical results for linear Fredholm integral equations of the first kind over surfaces in 3D1. Introduction

Definition 2 . 1 (

 21 Fredholm integral equation of the first kind)We denote the equation of the formΓ k(x, y)f (x) dΓ x = g(y), y ∈ Γa Fredholm integral equation of the first kind, where Γ is a closed and bounded surface in R 3 . The given functions k(x, y) and g(y) are called the kernel of the equation and the data, respectively. The solution f is an unknown function which is to be determined.

4 . 2

 42 Discretization of the integral operators K, K * and K * K

  discretization of K * h(Q) with h(Q) = Kf (P ) results in

  True solution q 2

  Calculated solution with the Tikhonov method, α = 5.8594• 10 -3 , and error δ = 0.1%.

Figure 1 .

 1 Figure 1. An ill-posed inverse problem over a spherical surface.

  True solution q 2

  Calculated solution with the Tikhonov method, α = 5.8594• 10 -3 , and error δ = 0.1%.

Figure 2 .

 2 Figure 2. An ill-posed inverse problem over an ellipsoidal surface.

  Calculated solution with the Tikhonov method, α = 4.8829• 10 -4 , and error δ = 0.1%.

Figure 3 .

 3 Figure 3. An ill-posed inverse problem over a cubic surface.

  True solution q 2

  Calculated solution with the Tikhonov method, α = 2.9297• 10 -3 , and error δ = 0.1%.

Figure 4 .

 4 Figure 4. An ill-posed inverse problem over a L-shaped surface.

  Calculated solution with the Tikhonov method, α = 7.3243• 10 -4 , and error δ = 0.1%.

Figure 5 .

 5 Figure 5. An ill-posed inverse problem over a tetrahedral surface.

  True solution q 2

  Calculated solution with the Tikhonov method, α = 1.9531• 10 -3 , and error δ = 0.1%.

Figure 6 .

 6 Figure 6. An ill-posed inverse problem over an elliptic paraboloidal surface.

f 5 (c 2 = p 2 x + p 2 y + p 2 z , c 3 = 5 -

 52235 9p x -9p y + p z , c 4 = p 2 x + p 2 y + p 2 z , c 5 = 5p x + 5p y -6p z ,where r denotes the side length of the tetrahedron.

f 6 ( 1 + c 2 + c 3 + √ 5 (c 4 +

 61254 c 5 + c 6 )), c 1 = 28644(p 3 x + p 3 y + p 3 z ), c 2 = 41283p x + 13761p y + 83226p z , c 3 = -27716 -82764p 2 z , c 4 = 23100(p 3 x + p 3 y + p 3 z ), c 5 = 12375p y + 37950p z , c 6 = 37125p x -49500p 2 z -10300. Page 16 of 16 URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.com International Journal of Computer Mathematics

Table 1 .

 1 Absolute maximum error between the calculated solution g N and the exact solution g and estimated order of convergence.

	N	Example 1		Example 2		Example 3
	12 48 192 768 3072	4.9224• 10 -1 6.9362• 10 -2 5.6772• 10 -3 3.8181• 10 -4 2.4314• 10 -5	2.8 3.6 3.9 4.0	9.6014• 10 -1 1.3301• 10 -1 1.1427• 10 -2 7.7630• 10 -4 4.9547• 10 -5	2.8 3.5 3.9 4.0	1.1394• 10 -1 1.6225• 10 -2 1.2309• 10 -3 8.1183• 10 -5 5.1457• 10 -6	2.8 3.7 3.9 4.0

Table 2 .

 2 The considered examples with different surfaces Γ, kernels k(P, Q), data f (P ) and solutions g(Q). qx) 10 + (pyqy) 10 + (pzqz) 10 f 1 (P )

	International Journal of Computer Mathematics

3 /3. Page 10 of 16 URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.com

Table 3 .

 3 Results

	Surface Γ (N )	B	α	ˆ		κ
	Sphere (768)	8	5.8594• 10 -3	0.00105482	0.0037	1.6563• 10 20
	Ellipsoid (768)	8	5.8594• 10 -3	0.00105368	0.0093	3.9516• 10 20
	Cube (1152)	10	4.8829• 10 -4	0.00101308	4.8387• 10 -4	1.7210• 10 23
	L-block (672)	9	2.9297• 10 -3	0.00106680	4.4179• 10 -4	4.8135• 10 22
	Tetrahedron (384)	11	7.3243• 10 -4	0.00104142	0.0493	7.1267• 10 19
	Paraboloid (768)	8	1.9531• 10 -3	0.00107592	0.0015	6.2739• 10 20

  3r 10 + 15r 8 c 1 + 110r 6 c 2 + 198r 4 c 3 + 99r 2 c 4 + 11c 5 , c 1 = p 2 x + p 2 y + 9p 2 z , c 2 = p 4 x + p 4 y + 7p 4 z , c 3 = p 6 x + p 6 y + 5p 6 z , c 4 = p 8 x + p 8 y + 3p 8 z , c 5 = p 10 x + p 10 y + p 10 z , where r denotes the radius of the sphere. f 2 (P ) = 4 429 ab 3 π c 1 + 143c 2 + 1430c 3 + 195c 4 + 2574b 4 c 5 + 1287c 6 + c 7 + c 8 , c 1 = 36b 10 + 3a 10 , c 2 = p 10 x + p 10 y + p 10 z , c 3 = a 6 p 4 x + b 6 p 4 y , c 4 = a 8 p 2 x + b 8 p 2 y , c 5 = p 6 y + 5p 6 z , c 6 = a 2 p 8 x + b 2 p 8 y , c 7 = 2574a 4 p 6 x + 3861b 2 p 8 z , c 8 = 10010b 6 p 4 z + 1755b 8 p 2 z , where a, b and c are constants describing the ellipsoid. (-465p x p y -465p x p z -465p y p z + 644p 2 y + 644p 2 x + 644p 2 z ), where r denotes the side length of the cube.

	January 11, 2011	11:56	International Journal of Computer Mathematics	ijcmAKrevision
		16	Andreas Kleefeld
		Appendix A. The data functions
		o r P e e r F o r P e r e f 1 (P ) = 4 33 r 4 π f 3 (P ) = 1 90 r 6 f 4 (P ) = 388 5 p 2 y + 1082 9 p 2 x -265 6 p x p z +	388 5	p 2 z -	389 6	p y p z -	265 6	p x p y
			R e v i e w R e v i e w
							O n O n
								l y l y

URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.comInternational Journal of Computer Mathematics

Acknowledgement

The author thanks Professor Kendall E. Atkinson for his source code that solves the Laplace equation by a boundary element method (BIEPACK). Special thanks goes to Professor Tzu-Chu Lin for numerous fruitful discussions.


Nikos Prekas
email: prekas|pl@co.umist.ac.uk

Pericles Loucopoulos

Colette Rolland

Georges Grosz

Farida Semmak

Danny Brash

Developping patterns as a mechanism for assisting the management of knowledge in the context of conducting organisational change

of conducting organisational change

Introduction

Today's turbulent business environment requires organisations to modify and extend their traditional practices. Businesses now have to undertake change and transform themselves into adaptive enterprises able to respond to increasing complexity and uncertainty. Conducting change in an organisation is not a straightforward task.

Experience has shown that change is only possible, when those involved engage in an incremental and managed change process.

Change management is primarily about affecting how people work. This places an emphasis on changing behaviour and sets a demand for adopting the right practices in order to achieve preferred behaviour. The organisational context which supports the preferred behaviour should be enhanced [Scarbrough 1996]; this move needs positive reinforcement, as everybody involved needs to be convinced of the advantages to themselves [Kautto-Koivula 1998]. In this demanding and ever evolving setting, the sharing and management of knowledge become essential, as change more often than not encompasses multiple aspects and domains of a business environment. A global and thorough understanding of the situations at hand is therefore required before embarking on the process of change, but also during the process and after its completion.

In this paper we advocate the use of patterns as a mechanism for assisting the management of knowledge in situations of change, and the dissemination of experience gained from similar situations. The framework that we propose is based on a view of patterns as organisational design proposals that can be repeatedly used in different enterprise settings. The pattern framework presented here draws on the existence of a wider knowledge management methodology, namely the Enterprise Knowledge Development methodology (EKD) [START_REF] Bubenko | [END_REF][START_REF] Loucopoulos | [END_REF]], and on the application of this methodology in the case of deregulation in the Electricity Supply Industry (ESI) sector. This research is part of the ongoing European research project ELEKTRA, which aims to tailor a methodology for managing change in ESI companies and to discover patterns of change management for reuse in the sector.

The paper is structured as follows: Section 2 presents an overview the pattern concept.

Section 3 presents the significance of patterns in the enterprise setting, while Section 4 completes this view by presenting how to encapsulate knowledge in the pattern template. In Section 5 we present the use of patterns in change management, and in Section 6 we explain the way-of-working for developing patterns. Section 7 gives an example from the case studies performed in ELEKTRA, while Section 8 concludes with a summary and our directions for future work.

Background to patterns

Recent years have witnessed an increasing interest in the use of patterns within the software development community and in particular by those advocating and practising object-oriented approaches and re-use. This recent interest in patterns has its origins in [Coad 1992] and has subsequently permeated into software programming [START_REF] Buschmann | [END_REF][START_REF] Beck | Smalltalk Best Practice Patterns[END_REF]], software and system design [START_REF] Coplien | [END_REF][START_REF] Gamma | [END_REF][START_REF] Vlissides | [END_REF][START_REF] Eriksson | [END_REF]], data modelling [START_REF] Hay | [END_REF]] and more recently into systems analysis [Fowler 1997].

Much of the contemporary work on patterns has been inspired by the work of

Christopher Alexander who wrote a seminal book on the use of patterns within the domain of architecture. This book, 'The Timeless Way of Building' [Alexander 1979] sets the scene on the importance of patterns in such a way that, in many respects, it transcends the field of architecture. Alexander presents in this book the main arguments for the discovery of patterns and their use for achieving quality of designs. He defines a pattern as describing "a problem which occurs over and over again in our environment and then describes the core of the solution to that problem, in such a way that you can use this solution a million times over, without ever doing it the same way twice" [Alexander et al. 1977]. Here, the emphasis is put on the fact that a pattern describes a recurrent problem and it is defined with its associated core solution.

The use of patterns outside architecture has been successful partly due to their interpretation by Alexander. According to this interpretation, buildings, towns, communities, enterprises, software, and so on, can be understood and developed in terms of their patterns of relationships. In terms of enterprise development and change management, Coplien argues that "patterns should help us not only to understand existing organisations but also to build new ones" [Coplien 1995a;Coplien 1995b].

The utility of the pattern concept lies primarily in the fact that it allows past experience to be made easily available. This makes patterns an ideal concept in human endeavours that involve design and the kind of decisions that come with it. As argued in [Coad 1992], "finding and applying patterns indicates progress in a field of human endeavour". The progress lies in the fact that by developing a pattern, domain practitioners condense part of their knowledge on current domain practice and make it available to others. In fact, a pattern can convey best practice in a domain, practice that has proved sound and efficient, and can therefore be disseminated. When other practitioners subsequently reuse this pattern, they benefit from the experience of the pattern designers.

Patterns as organisational solutions

Although patterns have been applied for some time to the software engineering domain, and in particular with respect to object technologies [Coad 1992;[START_REF] Coplien | [END_REF][START_REF] Gamma | [END_REF][START_REF] Eriksson | [END_REF]], there has not been any significant application in the organisational and business domain. The idea of developing and reusing patterns to solve recurring business problems is relatively new. There are no standard components for use in the enterprise world, mainly because there is no standard framework for that area. As more emphasis is put on the management of change, and as the focus of attention in the IT community is shifting from software engineering to business engineering and the integration of business with IT, this situation is starting to change. The analysis patterns, for instance, that are presented in [Fowler 1997], are as much business-oriented as they are software-oriented.

Patterns in the enterprise setting can be used to describe anything that presents regular, repeatable characteristics: formal organisational and contractual relationships, informal relationships, responsibilities, work practices etc. In electricity companies, for instance, there is usually a business process dedicated to the collection of data on customer consumption of electricity. We can represent this knowledge as the Meter Reading business process pattern.

While in the traditional pattern literature a pattern is described as a tested solution to a problem, in EKD we use this term with a slight variation in meaning. Our study is focused on the enterprise setting, therefore the patterns that we propose are essentially organisational designs. These designs are solutions to specific problems within the context of an organisation, problems that are important and recurring in a variety of cases. Whereas it is possible to argue that some of these designs are tried and tested throughout a sector, some others will only be proposals for organisational solutions, whose degree of applicability will need to be tested. Therefore, in the scope of our approach we place the emphasis on the fact that patterns are generic and abstract design proposals, solutions to recurring problems within the sector of interest, that can be easily adapted and reused (see Figure 1).

Encapsulating organisational knowledge in patterns

In developing patterns, we place equal emphasis on developing knowledge components that are meaningful in themselves and on providing enough information about these components so as to make them effectively reusable. Thus we aim to produce a set of patterns that propose useful solutions to recurring problems and at the same time to describe these patterns in a way that guides and supports the reuse process.

We therefore make the distinction between the body of a pattern and its descriptor (see Figure 2); together these two constitute the pattern template (see also [START_REF] Grosz | [END_REF]UP1-UMIST 1998]). The former is the part of the knowledge that is effectively reused (knowledge perspective) whereas the latter aims to provide sufficient information on the pattern and to describe the context in which the body of the pattern can be reused (usage perspective).

A pattern is thus completely defined by its descriptor and its body. In our framework the body of the patterns is in most cases an EKD model fragment, i.e. the patterns that we develop are cognisant of the semantics of the EKD methodology. In case that an EKD model is not available or feasible, the pattern can be described by natural language or a multimedia element (e.g. a drawing, a video etc.).

Figure 2: The pattern template

As far as the descriptor is concerned, it consists of

• the formal signature

• the informal signature, and

• the guidelines.

The formal signature is necessary during the reuse process in order to retrieve patterns which are appropriate for a given situation having a given usage intention in mind. A formal signature comprises the domain of activity where the pattern applies, the type of the pattern and the usage intention of the pattern. For describing the formal signature, we use formalised natural language.

The informal signature of a pattern contains detailed information about the problem at hand and the proposed solution; it is composed of mandatory and optional components.

The mandatory components include the name of the pattern, the context of its application, the problem that it is trying to solve, the forces characterising the problem and the solution to the problem. The optional components are the consequences of applying the pattern, the rationale behind the solution, related patterns or documents, hyperlinks and known applications of the pattern, as well as additional annotations.

Each pattern should reflect its applicability conditions within a real situation. This is achieved through pattern guidelines. The guidelines are usage tips to the potential user of the pattern about how the pattern can be tailored to fit into particular situations or needs. In a sense, using EKD terms, the guidelines aim to give an idea of how the pattern can be tailored to create an enterprise model. The informal signature and guidelines are represented using free natural language. Figure 3 presents an example of a pattern, the Meter Reading pattern. This pattern is of Actor-Role type, a modelling mechanism devised to represent organisational and operational affinities in an enterprise setting 2 . All the elements of the pattern template can be seen in this example (only the mandatory components of the informal signature are shown).

Meter Reading Management

Goals

Formal Signature

Guidelines : Depending on the target organisation, the actor «Distribution Division» and its associated role may be splitted into a set of actors and a set of associated roles reflecting the structure of the organisation. This also applies to the «Meter Reader» actor and its role. Coordination links will have to be defined between the roles resulting of the splitting. However, these new links have to conform to the coordination needs between the roles «Meter reading management» and «Meter reading» :

Guidelines

Figure 3: A complete description of the Meter Reading pattern

We have placed much emphasis in organising patterns using a meaningful and coherent mechanism. By organising and storing the patterns we develop a knowledge base for the domain of interest (what in other pattern frameworks has been termed the pattern language); thus for the ELEKTRA project, the outcome is a knowledge base for change management in ESI.

The mechanism that was preferred for pattern organisation is that of a hierarchy [START_REF] Grosz | [END_REF]]. The hierarchy of patterns is built using the formal signature of the patterns, and specifically by associating the usage intentions of patterns. The pattern hierarchy is therefore organised in an intentional manner. This mechanism facilitates the storage and retrieval of patterns through an on-line pattern repository (more on this issue can be found in [Rolland et al. 1998a;Rolland et al. 1998b]).

Using patterns in change management

There has been an increasing interest over the years in understanding organisational change and managing it in a structured and systematic manner [START_REF] Medina-Mora | [END_REF]Davenport 1993;[START_REF] Lee | Goal-Based Process Analysis: A Method for Systematic Process Redesign[END_REF]Yu 1994;Ould 1995;Wieringa 1996;[START_REF] Hsiao | [END_REF]].

The pressure on the enterprise world to move towards leaner and more efficient business solutions has led to the development of increasingly sophisticated change management frameworks.

In our approach [Kavakli and Loucopoulos 1998; Rolland et al 1998c], in order to deal with the task of consistently and systematically managing change, we impose upon the activity of conducting change the view of the process-product cycle (see Figure 4). The essence of this view is that an initial state of affairs gives rise to an intention for change;

this change intention can lead to a new state of affairs through a set of alternative ways of achieving change. The way of conducting the change is the process, while the states that we start from and arrive to are the products. This cycle is intention-driven in that it is triggered by the needs for change that characterise the initial state of affairs.

Figure 4: The product-process cycle in change management

What we want to achieve through the development of patterns is to ensure the repeatability of the process-product cycle. This involves identification of possible products (or states) that can be of interest to organisations in the domain of interest and of repeatable processes that can be followed to reach these states. In other words, we aim to disseminate the experience that we gain by applying the EKD methodology in a change management case. In ELEKTRA, our interest has focused on the ESI sector, and in particular on identifying 'generalised patterns of change management for re-using them in similar settings in other electric supply companies'.

EKD addresses both the description of the enterprise (in terms of organisational, operational, and informational components) and the description of the change process itself. This leads to the definition of two types of patterns:

• Patterns dedicated to modelling the change process in the domain of interest. We call these change process patterns. Change process patterns refer to the transition process, the choices and alternatives that one may have in reaching an objective. We use an extended goal model as a means to represent the body of change process patterns. The transition process, therefore, is viewed as a teleological process whereby the goal graph is a model of the design process.

• Patterns dedicated to modelling the domain of interest itself. We call these product patterns. Product patterns are further subdivided into types, according to the modelling aspect of EKD that they convey; thus, we can have goal patterns, actorrole patterns, business rule patterns etc.

This dual nature of patterns is illustrated in Figure 5 and directly reflects the view of change management outlined above. Product patterns on one hand, can be used to describe a state of affairs (whether it is the current or the future one), i.e. a way in which the enterprise can conduct its business. Change process patterns on the other hand, can express the ways in which a transition from one state to another can be achieved.

Product

(the current state)

Process

(the process of change)

Product

(the future state)

Described by change process patterns

Described by product patterns Let us consider an example from the case of change in the ESI sector. In the new international setting of the electricity market, an electricity company may operate in a competitive environment where multiple electricity producers and suppliers exist. The company's Supply division (dealing with the retailing of electricity to consumers) will need to establish a business process for procuring electricity from the producers in order to sell it to the consumers. Figure 6 shows the change process pattern Introduce the buying and selling of electricity, which illustrates the options of how the company can implement this process.

Figure 6:Change process pattern dealing with the trading of electricity

The pattern has the form of a goal graph where the leaf goals represent four alternative ways (mutually exclusive in this case) of organising an electricity procurement business process. The third leaf goal of the change process pattern, for instance, proposes the introduction of a purchasing activity from multiple sources. The details of how such an activity can be configured are presented in Figure 7; here a product pattern is shown that explains the buying-and-selling activity when multiple sources of electricity are involved. While each of these two patterns has its own complete meaning and can be used independently, together they constitute a design proposal for answering a greater problem in the electricity sector.

Distribution plays Electricity Distribution

Goals:

To operate Distribution network

To assure connection of customers to the network

Transmission plays Electricity Transmission

Goals:

To operate transmission network

Supplier plays Electricity Supply

Goals:

To procure electricity from producers

To sell electricity to customers

Customer plays Buying Electricity

Goals:

To procure electricity

The way-of-working for developing patterns

The main benefit of pattern development is reusability. A pattern is a component whose purpose is to be reused whenever a problem arises that matches the context of the pattern. The process of reuse can be divided into two activities [START_REF] Bellinzona | [END_REF][START_REF] Deantonellis | [END_REF]. The first one deals with the process of defining reusable components (known as Design for reuse), the second concerns the effective usage of the reusable components (known as Design by reuse). In this paper, we concentrate on the activity of 'design for reuse'.

Developing patterns that encapsulate enterprise knowledge is a task of increased complexity. Knowledge pertinent to a domain, sometimes only implicitly and vaguely defined, needs to be captured and correctly represented. When considering the case of change management in particular, additional constraints are prevalent. The conditions that characterise the change process need to be clearly identified, so that the applicability of the patterns in similar cases can be determined. Thus, the context of the particular situation at hand, and the potential directions for change have to be captured.

In order to tackle these demands, we devised a process of developing patterns that is iterative and involves domain experts and method experts (analysts) in close cooperation. The process consists of the following four steps:

1. Elicitation of candidates 2. Suitability evaluation 3. Documenting

Verification

We will now briefly examine each step of the process3 .

Step

1: Elicitation of candidates

Collecting candidate patterns aims at identifying potential change process patterns and product patterns that will be considered for inclusion in the knowledge base. The objective is to describe a pattern at a sufficient level of detail in order to proceed to its evaluation. The output of the elicitation process is a list of candidate patterns each having an initial description of major components: name, problem, solution and context along with a first draft of its formal signature and body.

Potential sources of knowledge for initial elicitation of patterns range from enterprisespecific documentation (e.g. current state models of a particular company) to domainwide documentation (e.g. documents describing practice in more than one settings). An important source of candidate patterns can be the not-yet documented knowledge of various domain experts involved in the process.

Step 2: Suitability evaluation

The candidate patterns obtained as a result of step one need to be evaluated by domain experts so that their further development can be decided upon. The evaluation consists of the following actions:

• Grading the pattern

• Deciding on the suitability of the pattern

• Correcting the pattern

Grading the pattern

Grading is based on three criteria: usefulness (the candidate pattern addresses an important problem in the domain of interest), quality (the proposed solution effectively addresses the problem) and cost (the cost of implementing the proposed solution). For each criterion, a set of sub-criteria is defined, which helps to evaluate the candidate in detail. The domain experts grade the pattern according to these criteria, giving values of "high", "medium" or "low". A set of decision tables is then used in order to summarise the voting results into a single score per criterion.

Deciding on the suitability of the pattern

The decision on whether a candidate pattern is accepted for further development or rejected is based on a global numerical score. Table 1 presents the translation of the scores of the previous step into numerical values.

Value Criterion

High

Medium Low

Usefulness

2 1 0 Quality 2 1 0 Cost 0 1 2

Table 1: Numerical values associated to the scoring of each criterion

The global rating of each pattern is the sum of the numeric values of each criterion, ranging from 0 to 6. The pattern is accepted if the global rating of a candidate pattern is greater than or equal to 4. The pattern needs to be corrected if the global rating of a candidate pattern is between 2 and 4. Finally, the pattern is rejected if the global rating of a candidate pattern is less than or equal to 2. These values can be negotiated and agreed upon among the participants.

Correcting the pattern

If the global rating of a pattern falls between 2 and 4, it needs to be corrected. For each criterion we define a set of corrective actions that give guidelines about how to improve the pattern. When the corrective actions have been performed, the pattern is reevaluated.

Step 3: Documenting At this stage the pattern template needs to be completed. More precisely, the domain experts, in co-operation with the analysts now provide the remaining elements of the informal signature (i.e. forces, rationale, consequences etc.) and a set of initial guidelines. The formal signature is also finalised at this stage, which allows the pattern to be included in a hierarchy based on its usage intention. The pattern is now part of the knowledge base and any relationships with other patterns become apparent and explicit.

Step 4: Verification

After a pattern has been fully described (i.e. all components of the pattern template have been described), the pattern is verified by domain experts. The wording of all elements in the pattern template is carefully studied and modified if necessary, as are interconnections between related patterns.

Applying the Methodology in the ELEKTRA Project

In the context of ELEKTRA, we have concentrated on the study of change in the domains of Distribution and Human Resource Management of electricity companies. In developing a pattern knowledge base we used information from the various phases of the ELEKTRA project (detailed in the form of EKD models in [Athena 1997;[START_REF] Carolus | [END_REF]Demetra 1998;Freja 1998]), knowledge about practice in the ESI sector world-wide [START_REF] Prekas | [END_REF]], and current understanding of domain experts involved in the project.

We have found one of the main contributions of patterns to the management of organisational change to be that they may supply decision-makers with semi-structured information in the face of choosing between alternatives. Since these alternatives are organised in an intentional hierarchical structure, browsing through them is more intuitive than, say, relational or object-oriented database searching. Problems and solutions are searched for according to hierarchies defined by the domain experts. This can make a difference in the case of participants that are not expert computer users, as is often the case with management personnel.

Description of the change problem -The case studies

Before the move for deregulation of the electricity sector started, it was often the case that electricity companies operated as utilities rather than as commercial enterprises, one of their main characteristics being vertical and horizontal integration. With the introduction of competition and the reduction of extensive regulation, this situation has started to change. Today, electricity companies in Europe and worldwide often need to reinvent themselves by building a profile based on market principles and/or by splitting into independent organisational entities, each dealing with a different aspect of the electricity market (e.g. Generation dealing with the production of electrical power, and Supply dealing with the retailing of power to consumers). This changing environment has created a pressure on electricity companies to adapt many practices and structures; these pressures have had an impact on almost all aspects of business, though expressed in different ways in Distribution and in Human Resource Management.

In ESI sector terms, Distribution deals with the low voltage network carrying electricity to the sites of consumers and often it also comprises the retailing of electricity to customers (a domain otherwise known as Supply). In integrated electricity companies, Distribution and Supply have often been bundled with other activity areas (e.g.

Generation, Transmission). The move for deregulation now requires them to become separate organisational entities or different companies altogether. As a consequence, the patterns that we developed for Distribution reflect the possible moves that a company starting from the status of a public utility (i.e. monopoly rights, extensive bundling) can carry out in order to adapt to an environment where monopoly is abolished and competition prevails.

As the electricity market in Europe is being deregulated, new skills, competencies and knowledge are needed; this greatly affects the way in which electricity companies conduct Human Resource Management. Changes need to be made in how companies traditionally deal with these issues in terms of organisation, processes and decisionmaking. The human resource management patterns developed so far reflect the need for change at all levels of the organisation, towards more integration, co-operation, and co-ordination. The relatively large size and degree of division of an electricity company can lead to the creation of "islands of knowledge" in terms of competency-related activities. In practice, there is a trade-off between an increased degree of divisional independence, which improves creativity and response times, and increased centralisation, which improves co-ordination and co-operation.

An example

As an example of applying the methodology, we shall detail the development of the change process pattern Introduce structural unbundling, taken from the Distribution case study.

This pattern addresses a significant problem of the Distribution division of an ESI company, with respect to deregulation and customer servicing. Specifically, an understanding is needed of how a horizontally and vertically integrated ESI company can undertake steps towards entering a deregulated and competitive electricity market. This is representative of the kind of thinking required for developing patterns in such an environment. The knowledge of the needs for reform gained during previous phases of the project offered the pattern developers an informed view of the issues that are likely to be of importance in this situation.

Elicitation

The choice of candidate pattern was driven by what was considered as being essential toward the evolution of the ESI sector with respect to the deregulation rules being applied in the European Union. Existing EKD models, developed during earlier phases of the project, were used as the source of knowledge for this pattern. Initially, the elements of the informal signature were identified as:

Problem: An electricity distribution company in a monopoly environment must enter into a competitive market.

Context:

The management of the wire and supply businesses of Distribution is currently fully integrated. All business-related activities are handled internally. The proposed services are restricted to electricity supply.

Forces:

The deregulation rules for the European electricity market issued by the Commission of the EU force an electricity distribution company functioning as a monopoly to enter a competitive market.

The initial version of the solution suggested considering four aspects of unbundling:

• wire business and customer services are unbundled,

• new technical services other than electricity are introduced,

• outsourcing of customer and wire services are considered,

• new customer services not related to electricity are introduced.

These can be seen in Figure 8, where the initial version of the pattern body is presented.

Summary -Future Work

Managing change in organisations is becoming a crucial issue due to the rapid change of Using the proposed framework, a set of patterns has been developed for managing change in the Distribution and Human Resource Management domains of electricity companies (presented in [Molière 1999]). We are currently working on defining how patterns can be effectively reused ('design by reuse' activity). The final objective of our work concerns the implementation of the proposed framework within a consultancy toolset for change in the ESI sector [START_REF] Ikarus | Ikarus: Design of the ESI Toolset ELEKTRA Project Deliverable Document[END_REF]]. We are also envisioning means for disseminating the knowledge embedded within the patterns through the WWW [Rolland et al. 1998a;Rolland et al. 1998b].

Acknowledgements

The research presented in this paper has been funded by the European research project ELEKTRA (ESPRIT project no. 22927). The authors wish to thank all the partners involved in the project for their cooperation and contribution to the results, and especially the participants from the two user companies, the Public Power Corporation of Greece and Vattenfall of Sweden. We also thank our colleague Vagelio Kavakli for her useful comments and recommendations.

Figure 5 :

 5 Figure 5: The role of patterns in change management

Figure 7 :

 7 Figure 7: Product pattern (Actor-Role) dealing with the trading of electricity with multiple sources of procurement

  today's business. To this end, we propose patterns as a means to assist in the management of knowledge in change situations. The result of our work takes the form of a framework where a pattern contains an organisational design proposal that can be re-used in different enterprise settings. These proposals are based on the EKD methodology. As illustrated in this paper, we have applied the framework in the context of the European research project ELEKTRA for supporting change management in two particular domains of the Electricity Supply Industry sector.A pattern couples a problem to a solution. The problem is described within the pattern descriptor whereas the solution, the organisational design proposal, constitutes the body of the pattern. We consider two types of patterns: change process patterns and product patterns. A change process pattern suggests how to conduct and manage the process of change in an organisation, whereas a product pattern describes a (part of the) solution implemented by the change. While each type of pattern has its own complete meaning and can be used independently, together they constitute a coherent and comprehensive solution for managing organisational changes. Patterns are organised within hierarchies that facilitate search and retrieval.

Figure 1: Reusability of patterns in different enterprise settings

	Meter Reading in
	PPC (Greece)
	Meter Reading Pattern
	(generic description)
	Meter Reading in
	Vattenfall (Sweden)

To be used as contact person.

More on Actor-Role Modelling can be found in[START_REF] Loucopoulos | [END_REF]].

A more detailed description can be found in[START_REF] Molière | Molière: ESI Knowledge Base Specification ELEKTRA Project Deliverable Document[END_REF]].

Evaluation

The pattern was evaluated by domain experts with knowledge of strategy issues. The pattern was deemed as an important contribution to the ESI knowledge base because it addresses an essential problem in deregulation of electricity companies. During the evaluation corrective actions were proposed, particularly with respect to introduction of new services (both wire-based and customer-oriented ones). Some information related to outsourcing of services was removed, as it was deemed trivial.

As an interesting result of the evaluation process, the participants noticed the need to introduce a completely new pattern, complementary in nature, namely Introduce new means to increase market share. This pattern was deemed important because it suggests a strategic view of how to face the results of structural unbundling and the introduction of competition. We will not, however, go into detail concerning this pattern here.

Documenting/ Verification

After further discussions it was decided that part of the pattern, dealing with the introduction of new services, contained sufficient information to be treated as separate patterns. The final version of the body of the pattern is presented in Figure 9. The remaining information is now contained in two additional change process patterns