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.
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Neumaier, A., Pownuk, A. Linear Systems with Large Uncertainties, with Applications to Truss Structures. Reliable Comput 13, 149–172 (2007). https://doi.org/10.1007/s11155-006-9026-1
- Large Uncertainty
- Truss Structure
- Trial Point
- Uncertain Linear System
- Interval Solution