Int. J. Reliability and Safety, Vol. 3, Nos. 1/2/3, 2009 235

Copyright © 2009 Inderscience Enterprises Ltd.

A global optimisation method for computing interval hull solution for parametric linear systems

Iwona Skalna* Department of Applied Computer Science, AGH University of Science and Technology, Krakow, Poland Email: skalna@agh.edu.pl *Corresponding author

Andrzej Pownuk Department of Mathematical Sciences, University of Texas at El Paso, Texas, El Paso, TX 79968 USA Email:andrzej@pownuk.com

Abstract: An algorithm for computing the interval solution for a parametric interval linear system is presented. The basic idea behind the research is to combine an interval global optimisation method with the Direct Method for Checking the Monotonicity (MCM) of the parametric solution. The MCM is used to perform the monotonicity test to speed up the convergence of the global optimisation. Other acceleration techniques such as subdivision direction selection rules, multisection and the midpoint test are involved as well and checked for usefulness. By using the proposed algorithm, several examples of parametric linear systems are solved. The research proves that the proposed monotonicity test is crucial for the convergence of the interval global optimisation used for computing the interval hull for parametric solution sets, whereas other accelerating techniques are not relevant. The presented algorithm can be useful for solving real-life problems concerning structure mechanics.

Keywords: global optimisation; monotonicity approach; parametric linear systems; interval hull solution.

Reference to this paper should be made as follows: Skalna, I. and Pownuk, A. (2009) ‘A global optimisation method for computing interval hull solution for parametric linear systems’, Int. J. Reliability and Safety, Vol. 3, Nos. 1/2/3, pp.235–245.

Biographical notes: Iwona Skalna obtained Masters degree in Mathematics, Masters degree in Computer Science and PhD degree from Faculty of Civil Engineering at the Silesian University of Technology in Gliwice, Poland. She is holding a post-doctoral position in the Faculty of Management at the AGH University of Science and Technology in Cracow. She is a Lecturer in the Department of Applied Computer Science. Her research focuses on modelling uncertainties with intervals and fuzzy numbers, and solving parametric linear systems with uncertain parameters. She has published many research papers in national and international journals, conference proceedings as well as chapters of books.

236 I. Skalna and A. Pownuk

Andrzej Pownuk obtained Masters degree in Applied Mechanics and PhD degree from Faculty of Civil Engineering at the Silesian University of Technology in Gliwice, Poland. He is a Faculty Member in the Department of Theoretical Mechanics of Civil Engineering at the Silesian University of Technology since 1995. He was employed in the Department of Mathematics at the Vienna University in Austria, during 2003–2004, and worked as a post-doctoral researcher at the University of Bergen in Norway. His research interests include interval applications to structural analysis, sensitivity analysis and mathematical modelling of structures.

1 Introduction

This paper focuses on solving parametric linear systems of structure mechanics with interval parameters. Parametric interval methods allow the engineering practice to account for uncertainty connected either to external factors, such as boundary conditions or applied loads, or to internal factors, such as mechanical or geometric characteristics (Lallemand et al., 2000; Aughenbaugh and Paredis, 2006; Muhanna and Erdolen, 2006; Muhanna et al., 2006; Zalewski et.al., 2006), and to calculate the very sharp bounds on the system response for all possible scenarios in a single analysis (Mullen and Muhanna, 2002).

Usually, a parametric solution set is not an interval vector (Neumaier, 1990). Hence, instead of the parametric solution set itself, an interval vector containing the parametric solution set, outer interval solution, is calculated. The tightest outer interval solution is called the interval hull solution. Some of the methods also produce an inner interval solution that is an interval vector contained in the interval hull. Most of the papers focus on inner and outer solutions (Chen et al., 2002; Kolev, 2004; Pownuk, 2004; Muhanna and Erdolen, 2006; Muhanna et al., 2006; Popova et al., 2006; Neumaier and Pownuk, 2007; Popova and Kraemer, 2007). The difference between inner and outer solutions is treated as an approximate measure of the overestimation. Often, for small diameters, reasonable inner and outer interval solutions can be rigorously computed in polynomial time. However, the overestimation, which is mainly due to the dependency problem, grows with the number of parameters and the size of the problem. Then, the interval hull solution might be required despite large computational time.

In general case, the problem of computing the interval hull solution is NP-hard (Rohn and Kreinovich, 1995). However, when a parametric solution is monotone, with respect to all interval parameters, the interval hull solution can be calculated by solving at most 2n + k real linear systems, where k is a number of interval parameters (e.g. sensitivity analysis; Pownuk, 2004).

The monotonicity of the parametric solution can be verified using the MCM proposed by Skalna (2008). The method is based on checking the sign of the derivatives of the solution with respect to uncertain parameters. Using this method a good outer approximation for the interval hull solution set can be obtained.

If the parametric solution is not monotone, the problem of computing an interval hull solution can be solved using optimisation techniques. To compute the hull, 2n constrained optimisation problems must be solved. Skalna (2006b) proposed the Evolutionary Optimisation Method (EOM) for approximating from below the hull solution. The main drawback of this method is that the result depends closely on the size of population and the number of generations. Each problem requires specific treatment,

A global optimisation method for computing interval hull solution 237

and it is difficult to predict the values of these two parameters. Hence, the amount of the underestimation is not known. To get reliable result, the Global Optimisation Method (GOM) (Hansen, 1992), which refers to finding the extreme value of a given nonconvex function in a certain feasible region, with the monotonicity test performed using the MCM method is proposed. The MCM method enables to handle considered optimisation problems with implicit objective functions. Other accelerating techniques such as subdivision direction selection rules and the midpoint test are involved and checked for usefulness.

The paper is organised as follows: Section 2 contains preliminaries on parametric interval linear systems with two disjoint sets of parameters. In Section 3, the optimisation problem is outlined. This is followed by description of the selected accelerating techniques and the global optimisation algorithm. Next, some illustrative examples of truss structures and the results of computational experiments are presented. The paper ends with summary conclusions.

2 Preliminaries

Real quantities will be in italic, while their interval counterparts will be in bold italic. Capital letters will be used to denote sets and matrices.

Let I denote the set of real compact intervals [ ] { }, |x x x x x x= = ∈x . For

two intervals a, b ∈ ,I , ,a b a b and a = b when ,a b a b and ,a b a b= ∧ = respectively. Then, nI will denote interval vectors, n n×I square interval matrices (Neumaier, 1990). For an interval [ ], ,x x=x some characteristics are defined:

the midpoint ˇ m( ) ( ) / 2,x x x= = +x the radius r( ) ( ) / 2x x= −x and the width w( ) .x x= −x They are applied to interval vectors and matrices componentwise.

Consider a linear algebraic system

( ) ( ) ( ), ,A p x p q b q= (1)

with coefficients that are linear functions of parameters { }1,..., ,kkp p p= ∈

{ }1,..., :llq q q= ∈

0 0( ) , ( ) ,T Tij ij ij j j ja p p b q qα α β β= + ⋅ = + ⋅ (2)

where αij0, βj0 ∈ , αij = {αijν} ∈ k, βj = {βjν} ∈ l, i, j = 1,..., n. Now assume that some of the parameters are unknown. The real vectors p and q are

then replaced by interval vectors p ∈ I k and q ∈ I l, (real elements are represented by

point intervals). This gives a family of the systems:

( ) ( , ) ( ), , ,A p x p q b q p q= ∈ ∈p q (3)

which is usually written in a symbolic compact form

( ) ( ) ( ),A x b=p p q q (4)

238 I. Skalna and A. Pownuk

and is called the parametric interval linear system. Parametric (united) solution set of the system (4) is defined (see e.g. Jansson, 1991; Rump, 1994; Kolev, 2004) as

{ }( , ) | , , ( ) ( , ) ( ) .S x p q A p x p q b q= ∃ ∈ ∃ ∈ =p q p q (5)

If the solution set S = S(p, q) is bounded, then its interval hull

S = [inf S, sup S] =∩{y ∈ I n | S ⊆ y}

exists. S is called the interval hull solution. In order to guarantee that the solution set is bounded, the matrix A(p) must be regular, i.e. A(p) must be non-singular for all the parameters p ∈ p.

3 Interval hull solution

3.1 Optimisation problem

The problem of computing the interval hull solution for parametric interval linear system (3) can be written as a problem of solving 2n constrained optimisation problems

min ( , ), 1,...,ipq

x p q i n∈∈

=p

q

(6)

and

max ( , ), 1,..., ,ipq

x p q i n∈∈

=p

q

(7)

where the objective functions : k lix × → are implicit functions given by formula

(1). If A(p) is non-singular then xi(p, q) = {A(p)–1b(q)}i.

Theorem 1: Let A(p) be regular, p ∈ I k. Let min maxi ix and x denote global solutions of the

i-th minimisation (6) and maximisation (7) problem, respectively. Then the interval vector

( )min max min max 1[ ,| ] , ( , ).

ni i

ix x x x S

=⎡ ⎤= = =⎣ ⎦x p q (8)

3.2 Global optimisation

Optimisation problems (6) and (7) are solved using a global optimisation method with selected acceleration techniques. The maximisation problem is transformed into the respective minimisation problem, hence the research focuses on minimisation. Global optimisation requires inclusion function of the objective function. In what follows, it is assumed that the inclusion functions have the isotonicity property; i.e. x ⊆ y implies F(x) ⊆ F(y), and that for all the inclusion functions holds

( )w ( ) 0 as w( ) 0.i iF x x→ → (9)

The Direct Method (DM) for outer interval solution (Skalna, 2006a) will be used to construct inclusion functions Xi(p, q) for xi(p, q). It can be easily shown that the method preserves the isotonicity property.

A global optimisation method for computing interval hull solution 239

3.3 Midpoint test

The midpoint test is used to reduce the number of intervals on the list L. The midpoint

value ˇ sup ( )f F c= is computed with mid( )c = y . Now, all pairs ( ),f ′ ′y satisfying

ˇf f ′< can be discarded from the list L. Also, a new pair ( ),f ′′ ′′y must only be entered

in the list L if ˇf f ′′ is satisfied.

3.4 Subdivision direction selection rules

Following Ratz and Csendes (1997), the interval subdivision direction selection rules have the following merit function:

{ }: min | {1,..., }and ( ) max ( ) ,i

k j j n D j D i= ∈ = (10)

where D(i) is determined by a given rule.

Rule A: The first rule to be applied is the interval-width-oriented rule (Hansen, 1980); it can also be applied to non-differentiable function. This rule chooses the coordinate direction with

D(i) = w(yi) (11)

and was justified by the idea that if the original interval is subdivided in a uniform way, then the width of the actual subintervals goes to zero most rapidly.

Rule B: Hansen (1980) described another rule (initiated by G.W. Walster) which selects the coordinate direction with

( )( ) w ( ) w( ).i iD i F ′= y y (12)

Rule C: The third rule was defined by Ratz (1992) with

( )( ) w ( )( m( )) .i i iD i F ′= −y y y (13)

3.5 Monotonicity test

The monotonicity test is used to figure out whether a function f is strictly monotone in a subbox y ⊆ x. Then, y cannot contain a global minimiser in its interior. Therefore, if f satisfies

( ) 0 ( ) 0,i i

f fx x∂ ∂

< ∨ >∂ ∂

y y (14)

then the subbox y can be reduced to one of its edges. The proper endpoint, degenerated subbox, is then entered into the list.

Monotonicity test is performed this time using the MCM method. The MCM method enables to handle the monotonicity test for optimisation problems (6) and (7) with implicit objective functions. Below, a brief description of the MCM method is presented.

240 I. Skalna and A. Pownuk

Let x(p, q) be defined by formula (1). Then enclosures for ( , ), ( , )m r

x xp q∂ ∂∂ ∂

p q p q are

obtained by solving the following parametric linear systems:

( ) ( *), 1,..., ; ( ) , 1,..., ,m r

m r

x xA b m k A b r lp q∂ ∂′ ′′= = = =∂ ∂

p x p (15)

where p ∈ I k, x* ∈ I n [an outer solution of system (4)] are vectors of parameters, *( *) ( * *), , 1,..., .m r

j i jm j j jrb x x x b j nβ′ ′′= −α ∈ = =x A detailed description of the MCM method can be found in work of Skalna (2008).

3.6 Algorithm

Below, the global optimisation algorithm with the monotonicity test, used to calculate i-th element of the interval hull solution for parametric solution set, is presented.

1 Set y = {p, q}T and f = min Xi(y)

2 Initialise the list L = {(f, y)} and the cutoff level z = max Xi(y)

3 Remove (f, y) from the list L

4 Choose the subdivision direction ν using one of the rules A, B, or C

5 Bisect y in the direction ν: y1 ∪ y2

6 Calculate Xi(yi), fi = min Xi(yi) (i = 1, 2), and { }min ,max ( )i iiz z X= y

7 Discard the pair (fi, yi) if fi > z (i = 1, 2) (cutoff test)

8 Perform the monotonicity test using the MCM method

9 Add any remaining pairs to the list L so that the list remains increasingly sorted by the function value; if the list becomes empty then STOP

10 If w( ) ε≤y (where ε is an algorithm parameter), then print Xi(y), y and STOP, else GOTO 3.

4 Examples and numerical results

To check the performance of the proposed approach, three two-dimensional linear systems with two interval parameters and three illustrative examples of structural mechanical systems of different sizes and with different number of uncertain parameters are described. The results of the global optimisation method are compared with the results of the method based on Sensitivity Analysis (SA) and the results of the DM for outer interval solution.

Example 1 A two-dimensional parametric linear system with two interval parameters:

1 2 21 2

1

1 2, , [0,1] [0,1].

3 3* 1 0p p p

p pp

+ +⎡ ⎤ ⎡ ⎤= = ∈ ∈⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎣ ⎦

A b (16)

A global optimisation method for computing interval hull solution 241

In this case, the monotonicity test speeded up the convergence of the optimisation approximately by 66%. The differences between the results of the GOM and the SA methods are underlined. The right bound of x2 is underestimated approximately by 3.92% (x2 is maximal at p1 = 5/3, p2 = 0). Table 1 Boundaries of the solution set of system (16) obtained by the global optimization

method and the sensitivity analysis

GOM SA x1 [−0.666667, 0.8] [−0.666667, 0.8] x2 [0.6, 2.05714] [0.6, 2]

Example 2 A two-dimensional parametric linear system with two interval parameters

1 2 21 2

1

1 3, , [0, 2] [0, 2].

3 3* 1 1p p p

p pp

+ +⎡ ⎤ ⎡ ⎤= = ∈ ∈⎢ ⎥ ⎢ ⎥− − −⎣ ⎦⎣ ⎦

A b (17)

The monotonicity test speeded up the convergence of the optimisation approximately by 33%. The differences between the results are greater than in the previous example. The right bounds of both x1 and x2 are underestimated. Table 2 Boundaries of the solution set of system (17) obtained by the global optimization

method and the sensitivity analysis

GOM SA x1 [−0.666667, 1.43798] [−0.6666667, 1.23077] x2 [0.17241, 3.03798] [0.17241, 3]

Example 3 Consider a two-dimensional system with two interval parameters. The width of the interval parameters depends on δ > 0 (Neumaier and Pownuk, 2007).

1 2 2

2 1 2

1 2

60, ,

61[1 , 1 ], [5 , 5 ], 0.

p p pp p p

p pδ δ δ δ δ

+⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥+ ⎣ ⎦⎣ ⎦∈ − + ∈ − + >

A b (18)

Table 3 contains the results of the global optimisation method, the sensitivity analysis and the DM for outer interval solution. The overestimation of the DM and underestimation of the sensitivity analysis grows rapidly with δ. Table 3 Boundaries of the solution set of system (18) obtained by the global optimisation

method with the monotonicity test, the sensitivity analysis and the Direct method for outer interval solution

δ CPU time GOM SA DM

x1 0.1 0.23s [4.8949, 5.1021] [4.8994, 5.0987] [4.8803, 5.1196] x2 [5.8085,6.2098] [5.8085, 6.2098] [5.7794, 6.2206] x1 0.3 0.45s [4.6396, 5.3191] [4.6994, 5.2758] [4.4570, 5.5430] x2 [5.4686, 6.7044] [5.4686, 6.7044] [5.1486, 6.8514] x1 0.5 0.82s [4.2608, 5.5556] [4.5066, 5.3684] [3.4285, 6.5714] x2 [5.1733, 7.3684] [5.1733, 7.3684] [3.9047, 8.0952]

242 I. Skalna and A. Pownuk

Example 4 7-bar planar truss structure

Figure 1 7-bar planar truss structure with four (a) and five (b) interval parameters

(a) (b)

For the 7-bar planar truss structure shown in Figure 1 the displacements of the nodes are computed. The truss is subjected to downward force P2 = 10[kN] at node 2. Young’s modulus Y = 2.0 × 1011[Pa], cross-section area C = 0.005[m2]. Lengths of the horizontal bars equal 2[m], lengths of the vertical bars equal 1[m]. The stiffness of four (a) and five (b) bars, depicted in the Figure with thick lines, is assumed to be uncertain by ±5% (which gives the total uncertainty of 10%). There are then k = 4 and k = 5 interval parameters, respectively. The results are presented in Table 4. Table 4 Boundaries of the node displacements of the 7-bar planar truss structure obtained by

the global optimisation method; case of four and five interval parameters

CPU time 1.04 s (k = 4) [×10−5] 1.25s (k = 5) [×10−5] x2 −2 −2 y2 [−0.303166, −0.196834] [−0.326975, −0.170518] x3 [−3.98297, −3.77031] [−4.00928, −3.7465] y3 −0.5 −0.5 x4 [−3.52409, −3.31143] [−3.57672, −3.26381] y4 [−1.30317, −1.19683] [−1.32948, −1.17302] x5 [−2.02977, −1.88799] [−2.05608, −1.86418]

The monotonicity test significantly speeded up the convergence of the global optimisation.

Example 5 21-bar planar truss structure

Figure 2 21 planar truss structure with the stiffness of all the bars uncertain by ±5%

A global optimisation method for computing interval hull solution 243

Consider the truss structure depicted in Figure 2 subjected to downward forces P1 = P2 = P3 = 30[kN], Young’s modulus Y = 7.0 × 1010[Pa], cross-section area C = 0.003[m2] and the length L = 2[m]. The stiffness of all the bars is assumed to be uncertain by ±5%. There are then 21 interval parameters.

The result of the GOM method (CPU time 4.56 minutes) and the best result of the EOM method (population size: 20, number of generations: 80, crossover probability: 0.2, mutation probability: 0.9), obtained from three successive runs, coincide. A single run of the EOM method (CPU time 3 minutes) performs faster, but usually several runs are required. Hence, the computational complexity of the GOM method is lower. Moreover, the result of the GOM method is reliable, while the underestimation of the EOM result is not known exactly.

Example 6 Baltimore bridge built in 1870

Figure 3 Baltimore bridge (built in 1870)

The planar truss structure shown in Figure 3 is considered. It is subjected to downward forces of P1 = 80[kN] and P2 = 120[kN]; length L = 1[m], cross-section area C = 0.004[m2] and Young’s modulus Y= 2.1 × 1011 [Pa]. Assume that the stiffness of 16 bars is uncertain by ±5%. Then, there are 16 interval parameters.

The results of the GOM method and the best result of the EOM method (population size: 20, number of generations: 80, crossover probability: 0.2, mutation probability: 0.9), obtained from the three successive runs, slightly differed. After increasing the population size to 26 and the number of generations to 120, the results coincided. GOM CPU time is 25.78 minutes.

5 Conclusions

The problem of computing the interval hull solution for parametric linear systems has been considered. The global optimisation method with the monotonicity test performed using the MCM has been proposed to calculate the interval hull solution and verified for usefulness. Numerical experiments proved that the monotonicity test significantly decreases the computational time and hence is a crucial accelerating technique for the proposed approach. For small two-dimensional examples, the monotonicity test speeded up the convergence approximately by 66%. In case of the 7-bar planar truss structure with four and five interval parameters, the monotonicity test speeded up the convergence of the global optimisation method approximately by 96.7% and by 97.5%, respectively. Other accelerating techniques such as different subdivision direction rule, multisection

244 I. Skalna and A. Pownuk

and the midpoint test turned out to be not relevant. The proposed approach can be very useful for solving real-life problem where the reliability and safety are most important. The computational complexity may be decreased using parallel computations. The parallelism can be easily incorporated to the computations. This will be a subject of future research.

References Aughenbaugh, J. and Paredis C. (2006) ‘Why are intervals and imprecisions important in

engineering design?’, in Muhanna, R.L. (Ed.): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC'06), 22–24 February, Svannah, Georgia, USA, pp.319–340.

Chen, S., Lian, H. and Yang, X. (2002) ‘Interval static displacement analysis for structures with interval parameters’, Journal for Numerical Methods in Engineering, Vol. 53, pp.393–407.

Hansen, E. (1980) ‘Global optimization using interval analysis – the multidimensional case’, Numerische Mathematik, Vol. 43, pp.247–270.

Hansen, E. (1992) Global Optimization Using Interval Analysis, Marcel Dekker, New York. Jansson, C. (1991) ‘Interval linear systems with symmetric matrices, skew-symmetric matrices and

dependencies in the right hand side’, Computing, Vol. 46, No. 3, pp.265–274. Kolev, L.V. (2004) ‘A method for outer interval solution of linear parametric systems’,

Reliable Computing, Vol. 10, No. 3, pp.227–239. Lallemand, B., Plessis, G., Tison, T. and Level, P. (2000, January) ‘Modal behaviour of structures

defined by imprecise geometric parameters’, SPIE Proceedings Series Vol. 4062, Proceedings of IMAC-XVIII: A Conference on Structural Dynamics, Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, pp.1422–1428.

Muhanna, R.L. and Erdolen, A. (2006) ‘Geometric uncertainty in truss systems: an interval approach’, in Muhanna, R.L. (Ed.): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC'06), 22–24 February, Svannah, Georgia, USA, pp.239–247.

Muhanna, R.L., Kreinovich, V., Solin, P., Cheesa, J., Araiza, R. and Xiang, G. (2006) ‘Interval finite element method: new directions’, in Muhanna, R.L. (Ed.): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC'06), 22–24 February, Svannah, Georgia, USA, pp.229–244.

Mullen, R. and Muhanna, R.L. (2002) ‘Efficient interval methods for finite element solutions’, HPCS '02: Proceedings of the 16th Annual International Symposium on High Performance Computing Systems and Applications, IEEE Computer Society, Washington, DC, USA, p.161.

Neumaier, A. (1990) 'Interval Methods for Systems of Equations', Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1990.

Neumaier, A. and Pownuk, A. (2007) ‘Linear Systems with large uncertainties, with applications to truss structures’, Reliable Computing, Vol. 13, No. 2, pp.149–172.

Popova, E., Iankov, R. and Bonev, Z. (2006) ‘Bounding the response of mechanical structures with uncertainties in all the parameters’, in Muhannah, R.L. and Mullen, R.L. (Eds): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC'06), 22–24 February, Savannah, Georgia, USA, pp.245–265.

Popova, E. and Kraemer, W. (2007) ‘Inner and outer bounds for parametric linear systems’, Journal of Computational and Applied Mathematics, Vol. 199, No. 2, pp.310–316.

Pownuk, A. (2004) ‘Efficient method of solution of large scale engineering problems with interval parameters based on sensitivity analysis’, Proceeding of NSF workshop on Reliable Engineering Computing, 15–17 September, Savannah, Georgia, USA, pp.305–316.

Ratz, D. (1992) Automatische Ergebnisverifikation bei globalen Optimierungsprob-lemen, PhD Thesis, Universit¨at Karlsruhe, Karlsruhe, Germany.

A global optimisation method for computing interval hull solution 245

Ratz, D. and Csendes, T. (1997) ‘Subdivision direction selection in interval methods for global optimization’, SIAM Journal on Numerical Analysis, Vol. 34, No. 3, pp.922–938.

Rohn, J. and Kreinovich, V. (1995) ‘Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard’, SIAM Journal on Matrix Analysis and Applications (SIMAX), Vol. 16, pp.415–420.

Rump, S.M. (1994) ‘Verification methods for dense and sparse systems of equations’, in Herzberger, J. (Ed.): Topics in Validated Computations – Studies in Computational Mathematics, Elsevier, Amsterdam, pp.63–136.

Skalna, I. (2006a) ‘A method for outer interval solution of parametrized systems of linear interval equations’, Reliable Computing, Vol. 12, No. 2, pp.107–120.

Skalna, I. (2006b) ‘Evolutionary optimization method for approximating the solution set hull of parametric linear systems’, LNCS: Numerical Method and Applications, Vol. 4310, pp.361–368.

Skalna, I. (2008) ‘On checking the monotonicity of parametric interval solution’, Proceedings of the 7th International Conference on Parallel Processing and Applied Mathematics (PPAM 2007), Workshop on Scheduling for Parallel Computing, LNCS, Vol. 4967, Springer-Verlag.

Zalewski, B., Muhanna, R.L. and Mullen, R. (2006) ‘Bounding the response of mechanical structures with uncertainties in all the parameters’, in Muhanna, R.L. (Ed.): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC'06), 22–24 February, Svannah, Georgia, USA, pp.439–456.