Reliable Computing (2007) 13: 149–172
Linear Systems with Large Uncertainties, with
Applications to Truss Structures
ur Mathematik, Universit
at Wien, Nordbergstr. 15, A-1090 Wien, Austria,
Department of Theoretical Mechanics, Faculty of Civil Engineering,
Silesian University of Technology, ul. Krzywoustego 7, 44-100 Gliwice, Poland,
(Received: 26 November 2004; accepted: 12 November 2005)
Abstract. Linear systems whose coefﬁcients have large uncertainties arise routinely in ﬁnite element
calculations for structures with uncertain geometry, material properties, or loads. However, a true
worst case analysis of the inﬂuence of such uncertainties was previously possible only for very small
systems and uncertainties, or in special cases where the coefﬁcients do not exhibit dependence.
This paper presents a method for computing rigorous bounds on the solution of such systems, with
a computable overestimation factor that is frequently quite small. The merits of the new approach are
demonstrated by computing realistic bounds for some large, uncertain truss structures, some leading
to linear systems with over 5000 variables and over 10000 interval parameters, with excellent bounds
for up to about 10% input uncertainty.
Also discussed are some counterexamples for the performance of traditional approximate methods
for worst case uncertainty analysis.
Linear systems of equations are among the most frequently used tools in applied
mathematics. In realistic applications, the data entering the coefﬁcients of these
equations are generally uncertain. Since linear equations become nonlinear when
coefﬁcients are uncertain and become variable, traditional sensitivity analysis
remains valid only for sufﬁciently small errors. Unfortunately, it is usually unclear
when the errors are sufﬁciently small for its validity: For errors larger than some
unknown, problem-dependent margin, sensitivity analysis may be severely biased,
since it does not account for the nonlinearities in the problem.
The traditional remedy for assessing the accuracy of nonlinear computations with
uncertain data are Monte Carlo calculations. For problems of signiﬁcant size, these
are expensive (and therefore usually quite incomplete) since similar computations
must be done over and over again, many thousand times. Moreover, by their nature,